c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 238–241, 2011 http://ijamc.psit.in

The modified extended tanh method with the Riccati equation for solving (3 + 1)-dimensional Kadomtsev-Petviashvili (KP ) equation Nasir Taghizadeh, Mohammad Ali Mirzazadeh Department of Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O.Box 1914, Rasht, Iran Email: taghizadeh@guilan.ac.ir,mirzazadehs2@guilan.ac.ir

Abstract: In this paper, the modified extended tanh method is used to construct new exact traveling wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvili equation. The modified extended tanh method is one of most direct and effective algebraic method for obtaining exact solutions of nonlinear partial differential equations. The method can be applied to nonintegrable equations as well as to integrable ones.

Key words: Modified extended tanh method; Riccati equation; (3+1)-dimensional KadomtsevPetviashvili (KP) equation.

1

Introduction

The investigation of the exact traveling wave solutions of nonlinear evolutions equations paly an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in fluid dynamics, elastic media, optical fibers, etc. In recent years, there was interest in obtaining exact solutions of nonlinear partial differential equation by the extended tanh method. The standard tanh method is developed by Malfliet [1]. Recently, Wazwaz investigated exact solutions of nonlinear partial differential equation by the extended tanh method [2-4]. Fan in [5] presented the generalized tanh method for constructing the exact solutions of nonlinear partial differential equation, such as, the (2+1)-dimensional sineGordon equation and the double sine-Gordon equation. The aim of this paper is to find exact solutions of the (3+1)-dimensional Kadomtsev-Petviashvili equation by modified extended tanh method with the Riccati equation.

2

Modified extended tanh method

Let us describe the modified extended tanh method. For given a nonlinear equation F (u, ux , uy , ut , uxx , uxy , uxt , ...) = 0,

(2.1)

when we look for its traveling wave solutions, the first step is to introduce the wave transformation u(x, y, t) = u(ξ), ξ = sx + ly + nt + d, and change Eq. (2.1) to an ordinary differential equation(ODE) H(u, u0 , u00 , u000 , ...) = 0. (2.2) The next crucial step is to introduce a new variable φ = φ(ξ), which is a solution of the Riccati equation dφ = k + φ2 . (2.3) dξ

239

The modified extended tanh method admits the use of the finite expansion: u(x, y, t) = u(ξ) =

N X

ai φi (ξ) +

i=0

N X

bi φ−i (ξ),

(2.4)

i=1

where the positive integer N is usually obtained by balancing the highest-order linear term with the nonlinear terms in Eq. (2.2). Expansion (2.4) reduces to the generalized tanh method [5] for bi = 0, i = 1, ..., N. Substituting (2.3) and (2.4) into Eq. (2.2) and then setting zero all coefficients of φi (ξ), we can obtain a system of algebraic equations with respect to the constants k, s, l, n, a0 , ..., aN , b1 , ..., bN . Then we can determine the constants k, s, l, n, a0 , ..., aN , b1 , ..., bN . The Riccati equation (2.3) has the general solutions: If k < 0 then √ √ (2.5) φ(ξ) = − −k tanh( −kξ), √ √ φ(ξ) = − −k coth( −kξ). If k = 0 then

1 φ(ξ) = − . ξ

If k > 0 then

(2.6)

√

√ k tan( kξ), √ √ φ(ξ) = − k cot( kξ). φ(ξ) =

(2.7)

Therefore, by the sign test of k, we can obtain exact solutions of Eq. (2.1).

3

The (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation

For the (3 + 1)-dimensional KP [6] (ut + 6uux + uxxx )x − 3uyy − 3uzz = 0.

(3.1)

Let us consider the traveling wave solutions u(x, y, z, t) = u(ξ),

ξ = sx + ly + mz + nt + d,

(3.2)

where s, l, m, n and d are constants. Substituting (3.2) into (3.1), then (3.1) is reduced to the following nonlinear ordinary differential equation s4 u0000 + (sn − 3(m2 + l2 ))u00 + 6s2 (uu0 )0 = 0. (3.3) Integrating twice of Eq. (3.3), setting the constants of integrating to zero, we obtain s4 u00 (ξ) + (sn − 3(m2 + l2 ))u(ξ) + 3s2 u2 (ξ) = 0.

(3.4)

It is easy to show that N = 2, if balancing u00 with u2 . Therefore, the modified extended tanh method (2.4) admits the use of the finite expansion: u(ξ) = a0 + a1 φ(ξ) + a2 φ2 (ξ) +

b2 b1 + 2 . φ(ξ) φ (ξ)

(3.5)

Thus, by Eq. (2.3), we get u00 (ξ)

= +

6a2 φ4 (ξ) + 2a1 φ3 (ξ) + 8ka2 φ2 (ξ) + 2ka1 φ(ξ) + 2b2 + 2k 2 a2 2kb1 8kb2 2k 2 b1 6k 2 b2 + 2 + 3 + 4 . φ(ξ) φ (ξ) φ (ξ) φ (ξ)

(3.6)

240

u2 (ξ)

= +

a22 φ4 (ξ) + 2a1 a2 φ3 (ξ) + (2a0 a2 + a21 )φ2 (ξ) + (2b1 a2 + 2a0 a1 )φ(ξ) + 2b2 a2 2b2 a1 + 2b1 a0 2a0 b2 + b21 2b1 b2 b2 2b1 a1 + a20 + + + 3 + 42 . (3.7) 2 φ(ξ) φ (ξ) φ (ξ) φ (ξ)

Substituting Eqs. (3.5) and (3.6) and (3.7) into Eq. (3.4), and equating the coefficients of like powers of φi (i = −4, −3, −2, −1, 0, 1, 2, 3, 4) to zero yields the system of algebraic equations to a0 , a1 , a2 , b1 , b2 , s, l, m, n and k φ4 : 6s4 a2 + 3s2 a22 = 0, φ3 : 6s2 a1 a2 + 2s4 a1 = 0, φ2 : 3s2 a21 + 6s2 a0 a2 + (sn − 3(m2 + l2 ))a2 + 8s4 ka2 = 0, φ1 : (sn − 3(m2 + l2 ))a1 + 6s2 a0 a1 + 2s4 ka1 + 6s2 b1 a2 = 0, φ0 : 6s2 b1 a1 + 6s2 b2 a2 + 3s2 a20 + (sn − 3(m2 + l2 ))a0 + 2s4 b2 + 2s4 k 2 a2 = 0, φ−1 : 6s2 b2 a1 + 6s2 b1 a0 + 2s4 kb1 + (sn − 3(m2 + l2 ))b1 = 0, φ−2 : 3s2 b21 + 6s2 b2 a0 + (sn − 3(m2 + l2 ))b2 + 8s4 kb2 = 0, φ−3 : 6s2 b1 b2 + 2s4 k 2 b1 = 0, φ−4 : 6s4 k 2 b2 + 3s2 b22 = 0. Solving the resulting system, by using Maple, we find the following solutions −sn + 3l2 + 3m2 −sn + 3l2 + 3m2 , a = − , a1 = 0, a2 = −2s2 , b1 = b2 = 0. (3.8) 0 4s4 6s2 −sn + 3l2 + 3m2 −sn + 3l2 + 3m2 , a = , a1 = 0, a2 = −2s2 , b1 = b2 = 0. (3.9) k=− 0 4s4 2s2 −sn + 3l2 + 3m2 −sn + 3l2 + 3m2 k=− , a = , a1 = 0, a2 = −2s2 , b1 = 0, (3.10) 0 16s4 4s2 s2 n2 − 6snl2 − 6snm2 + 9l4 + 18m2 l2 + 9m4 b2 = − . 128s6 −sn + 3l2 + 3m2 −sn + 3l2 + 3m2 k= , a0 = , a1 = 0, a2 = −2s2 , b1 = 0, (3.11) 4 16s 12s2 s2 n2 − 6snl2 − 6snm2 + 9l4 + 18m2 l2 + 9m4 b2 = − . 128s6 −sn + 3l2 + 3m2 −sn + 3l2 + 3m2 k= , a0 = − , a1 = a2 = b1 = 0, (3.12) 4 4s 6s2 (−sn + 3l2 + 3m2 )2 b2 = − . 8s6 −sn + 3l2 + 3m2 −sn + 3l2 + 3m2 k=− , a0 = , a1 = a2 = b1 = 0, (3.13) 4 4s 2s2 (−sn + 3l2 + 3m2 )2 . b2 = − 8s6 By using (2.5) and (2.7), the sets (3.8) -(3.13) give the following solitons solutions: r sn − 3l2 − 3m2 −sn + 3m2 + 3l2 2 u1 (x, y, z, t) = (1 + 3 tan ( (sx + ly + mz + nt + d))). 2 6s 4s4 r sn − 3l2 − 3m2 −sn + 3m2 + 3l2 2 u2 (x, y, z, t) = (1 + 3 cot ( (sx + ly + mz + nt + d))). 2 6s 4s4 k=

241

r −sn + 3m2 + 3l2 −sn + 3l2 + 3m2 2 sec h ( (sx + ly + mz + nt + d)). u3 (x, y, z, t) = 2s2 4s4 r −sn + 3l2 + 3m2 −sn + 3m2 + 3l2 2 u4 (x, y, z, t) = − csc h ( (sx + ly + mz + nt + d)). 2s2 4s4 u5 (x, y, z, t)

u6 (x, y, z, t)

4

−sn + 3l2 + 3m2 −sn + 3l2 + 3m2 − 2 4s 8s2 r −sn + 3m2 + 3l2 × tanh2 ( (sx + ly + mz + nt + d)) 16s4 s2 n2 − 6snl2 − 6snm2 + 9l4 + 18m2 l2 + 9m4 − 8s2 (−sn + 3m2 + 3l2 ) r −sn + 3m2 + 3l2 (sx + ly + mz + nt + d)). × coth2 ( 16s4 =

−sn + 3l2 + 3m2 −sn + 3l2 + 3m2 − 2 8s2 r12s −sn + 3m2 + 3l2 (sx + ly + mz + nt + d)) × tan2 ( 16s4 s2 n2 − 6snl2 − 6snm2 + 9l4 + 18m2 l2 + 9m4 − 8s2 (−sn + 3m2 + 3l2 ) r −sn + 3m2 + 3l2 × cot2 ( (sx + ly + mz + nt + d)). 16s4

=

Conclusion

In this paper, the modified extended tanh method has been successfully applied to find the solutions of the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation. The modified extended tanh method is used to find new exact traveling wave solutions. Thus, we can say that the proposed method can be extended to solve the problems of nonlinear partial differential equations which arising in the theory of solitons and other areas.

References [1] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992; 60(7) : 650 − 654. [2] Wazwaz AM. The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations. Chaos Solitons Fractals 2005; 25(1) : 55 − 63. [3] Wazwaz AM. The extended tanh method for new soliton solutions for many forms of the fifth-order-KdV equation. Appl Math Comput 2007 : 184(2); 1002 − 1014. [4] Wazwaz AM. The extended tanh method: exact solutions of the Sine-Gordon and Sinh-Gordon equations. Appl Math Comput 2005; 167 : 1196 − 1210. [5] Fan E, Hon YC. Generalized tanh method extended to special types of nonlinear equations. Z.Naturforsch 2002; 57a : 692 − 700. [6] Alagesan T, Uthayakumar A, Porsezian K. Painlev analysis and Backlund transformation for a threedimensional Kadomtsev-Petviashvili equation. Chaos Solitons Fractals 1997; 8 : 893 − 895.

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 242–249, 2011 http://ijamc.psit.in

A mathematical approach on field equations with a case for higher dimensional cosmological models R K Dubey1 , Abhijeet Mitra2 and Bijendra Kumar Singh3 1,2,3

Department of Mathematics, Govt. Science P G College (Affiliated to A P S University Rewa), Rewa (MP) India Email: 1 rkdubey2004@yahoo.co.in

Abstract: A mathematical solution to Einstein’s field equations with a perfect fluid source, with variable gravitational constant G and Cosmological constant Λ for FRW space-time in higher dimensions is obtained and case study has also been done where the values of ρ(t), G(t), Λ(t), q(t), and dH (t) has been obtained and their nature is also analyzed. Key words: Cosmology; Higher dimension; Variable gravitational coupling (G) and Cosmological Constant term (Λ).

1

Introduction

The formation of our universe i.e. study of modern cosmology took off in 1917 with a paper by Albert Einstein that attempted to describe the universe by means of a simplified mathematical model. Five years later Alexander Friedmann constructed models of the expanding universe that had their origin in big-bang theory [1]. he idea of a static universe or ’Einstein Universe’ is one which demands that space is not expanding nor contracting but rather is dynamically stable. Albert Einstein proposed such a model as his preferred cosmology by adding a cosmological constant (Λ) to his equations of general relativity to counteract the dynamical effects of gravity which in a universe of matter would cause the universe to collapse. ’Einstein Universe’ is one of Friedman’s solutions of Einstein’s field equations for the value of cosmological constant (Λ).This is only stationary solution of all Friedman’s solutions, and because it is stationary, it is thought to be non-physical by majority of astronomers [2, 3]. Those astronomers think that universe is expanding because there is observed a phenomena of Hubble redshift and it is interpreted by those astronomers as a Doppler’s shift caused by galaxies moving away from our own Galaxy. Therefore, it is thought that the real solution of Einstein’s field equations cannot be stationary. As discussed earlier by many researchers [4] that a constant (Λ) cannot explain the huge difference between the cosmological constant inferred from observations and energy density resulting from quantum field theories. In the year 1930 and onwards eminent cosmologists such as A.S. Eddington and Abbe Lemaitree [5, 6] felt that the -term introduced certain attractive features into cosmology and that models based on it should also be discussed. In modern cosmology the reception given to the Λ-term has varied from hostile to the ecstatic. The term is quietly forgotten if the observational situation does not depend on models based on it. It is resurrected if it is found that the standard Friedmann models without this term are being severely constrained by observations.

Corresponding author: R K Dubey

243

To solve the above discussed problem, variable Λ was introduced such that Λ was large in the early universe and then decayed with evolution. A number of models with different decay laws for the variation of cosmological term were investigated during last two decades (Chen and Wu [7] ,Pavon [8] Carvalho, Lima and Waga [9], Lima and Maia [10] Lima and Trodden [11] Arbab and Abdel- Rahaman [12] Vishwakarma [13] Cunha and Santos [14] Carneiro and Lima [15]. It was Bertolami [16, 17] who obtained Cosmological Models with time dependent G and Λ terms and suggested Λ ∼ R−2 ∼ t−2 . Similarly the gravitational constant G couples geometry to matter in the Einstein field equations, and in expanding universe it is considered as a function of time. A possible time variable G was suggested by Dirac [18]. Many authors [19, 20] have proposed linking of the variation of G with that of Λ within the rules of general relativity theory. This new thought leaves Einstein’s equations nearly unchanged as a variation in Λ is accompanied by the variation in G. In this paper we have discussed a case with the above considerations to get some interesting results.

2

The Field equations

Assuming a homogeneous and an isotropic (perfect or neutral) higher dimensional universe given by Friedmann- Liematrie Robertson- Walker (FLRW) space-time metric dr2 2 2 2 2 2 2 + r (dxm ) (2.1) ds = c dt − a (t) 1 − kr2 where dx2m = da21 + sin2 a1 da22 + . . . + sin2 a1 sin2 a2 . . . sin2 am−1 da2m . Here a(t), k = 0, ±1 and D = m + 2 stand for scale factor, curvature parameter and total number of space time dimensions respectively. The Einstein field equations with time varying cosmological and gravitational ’constants’, 1 aik − agik = 8πG [(ρ + p)ui uk − ρgik ] + Λgik 2 For the metric (1) yields two independent equations, " # 2 m(m + 1) a˙ k + 2 = 8πGρ + Λ 2 a a m¨ a m(m − 1) + a 2

" # 2 a˙ k + 2 = −8πGP + Λ a a

(2.2)

(2.3)

(2.4)

Here equation (3) is time-time component and equation (4) the space- space component of the field equation (2). The over-dot denote derivative with respect to time. Solving equations (3) & (4), we get the continuity equation From eqn. (3) we get 2 a˙ k 2(8πGρ + Λ) + 2 = a a m(m + 1) Differentiating w.r.t. t (time co-ordinate) a˙ a¨ a − (a) ˙ 2 2k 2 ˙ + Λ) ˙ 2 − 3 a˙ = (8πGρ˙ + 8π Gρ 2 a a a m(m + 1)

1 a3

˙ + Λ) ˙ (8πGρ˙ + 8π Gρ a ¨aa ˙ − a˙ 3 − k a˙ = m(m + 1)

(2.5)

244

a ¨aa ˙ = a3 a ¨=

˙ + Λ) ˙ (8πGρ˙ + 8π Gρ + a˙ 3 + k a˙ m(m + 1)

˙ + Λ) ˙ a3 (8πGρ˙ + 8π Gρ a˙ 3 k a˙ + + aa ˙ m(m + 1) aa ˙ aa ˙

(2.6)

Substituting the value of a ¨ obtained in (6) to (4) we get # " # " 2 ˙ + Λ) ˙ m a2 (8πGρ˙ + 8π Gρ a˙ 2 k m(m − 1) a˙ k + + + + 2 = −8πGP + Λ a m(m + 1)a˙ a a 2 a a # " 2 ˙ + Λ) ˙ a(8πGρ˙ + 8π Gρ m(m − 1) a˙ k + m+ + 2 = −8πGP + Λ (m + 1)a˙ 2 a a 2 Substituting the value of aa˙ + ak2 from (5) ˙ + Λ) ˙ a(8πGρ˙ + 8π Gρ m(m + 1) 2 (8πGρ + Λ) + = −8πGP + Λ (m + 1)a˙ 2 m(m + 1) ˙ + Λ˙ a 8πGρ˙ + 8π Gρ + 8πGρ + 8πGP = 0 (m + 1)a˙ Dividing the entire equation above by 8πG we get aρ˙ G˙ ρa aΛ˙ + + +P +ρ=0 a(m ˙ + 1) G (m + 1)a˙ 8πG(m + 1)a˙ Multiply the above equation by ρ˙ + ρ

a(m+1) ˙ a

we get

G˙ Λ˙ a˙ + + (P + ρ)(m + 1) = 0 G 8πG a

(2.7)

From equation (7) we observe that energy density is not conserved for matter field due to varying nature of scalars G & Λ [34, 35]. Since the Principle of equivalence requires only gik , Λ & G should not involve. So in this case law of Conservation of energy-momentum holds and it shows from equation (7) a˙ =0 a

(2.8)

−Λ˙ Λ˙ ρG˙ = ⇒ G˙ = − G 8πG 8πρ

(2.9)

ρ˙ + (m + 1)(ρ + P ) Using equation (8), equation (7) gives

Using equation of state P = (γ − 1)ρ. On putting P = (γ − 1)ρ in (8) ρ˙ a˙ = −(m + 1)γ ρ a

(2.10)

Integrating we get log ρ = −(m + 1)γ log a + log b2 ρ = b2 a−(m+1)γ Where b2 =

+(m+1)γ ρ0 a 0

and suffix 0 represents the present value of the parameters.

(2.11)

245 a˙ a

From equation (10) we get

ρ˙ 1 = − (m+1)γ P , Substituting this value of

a˙ a

in eqn. (3) we get

ρ˙ 2 k m(m + 1) m(m + 1) 1 = 8πGρ + Λ − 2 × 2 (m + 1)2 γ 2 ρ2 a 2 ρ˙ 2 16πG 2Λ k 2 2 = (m + 1) γ + − ρ3 m(m + 1) m(m + 1)ρ a2 ρ Again differentiating (12) and using equation (9) " 16π G˙ 2 ρ3 2¨ ρρ˙ − ρ˙ 2 .3ρ2 ρ˙ 2 2 = (m + 1) γ + 3 2 (ρ ) m(m + 1) m(m + 1)

ρΛ˙ − Λρ˙ ρ2

!

−k

(2.12)

2aaρ ˙ + a2 ρ˙ (a2 ρ)2

#

" # ˙ ˙ 2¨ ρρ˙ 3ρ˙ 2 ρ˙ 16π G k2 2 Λ 2 ρΛ ˙ a ˙ k ρ ˙ − 4 = (m + 1)2 γ 2 − 2 + − − 2 2 ρ3 ρ m(m + 1) m(m + 1)ρ m(m + 1)ρ2 a p a a ρ 2¨ ρρ˙ −3 ρ3

2 2k.1 ρ˙ k ρ˙ ρ˙ −2Λρ˙ ρ˙ 2 2 = (m + 1) γ + − 2 2 ρ ρ2 m(m + 1)ρ2 (m + 1)γ ρ2 a2 a ρ

Above equation is obtained using equation 9. So, we have 2¨ ρ −3 ρ 2¨ ρ −3 ρ Since from eqn. (10)

a˙ a

=

2 ρ˙ 2Λ 2 2k − (m + 1)γk = (m + 1)γ − ρ γa2 m

2 ρ˙ 2Λ 2 k (2 − (m + 1)γ) = (m + 1)γ − ρ γa2 m

(2.13)

ρ˙ −1 (m+1)γ ρ

ρ˙ −1 (m + 1)γ ρ −1 ρ¨ ρ − ρρ ˙ . H˙ = (m + 1)γ ρ2 2 ρ¨ 1 ρ˙ −1 + = (m + 1)γ ρ (m + 1)γ ρ 2 ρ˙ −2¨ ρ 2 ˙ 2H = + (m + 1)γρ (m + 1)γ ρ H=

(2.14)

From eqn. (13) we get 3 −2¨ ρ + (m + 1)γρ (m + 1)γ

2 ρ˙ ((m + 1)γ − 2) 2Λγ = k+ ρ a2 m

Substituting eqn. (14) in eqn. (13) we get 2H˙ + (m + 1)γH 2 + where H =

a˙ a

is the Hubble parameter.

(2 − (m + 1)γ)k 2Λγ − =0 a2 m

(2.15)

246

3

Solution and their analysis

Many authors have suggested that Λ ∼ a−2 by making different assumptions. Chen and Wu have suggested that in the relation Λ = αa−2 , the constant is related to the curvature parameter k and hence we may assume Λ = b1 a−2

(3.1)

2H˙ + (m + 1)γH 2 = 0

(3.2)

In order to have

we make specific assumption of Λ given by eqn. (16) so that the last two terms of eqn. (15) get canceled. i.e, 2Λγ k =0 (2 − (m − 1)γ) 2 − a m or, Λ=

m k m (2 − (m + 1)γ) 2 ⇒ (2 − (m + 1)γ)k = b1 2γ a 2γ

(3.3)

Now from equation (17) we obtain 2H˙ = −(m + 1)γ H2 which on integration provides − Let

(m+1)γ t 2

= u and H =

2 1 (m + 1)γ = −(m + 1)γt ⇒ = t H H 2

(3.4)

a˙ a

(m + 1)γ dt = du 2 Multiplying both sides by H and integrating we get H Since H =

a˙ a

(m + 1)γ dt = H.du 2

and H = u1 . Therefore, 1 (m + 1)γ a˙ dt = du 2 a u

Integrating, a=

(m + 1)γ b3 t 2

2 (m+1)γ

(3.5)

where b3 is an integrating constant. By using eqn. (20, 11, 16), we get ρ = b2

−2 (m+1)γ (m+1)γ (m + 1)γ b2 × 4 b3 t = 2 ((m + 1)γb3 )2 t2 −4 (m+1)γ (m + 1)γ Λ = b1 b3 t 2

(3.6)

(3.7)

247

Substituting the value of ρ and Λfrom eqn. (21) and (22) in equation (9) and integrating, we obtain 4 1− (m+1)γ 2b1 b3 (m + 1)γ ˙ G= b3 t 8πb2 2 On integrating, we obtain, 4 2− (m+1)γ 2b1 1 (m + 1)y G = b4 + b3 t 8πb2 ((m + 1)γ − 2) 2

Using equation (18) we get 4 2− (m+1)γ mk (m + 1)γ G = b4 + b3 t 8γπb2 2

(3.8)

where b4 is the constant of integration. In order to satisfy all the field equations by new values of a, ρ, Λ, G we obtain relation between constants as b4 =

m(m + 1)b23 , 16πb2

which is obtained by putting equations (20, 21, 22, 23) in equation (12). From eqn. (19) we can calculate the age of universe in terms of H as t=

2 (m + 1)γH

The deceleration parameter q for the present model takes the form q=−

Since a =

(m+1)γ b3 t 2

2 (m+1)γ

a¨ a (a) ˙ 2

. Differentiating with respect to t, we get

2 2 a˙ = (m + 1)γb3 (m + 1)γ

(m + 1)γ γb3 2

2 (m+1)γ −1

Again differentiating with respect to t, a ¨=

2 (m + 1)γb3

2

2 (m+1)γ −2 2 2 (m + 1)γ [ − 1][ γb3 ] (m + 1)γ (m + 1)γ 2

a¨ a Putting value of a, a, ˙ a ¨in equation q = − (a) 2. , we get 2 − 1 (m+1)γ 1 q=− = [(m + 1)γ − 1] 2 2 (m+1)γ

(3.9)

Equation (24) shows that q is constant in this model and depends on the dimensionality of the γ−1 space-time. The relation between temperature and energy density as T ∼ ρ γ has been widely accepted in literature. With the help of eqn. (21) we obtain ρα and

1 t2

248

T ∼ρ

γ−1 γ

1 2 ∼ = ρ1− γ ⇒ T ∼ t−2+ γ

(3.10)

The horizon distance dH (t) at time t is the proper distance travelled by light emitted at t = te Z t dt, dH (t) = a(t) lim te→0 (3.11) , te a(t ) 2 (m+1)γ Z t dt0 (m + 1)γ lim te → 0 b3 t 2 (m+1)γ 2 (m+1)γ te 0 b t 3 2 2 1− (m+1)γ (m+1)γ 2 (m+1)γ b3 t 2 (m + 1)γ 2 = b3 t 2 2 (m + 1)γb3 1−

dH (t) =

(m+1)γ

=

(m+1)γb3 t (m+1)γb3 (m+1)γ−2 (m+1)γ

=

(m + 1)γ (m + 1)γ − 2

t

(3.12)

From equation (27) it is clear that distance between two observers depend on the dimensionality of space-time.

4

CONCLUSION

In this paper we have tried to present a case of cosmological model with varying gravitational coupling G and cosmological term Λ in higher dimensions. It is clear from the values obtained in the case for ρ(t), G(t), Λ(t), T (t), q(t) and dH (t) that these quantities depend on the dimensionality of space-time. The results obtained in this paper are in favour of the views of astronomical observations [36, 37]. The cosmological term Λ decides the behaviour of the universe in the model. In this paper we have discussed a case, in which we obtain a positive value of Λwhich will correspond to a negative effective mass density. So in this case we can expect that expansion of universe will tend to accelerate. The observations on magnitude and red shift of type Ia Supernova [38, 39, 40] suggest that our universe may be an accelerating one or otherwise with induced cosmological density, through the cosmological Λ-term. Thus our models are consistent with the results of the observations made in recent times.

5

Acknowledgement

I acknowledge my grateful thanks to CRO, Bhopal of University Grant Commission New Delhi for providing financial assistance under Minor Research Project.

References [1] Narlikar, J., An Introduction to Cosmology, Cambridge University Press (2002). [2] Alam, U. and Sahni, V.; Phys. Rev, D 73, 084024 (2006). [3] Astier, P. et al.; Astrono. Astrophys. 447, 31 (2006). [4] Dolgov, A.D: In the very early Universe. Gibbons, G.W., Hawking, [5] S.W. and Siklos, S.T.C.; Cambridge University press Cambridge, p. 449 (1983). [6] Eddington, A.S.; Mon. Not. Roy. Astron. Soc. 90, 668. (1930) [7] Lemaitre, Abbe, G.; Mon. Not. Roy. Astron. Soc. 91, 483(1931).

249 [8] Chen, W. and Wu,Y,; Physics, Rev. D 48, 695 (1990) [9] D. Pavon.; Physics. Rev. D43, 375 (1991) [10] Lima, J.A.S. and Carvalho, J.C.; Gen. Rel. Grav. 26, 909 (1994). [11] Lima, J.A.S. and Maia, J.M.F.; Phys. Rev. D 49, 5579 (1994) [12] Lima, J.A.S. and Trodden, M.; Phys. Rev. D 53, 4280 (1996) [13] Arbab, A.I. and Abdel Rahaman, A.M.M.; Phys. Rev. D. 50, 7725 (1994). [14] Vishwakarma, R.G.; Gen. Relativ. Gravit. 33, 1973 (2001). [15] Cunha, J.V. and Santos, R.C.; Int. J. Mod. Phys. D 13, 1321 (2004). [16] Carneiro, S. and Lima, J.a.S.; Int. J. Mod. Phys. A 20, 2465 (2005). [17] Bertolami, O.; Nuovo Cimento 1393 (36 (1986). [18] Bertolami, O.; Fortschr. Phys. 34, 829 (1986) [19] P.A.M. Dirac.; Nature, 61, 323 (1937). [20] Abdel- Rahaman, A.M.M.; Gen. Rel. Grav. 22, 665 (1990) [21] Berman, M.S.; Gen. Rel. Grav. 23, 465 (1991). [22] Berman, M.S.; Phys. Rev., D 43, 1075 (1991). [23] Sistero, R.F.; Gen. Rel. Grav., 23, 1265 (1991). [24] Kalligas, D., Wesson, P., Everitt, C.W.f.; Gen., Rel. Grav., 24, 315 (1992). [25] Beesham, A.; Phys. Rev., D 48, 3539 (1993). [26] Beesham, A.; Gen. Rel. Grav., 26, 159, (1994). [27] Abbussattar, Vishwakarma, R.G.; Class. Quantum Grav., 14, 945 (1997). [28] Harko, T., Mak, M.K.; Gen. Rel. Grav., 31, 849 (1999). [29] Chakraborty, I., Pradhan, A., Grav. & Cosmo.; 7, 55 (2001). [30] Pradhan, A., Yadav, V.K.; Int. J. Mod. Phys., D 11, 893 (2002). [31] Pradhan, A., Pandey, P., Singh, G.P.; Deshpandey, R.V., Spacetime & Sulstance, 6 (116) (2003) [32] Pradhan, A., Singh, A.K., Otarod, S.; rom. J. Phys., 52, 415 (2007). [33] Singh, C.P., Kumar, S., Pradhan, A.; Class. Quantum Grav., 24, 455 (2007). [34] Singh, J.P., Pradhan, A., Singh, A.K.; Gr. Qe./0705-0459 (2007) [35] Singh, G.P., Kotambkar, S. and Pradhan, A.; Int. J. Mod. Phys., D 12, 941 (2003). [36] Singh, G.P. and Kotambkar, S.; Gen. Rel. Grav. 33 , 621 (2001). [37] Perlmutter, S. et al.; Astrophys. J. 483, 565 (1997). [38] Hu, Y., Turner, M.S. and Weinberg, E.J.; Phys. Rev. D 49, 3830 (1994). [39] Ratra, B. and Peebles, P.J.E.; Phys. Rev. D 37, 3406 (1988). [40] Riess, A.G.; et al., Astrophys. J., 607, 665 (2004). [41] Knop, R.K.; et al. Astrophys. J., 598, 102 (2003).

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 250â€“256, 2011 http://ijamc.psit.in

Quadrature method for cylindrical wire antenna M.P.Ramachandran Mission Development Group, ISRO Satellite Centre, Bangalore 560 017, INDIA Email: 1 mprama@isac.gov.in

Abstract: The distributed current in the straight cylindrical antenna can be obtained by solving the Hallen equation with certain unknown constants. In this paper the Hallen equation is reduced to a Cauchy singular integral equation (CSIE). Quadrature method is then applied to the CSIE to obtain a linear system of equations. This approach enables to resolve the unknown constants with the condition that the current vanishes at the ends. This alternative approach is now well posed. A couple of examples are worked out and distributed current is computed. Key words: Quadrature; cylindrical antenna; integral equation

1

Introduction

Consider the cylindrical antenna integral equation for a perfectly conducting tube of length 2h, with a radius a and described by Hallen equation [1] Z âˆ’jĎ‰ z A(z) = E(z 0 ) sin k(z âˆ’ z 0 ) dz 0 + C1 cos kz + C2 sin kz |z| â‰¤ h (1.1) k 0 where A(z) =

1 4Ď€

Z

+h

K(z, z 0 )I(z 0 )dz 0

(1.2)

âˆ’h

We need to solve for the total axial current I(zâ€™) in (1.1). The kernel in (1.2) is K(z, z 0 ) =

1 2Ď€

Z

+Ď€

âˆ’Ď€

eâˆ’jkP 0 dĎ† P

(1.3)

q 0 where P = P (z âˆ’ z 0 , Ď†0 ) = [(z âˆ’ z 0 )2 + 4a2 sin2 ( Ď†2 )]. Here k is the wave number, Ď‰ is the angular frequency and characterises the medium. The antennaâ€™s length is aligned along the zaxis and the current flows along its length. Literature has extensively discussed the determination of the distributed current either from (1.1) or from the Pocklington equation (see [2, 3]). In (1.1), C1 = A1 (0) and C2 k = A1 (0) (1 denotes the derivative). These constants however depend on the unknown function. Rynne [2] has observed that the Hallen approach becomes well posed and is equivalent to the solution of the Pocklington equation provided the constants have certain specific

Corresponding author: M.P.Ramachandran

251

values and that the current vanishes at the ends. This suggestion is satisfied here while solving the CSIE using quadrature methods. This is the proposal in the paper. In Section 2 we first briefly discuss the kernel and obtain equivalent CSIE from (1.1). To make I(z) vanish at the ends of the interval, an appropriate condition is obtained, which depends on current, the incident field and the kernel. This condition helps to determine the unknown constants appearing in the Hallen equation. Quadrature method is then applied to the CSIE in Section 3. The integral equation is finally replaced by a linear system of equations and the solution is the distributed current. In Section 4 of the paper we suggest separating the incident field E(z) into odd and even parts. Subsequently the unknown constants in the Hallen equation are either eliminated or determined by using the boundary condition. The computed current is illustrated for convergence against various dimensions of the linear system.

2

Cauchy Singular Integral Equation

p −jkr The kernel in equation (1.2) can replaced as K = Kr = e r here r = [(z − z 0 )2 + a2 ] for cylindrical antenna while a << h and a << λ. The equation (1.1) then becomes ill-posed [4]. The kernel in (1.3) has a logarithmic singularity as suggested by Schelkunoff [1] and derived by Pearson [5]. This singularity should be incorporated to obtain sensible solution. We extend the approach of Jones [6] to decompose K as: K = K1 + K2 + K3 where K1 = 1 K2 = π

Z

1 π

Z

π

(u2 + a2 φ02 )−1/2 dφ0

π

[(u2 + 4a2 sin2 (

0

φ0 −1/2 )) − (u2 + a2 φ02 )−1/2 ] dφ0 2

and K3 =

(2.1)

0

−1 2π

Z

π

−π

1 − ejkP 0 dφ P

(2.2)

(2.3)

The integral in (2.1) has a closed form expression: K1 = Ks + K1b where Ks =

−1 ln |z − z 0 | aπ

1 u {ln a + ln |π + [( )2 + π 2 ]1/2 |} aπ a The kernel can thus be expressed as a sum of singular and non-singular part: K1b =

(2.4)

(2.5) (2.6)

K = Ks + KB

(2.7)

KB = K1b + K2 + K3

(2.8)

Here, The term Ks has the logarithmic singularity. In absence of any closed form expression for K2 , it is replaced by applying trapezoidal rule and the expression in [7] to evaluate K3 . Alternative approximation for KB is given in [4]. Note that the first derivative of the bounded part KB can be shown to be bounded and continuous, while its second derivative can be shown to have logarithmic singularity (see [3, 6]).

252

The equation (1.1) defined over [−h, h] is normalized to [−1, 1] in order to facilitate the application of the quadrature formulae. Differentiating the equation (1.1) with respect to z, and denoting ∂K(z,z 0 ) as K 1 (z, z 0 ), we have the CSIE ∂z Z

+1

(1/4π)

K 1 (z, z 0 )I(z 0 )dz 0 = (−jεω)

−1

Z

z

E(z 0 ) cos k(z − z 0 )dz 0 − C 1 k sin kz + C2 k cos kz (2.9)

0

The analytic theory of CSIE is discussed in detail in [8] as a Riemann-Hilbert problem. We shall use a more general notation instead of (2.9) for simplicit Z +1 Z +1 R(z, z 0 )I(z 0 )dz 0 = F (z)jzj = 1 (2.10) I(z 0 )/(z − z 0 )dz 0 + (−1/πa) −1

−1

where by using (2.7) and (2.8) we have R = KB1 Z F (z) = (−4πjεω)

(2.11)

z

E(z 0 ) cos k(z − z 0 )dz 0 − C1 k sin kz + C2 k cos kz

(2.12)

0

The solution of (2.10) has integrable singularity at the ends of the interval. Z +1 2 −1/2 I(z) = (1 − z ) c − (a/π) χ(t)(1 − t2 )1/2 /(z − t)dt

(2.13)

−1

where Z

+1

I(z 0 )dz 0

c = (1/π)

(2.14)

−1

Z

+1

χ(t) = F (t) −

R(t, t0 )I(t0 )dt0

(2.15)

−1

In physical problems, ‘c’ in (2.14) is usually known and hence closed form solution can be derived when R(z, z 0 ) is zero. The numerical solution of CSIE has a prolific literature (see [9]) ever since the work of Erdogan [10] appeared. The current that is confined to the wire is bounded and vanishes in the neighborhood of the ends. This physical condition implies that, I(±1) = 0

(2.16)

To enable the current I(z 0 ) satisfy this relation we write [8]: I(z 0 ) = (1 − z 02 )1/2 Ψ(z 0 ),

(2.17)

The following condition needs to be satisfied by the solution of (2.13) (with (2.14) and (2.15)) in order to satisfy (2.16). This equation is given below which contains the constants C1 and C2 and specifies them. Z +1

(1 − z 2 )−1/2 χ(z)dz = 0

(2.18)

−1

The numerical solution of CSIE wherein the solution vanishes at the ends of the interval and in the absence of any unknown constants has been discussed in [11] - [14]. It needs to be mentioned that Tan [15] has also considered the Cauchy conversion (2.3) and has approximated the solution by Chebychev polynomials as in [10] while recently Bruno [16] has used them while solving (1.1). While differentiating (1.1) and deriving the CSIE (2.9), the derivatives of the kernel in (2.7) and (2.8) are obtained as follows. K 1 s = (−1//πa)(1/u)

(2.19)

253

K 1 1b = [{π + ((u/a)2 + π 2 )−1/2 }(u/a2 ){(u/a)2 + π 2 }−1/2 ](1/πa)

(2.20)

K2 is approximated using trapezoidal rule on the variable ϕ’ over [-1,1] and then its derivative is obtained. Also, the derivative of K3 (composed of real and imaginary parts) is first obtained and approximated to a known accuracy by applying Mclaurin expansion (see [7]). As mentioned earlier other alternative approximate form for KB 1 as derived from [4] can also be used.

3

Quadrature and numerical solution

The Cauchy integral in (2.10) is approximated as follows [11] : Z

+1

(1 − z 02 )1/2 Ψ(z 0 )/(z 0 − z)dz 0 =

Z

−1

+1

(1 − z 02 )1/2 (Ψ(z 0 ) − Ψ(z))/(z 0 − z)dz 0 − zΨ(z)

−1

≈ (π/(n + 1))

n X

(1 − tk 2 )Ψ(tk )/(tk − −z) − Ψ(z){−z + T n+1 (z)/U n (z)} − zΨ(z)

(3.1)

k=1

here U n (tk ) = 0; tk = cos(kπ/(n + 1)).; k = 1, 2, . . . n

(3.2)

When we select z such that, z = xr , T n+1 (xr ) = 0; xr = cos((2r − 1)π/2(n + 1)); r = 1, 2 . . . , (n + 1)

(3.3)

We notice the terms trailing the summation in (3.1) vanishes. The Gauss-Chebychev quadrature is applied to the regular integral in (2.10) for z = xr to obtain Z +1 n X (1 − z 02 )1/2 R(xr , z 0 )Ψ(z 0 )dz 0 = π/(n + 1) (1 − tk 2 )R(xr , tk )Ψ(tk ) (3.4) −1

k=1

Thus effectively we have reduced (2.10) as (1/a(n + 1))

n X

2

(1 − tk )Ψ(tk )/(xr − tk ) + π(n + 1)

k=1

n X

(1 − tk 2 )R(xr , tk )Ψ(tk ) = F (xr )

(3.5)

k=1

where r = 1, 2..., (n + 1). The above linear system has (n + 1) equations in (n + 2) unknowns, namely Ψ(tk ) , k = 1, 2, . . . n, C1 and C2 . The current I(z 0 ), at z = tk can be obtained using (2.17). Quadrature methods have the advantage of being simpler and eliminate the need of the evaluation of integrals though the collocation points are restricted while arriving at the linear system of equations. Next we use Z +1 n+1 X (1 − z 02 )−1/2 F (z 0 )dz 0 = π/(n + 1) F (xr ) (3.6) −1

r=1

along with the approximation (3.4) in (2.18) to get π/(n + 1)

n+1 X

F (xr ) − (π/(n + 1))

r=1

n X

(1 − tk 2 )R(xr , tk )Ψ(tk ) = 0

(3.7)

k=1

Interestingly while summing all the (n+1) equations in (3.5) and using the formula n+1 X r=1

1/(tk − xr ) = 0

(3.8)

254

Figure 1: Quadrature Method : 2h = 1.2 λ and a = 0.001λ

The first summation term in (3.5) vanishes and we arrive at (3.7). Thus, we have an important observation that the solution of (3.5) for Ψ (tk ) , k = 1,2,. . . .,n , C1 and C2 obtained by discretising (2.10) also satisfies the equation (3.7) which is nothing but the discretisation of the boundary condition (2.18). Thus unknown constants in (3.5) satisfy boundary condition in (2.18) and attain specific values as suggested by Rynne in being well posed. Next, as any constant term (from the even component of the right side terms) appearing in (1.1) vanishes on differentiation to (2.9), we propose the following to establish the equivalence of CSIE to (1.1). An additional condition is obtained after multiplying either side of (1.1) by (1 − z 02 )−1/2 and then integrating over [−1, 1] in order to remove the singularity by using the formula Z

+1

lnjz − z 0 j(1 − z 2 )−1/2 dz = − ln 2,

(1/π)

(3.9)

−1

We have Z +1 Z (1/4π) {−ln2/a + −1

+1

KB (1 − z 2 )−1/2 dz}I(z 0 )dz 0 =

−1

Z

+1

{g(z) + C1 cos kz(1 − z 2 )−1/2 dz

−1

(3.10) where g(z) denotes the first right side term in (1.1). This equivalence approach has been suggested in [17] where the logarithmic integral equation has a solution that possesses integrable singularity. Notice that the term containing C2 vanishes. Then applying the quadrature (3.6) for the integrals in (3.10) we get n X

(1/4(n + 1)) − ln2/a + π/(n + 1)

n+1 X r=1

k=1

n+1 X

K B (xr , tk )(1 − tk 2 )Ψ(tk ) = π(n + 1){

g(xr ) + C1 cos kxr }

r=1

(3.11) The linear system in (3.5) along with that in (3.11) has (n+2) equations to solve for as many unknowns. Alternatively the value of C1 from (3.11) can be substituted in (3.5) to solve for the remaining (n+1) unknowns in (n+1) equations. This shall become clear in the next section.

4

Case Study

The external source field in (1.1) could in general be separated as a sum of even and odd parts. The incident field is set to be a plane wave and that a constant say C0 (an even function). In (2.10), it can be shown that R(v) = −R(u) , where v = −(u). Whenever the input function in (2.10) is odd, the solution Ψ(z) is then an even function. In (1.1) the constant C2 is absent and hence can be written as

255

Figure 2: Quadrature Method : 2h = Îť /2 and a = Îť/5000

Figure 3: Convergence I(0)

jĎ‰ (1 âˆ’ cos kz) + C1 cos kz k2 The differentiation leads to a CSIE which is Z +1 1 jĎ‰ K 1 (z, z 0 )I(z 0 )dz 0 = C0 2 (k sin kz) âˆ’ C1 k sin kz 4Ď€ âˆ’1 k A(z) = (C0

(4.1)

(4.2)

The additional equation in (3.11) is Z

+1 0

0

Z

+1

I(z ) dz {âˆ’ln2/a + âˆ’1

K B (1 âˆ’ z 2 )âˆ’1/2 dz} = (4Ď€Co jĎ‰Îľ/k 2 )(Ď€ âˆ’ d) + C1 d

(4.3)

âˆ’1

R +1 where d = âˆ’1 (cos kz)(1âˆ’z 2 )âˆ’1/2 dz. Equation (4.3) enables to eliminate C1 in (4.2). The equation (4.2) is reduced to a linear system as in (3.11). It contains (n + 1) equations with only n unknowns; Î¨(tk ), k = 1, 2, . . . , n. It is interesting to note that if n is an even number, then when z = 0 (that is xr = 0) the particular linear equation in (3.11) is trivial as Î¨(z) is an even function. Ignoring that equation, the remaining n equations enable to solve uniquely for Î¨(tk ), else if n is an odd number then we end up having an over determined system of equations and adopt the suggestion in [13] to obtain an optimal solution. We give the outline of the convergence of this solution. After eliminating C1 , the system in 1 is continuous and that the input function is also (3.5) is obtained. We find in (4.2) that R = KB continuous. Then the theorem in Elliot [18] (p142) assures that the numerical solution of (3.5) converges to the exact solution at the discrete points tk . 2 to make We considered an example where the antenna length, 2h is 1.2Îť. We choose Îť = 1.2 h = 1. The radius is 0.001Îť and the incident plane wave is constant. Behaviour of the absolute value of the current (in milli Amps) is depicted in Fig 1 for various values of nodes, n. For values of n higher than 94, there was no change in the second decimal. 5Îť Next, when we change Îť = 4 and set radius equal to ( 1000 ). This is to solve (1.1) as in [19, 20] and is a case of a cylinder whose length is half wave length. The result when n is 18 or 72 is given in Fig 2 (a) and is normalized to Îť = 1 and this agrees with that in [19, 20]. The value of I(0), obtained by interpolation, is seen in Fig 2 (b) converging for n greater than 70 to 3.486. This value is 3.485 when n is 120 or 150. Before concluding we notice that whenever the source field is an odd function, the additional condition (4.1) is trivial and C1 = 0. Equations in (3.5) directly determine the current and constant C2 . However, the convergence analysis mentioned earlier is not applicable because of the presence of the constant C2 and is beyond the scope of this paper (see [14]).

256

5

Conclusion

The Cauchy singular integral equation (CSIE) with the pair of unknown constants and an additional equation are deduced from the Hallen equation having a kernel containing logarithmic singularity. The quadrature method based method to solve the CSIE is formulated. This approach allows in resolving the unknown constants besides satisfying the boundary condition while computing the distributed current. The proposed alternative approach is thus well posed [2]. Results are presented for a couple of applications.

References [1] S.A.Schelkunoff, Advanced Antenna Theory, John Wiley placeStateNew York, 1952. [2] B.P.Rynne , ‘ The well-posedness of the integral equations for the thin wire antennas’, IMA J Appl Math 49 :35-44 (1992). [3] T.K.Sarkar, ‘ A study of various methods for computing electromagnetic fields utilizing thin wire integral equation, Radio Science 18 : 29-38 (1983) . [4] P.J.Davies, D.B.Duncan and S.A.Funken,’ Accurate and efficient algorithms for frequency domain scattering from thin wire’, Jnl of Compt Phys 168 : 155-183 (2001). [5] L.W.Pearson , A separation of logarithmic singularity in the exact kernel of the cylindrical antenna integral equation , IEEE Trans Antennas and Propagation, AP 23: 256-258 (1975). [6] D.S.Jones, ‘Note on the integral equation for a straight wire antenna’, IEE Proc Microwave, Antennas and Propagation, 128 :114-116 (1981) [7] M.P.Ramachandran, ’On the bounded part of the kernel in the cylindrical antenna integral equation’, Applied Computational Electromagnetics Society Journal, 13 : 71- 77 (1998). [8] N.I.Muskelishvilli, Singular Integral Equations, P.Nordhoff, placeCityGroningen, (1953) [9] M.A.Golberg, Introduction to numerical solution of Cauchy-type singular integral equation, in ‘Numerical Solution of Integral equations’ , Ed M.A. Golberg, (1990) Plenum Press, New York. [10] F.Erdogan ‘Approximate solution of systems of singular integral equation ‘, SIAM J Appl Math 17 :1041 -1059 (1969). [11] F.Erdogan and G.D.Gupta ,’ On numerical solution of singular integral equation ‘, Quart Appl Math 30 : 525-534 (1972). [12] N.I.Iokimidis, ‘ Some remarks on the numerical solution of Cauchy type singular integral equations with index equal to -1’, Computers and Structures , 14: 403-407 (1981). [13] E.Jen and R.P . Srivastav, ‘ Solving singular integral equations using Gaussian quadrature and over determined systems’, Comp and Math with applics, 9: 625-632 (1983). [14] J.A.Cuminato, ‘Numerical solution of cauchy-type integral equations of index -1 by collocation methods,’ Advances in Computational Mathematics, 6 : 47-64 (1996). [15] S.H.Tan , ‘Modelling of electrically thick cylindrical antennas’, International Jnl of Numerical Modelling: Electronic Networks, Devices and Fields, 3 : 195-206 (1990) . [16] O.P.Bruno and M.C.Haslam ,’Regularity theory and superalgebraic solvers for wire antenna problems’, SIAM J Sci Compt, 29 :1375-1402 (2007). [17] M.P.Ramachandran,’ Numerical solution of an integral equation with logarithmic singularity, ‘ Comp and Math Applics, 26 :51-57 (1993) . [18] D.Elliot,’ Rates of convergence for the method of classical collocation for solving singular integral equations’, SIAM J Numer Analy , 21 : 136-148 (1984). [19] S.J.Orfanidis , Electromagnetics waves and Antennas , [Online] www.ece.rutgers.edu/˜orfanidi/ewa/ch21.pdf [20] M.C.van Beurden and A.Tijhuis,’ Analysis and regularization of the thin-wire integral equation with reduced kernel’, IEEE Trans Antennas and Propogation, AP 55: 120-129 (2007) .

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 257–260, 2011 http://ijamc.psit.in

On finite π - directable automata Milena Bogdanovic Teacher Training Faculty in Vranje, Serbia Email: mb2001969@beotel.net milenab@ucfak.ni.ac.rs

Abstract: Directable automata, known also as synchronizable, cofinal and reset automata, are a significant type of automata with very interesting algebraic properties and important applications in various branches of Computer Science. The central concept that we introduce and discuss in this paper is the concept of π-directable automata as a concomitant specialization concept of the directable automata and generalization of the concept definite automata. Are introduced and a new class of automata, such as trapπ-directable automata, the local trap π-directable automata, uniformly locally trap π-directable automata, finite π-directable automata. Key words: π-directable automata; trapπ-directable automata; the local trap π-directable automata; uniformly locally trap π-directable automata; finite π-directable automata.

1

Indtroduction and basic concepts

Directable automata, known also as synchronizable, cofinal and reset automata, are a significant type of automata with very interesting algebraic properties and important applications in various branches of Computer Science (synchronization of binary messages, symbolic dynamics, verification of software, etc.). They have been a subject of interest of many eminent authors since 1964, when they were introduced by J. Cerny in [3], although some of their special types were investigated even several years earlier. Various specializations and generalizations of directable automata have appeared recently. T. Petkovic, M. Ciric and S. Bogdanovic in [4] introduced and studied trapdirectable, trapped, monogenically, locally and generalized directable automata, as well as other related kinds of automata. These automata have been also studied by Z. Popovic, S. Bogdanovic, T. Petkovic and M. Ciric in [5] and [6]. We also refer to the survey paper by S. Bogdanovic, B. Imreh, M. Ciric and T. Petkovic [2], devoted to directable automata, their generalizations and specializations. The purpose of this paper is to study the concept of π-directable automata as a concomitant specialization concept of the directable automata and generalization of the concept definite automata. Automata be referred the π-directable automata if for each input word u ∈ X + there is k ∈ N so that there is uk ∈ DW (A). Similarly, if for each input word u ∈ X + there is k ∈ N so that uk ∈ LDW (A) , uk ∈ GDW (A) , uk ∈ T DW (A) , uk ∈ LT DW (A) , uk ∈ T W (A) , then we call the automata A the locally-π-directable automata, the general π-directable automata, the trap-π-directable automata, the uniformly locally trap-π-directable and the π-trapped automata, respectively. To mark a class consisting of these automata will use tags that are being introduced following table.

Corresponding author: M. Bogdanovic

258 Table 1

Mark πDir U LπDir GπDir

2

Class of automata π−directable uniformly locally π-directable general π-directable

Mark T πDir U LT πDir πT rap

Class of automata trap-π-directable uniformly locally trap π-directable π-trapped

Characterization of the π-directable automata

The automata A is colled general π-directable automata if for each word u ∈ X + , there is k ∈ N , so that uk is the general directing word, ie. uk ∈ GDW (A) [1]. Lemma 2.1. For an arbitrary automaton A, sets of TDW (A), LTDW (A), TW (A), DW (A), LDW (A) and GDW (A) are the ideals of free monoids X ∗ and holds conditions: (i) T DW (A) 6= ∅ =⇒ T DW (A) = LT W (A) = T W (A) = DW (A) = LDW (A) = GDW (A); (ii) LT DW (A) 6= ∅ =⇒ LT DW (A) = LDW (A) = T W (A) = GDW (A); (iii) T W (A) 6= ∅ =⇒ T W (A) = GDW (A); (iv) DW (A) 6= ∅ =⇒ DW (A) = LDW (A) = GDW (A); (v) LDW (A) 6= ∅ LDW =⇒ (A) = GDW (A). Theorems that follow provide a variety of characteristics of π-directable automata. Theorem 2.1. For the automata A the following conditions are equivalent: 1) A is generally π-directable automata; 2) A is an extension of local π-directable automata using trap π-directable automata; 3) S is a nil-extensions of rectangular bands. Proof: 1) → 2). Let the automaton A is generally π-directable. Then, it is the extension of the locally directable automata B using trap-directable automata C. As the automata B is locally directable, then LDW (A) 6= ∅, and based on Lemma 2.1, we have LDW (B) = GDW (B). Thus, for every word u ∈ X ∗ there is k ∈ N such that uk ∈ GDW (B) = LDW (B), and B is uniformly locally π-directable automata. On the other hand, since C is the trap directable automata, it is T DW (A) 6= ∅, and by the Lemma 2.1 it follows that T DW (C) = GDW (C). Therefore, for every word u ∈ X + , there exists k ∈ N , so that uk ∈ T DW (C) = GDW (C), and the automata C is an trap directable automata. This we have proved the implication 1) → 2). 2) → 1). Let A is the extension of uniformly locally π-directable automata B using the trap π-directable automata C. Consider an arbitrary word u ∈ X + . Then exist k, l ∈ N such that uk ∈ LDW (B) and ul ∈ T DW (C) . Based on the feature of the sets of directing words, applies uk+l = uk ul ∈ GDW (A) . Thus we have proved that A is the generally π-directable automata. 1) → 3). Notice, an arbitrary word u ∈ X + . Then there exists k ∈ N such that uk ∈ GDW (A), which means that ηuk = ηuk is an bi-zero in S(A). Let E be the set of all-zero of S(A). Then we have that E is a rectangular bar and the ideal of S(A), and how we proved that for every ηu ∈ S(A) there exists k ∈ N such that ηuk ∈ E, this conclude that the S(A) is the nil-extensions of rectangular bands E. 3) → 1). Let S(A) is the nil-extensions of rectangular bands E. Consider arbitrary word u ∈ X + . By assumption, there exists k ∈ N such that ηuk ∈ E, ie. ηuk ∈ E. This means that ηuk is the bi-zero of S(A), which implies that the uk ∈ GDW (A). Thus we have proved that A is the generally π-directable automata. This is proof of the theorem is complete. Let A is the π-directable automata. Then, for each word u ∈ X + is there n ∈ N so that is un ∈ DW (A). The smallest number n ∈ N such that un ∈ DW (A) , is the level of directing word u. Clearly, the directing words of the automata A has the same level of guidance 1. Following theorems are fully proven in [1]. Now we describe the uniformly locally trap-π-directable automata.

259

Theorem 2.3. For the automata A the following conditions are equivalent: 1) A is uniformly locally π-directable automata; 2) A is the direct sum of π-directable automata, Aα , α ∈ Y and every word u ∈ X + has a limited level of guidance in automata Aα , α ∈ Y ; 3) S(A) is a nil-extension right zero bands. Automata A is the π-trapped, if for each word u ∈ X + , for which there is k ∈ N, hold uk ∈ T W (A). The next theorem describes, among other things, the structure of the transition semigroup of the π-trapped automata. Theorem 2.4. For the automata A the following conditions are equivalent: 1) A is the π-trapped automata; 2) A is an extension of discrete automata with trap π-directable automata; 3)A is a nil-extension of left zero bands. The following theorem gives a complete characterization of uniformly locally trap π-directable automata. Theorem 2.6. For the automata A the following conditions are equivalent: 1) A is uniformly locally trap π-directable automata; 2) A is the extension retractiveof the discrete automata with the trap πdirectable automata; 3) A is the direct sum of trap π-directable automata, Aα , α ∈ Y and every word u ∈ X + has a limited level of guidance in automata Aα , α ∈ Y ; 4) A is the product subdirect of a discrete automata and a trap π-directable automata; 5) A is the parallel composition of a discrete automata, and a trap π-directable automata; 6) S(A) is a nil-semigroup.

3

Finite π-directable automata

In this section we prove that in the case of finite automata is no difference between the π-directable automata and the definite automata. Similar features will be demonstrated for other types of automata that are discussed in the previous sections. Theorem 3.1. The finite automata A is a trap π-directable automata if and only if it is nilpotent automata. Proof: Let the automata A is the trap π-directable automata. This means that there is a state a0 ∈ A, so that for all a ∈ A and for every word u ∈ X + , for which there is n ∈ N such that un ∈ DW (A), holds au = a0 . The transition semigroup S(A) of the automata A is a nil-semigroup. On the other hand, A is a finite automata, so the transition semigroup S(A) is finite. Any finte nil-semigroup is nilpotent, then S(A) is nilpotent semigroup. The automata A is the direct sum of the automata nilpotent Aα , α ∈ Y . However, A is the trap-directable automata, and it is the indecomposable of the direct sum. This means that |Y | = 1, ie. Y = {α} , and A = Aα . So, A is the nilpotent automata. The reversal of the theorem is clear. The proof that a finite automata who is the π-directable, it is also definite, given the following theorem. Theorem 3.2. The finite automata A is a π-directable automata if and only if it is definite automata. Proof: Let the automata A is the π-directable automata. Then, A is a locally π-directable automata and the transition semigroup of the automata A is nil-extension of the right zero bands. However, as A is a finite automata, then S(A) is finite semigroup, so it must be a nilpotent extension of right zero bands. From this fact it follows that A is a direct sum of the automata definite, with the same degree of definiteness. Furthermore, the automata A is indecomposable in direct sum, because it is the π-directable, so it must be an automata definite. The reversal of the theorem is clear. We can prove the following two theorems with analogous considerations.

260

Theorem 3.3. The finite automata A is generally π-directable if and only if it is the general definite. Theorem 3.4. The finite automata A is a uniformly locally π-directable if and only if it is uniformly locally definite. Last theorem shows that, for finite automata, the concept of uniform local trap π-directability is the same concept as uniform local nilpotentility. Theorem 3.5. The finite automata A is a uniformly locally trap π-directable if and only if it is uniformly locally nilpotent. Proof : Let A be a uniformly locally nilpotent automata. Then each monogenic subautomata of A is nilpotent, and how this monogenic subautomata can be finite, according to Theorem 3.2. we have that every monogenic subautomata of A is the trap π-directable. Therefore, A is locally trap π-directable automata, and as a finite automata, that it is uniformly locally trap π-directable automata.

4

Conclusion

In the second half of the twentieth century, the concepts of information and the processing and transfer of information, have become central in many areas of modern science. The very mathematical abstraction of these concepts plays an important role in ensuring that their the study apply exact mathematical models. As one of those mathematical abstraction, forties, fifties years of this century, came the notion of automata. The automata are viewed as systems that can be used for processing and transmission of certain kinds of information. Sixties and later, there is a considerable number of books on the Theory of Automata, which resulted in the development of this area as one of the most important in the field of Computre Science. Here we introduce the notion of the π-directable automata, which is also the concept of specialization and generalization of idea automata directable and definite automata. Also, we introduce and describe the concepts of the trap π-directable automata, locally and uniformly locally π-directable automata, locally and uniformly locally trap π-directable automata etc..

References [1] Milena Bogdanovic, Directable Automata, Their Generalizations and Specializations (Direktabilni automati, njihova uoptenja i specijalizacije), (in Serbian), MSc thesis, University of Ni, Faculty of Sciences and Mathematics, 2001. [2] S. Bogdanovic, B. Imreh, M. Ciric and T. Petkovic, Directable automata and their generalizations – A survey, in: S. Crvenkovic and I. Dolinka (eds.), Proc. VIII Int. Conf. ”Algebra and Logic” (Novi Sad, 1998), Novi Sad Journal of Mathematics 29 (2) (1999), 31-74. [3] J. ernı, Poznmka k homognym experimentom s konecinımi automatami, Mat.-fyz. cas. SAV 14 (1964), 208–215. [3] T. Petkovic, M. Ciric and S. Bogdanovic, Decompositions of automata and transition semigroups, Acta Cybernetica (Szeged) 13 (1998), 385-403. [4] Z. Popovic, S. Bogdanovic, T. Petkovic, and M. Ciric, Trapped automata, Publicationes Mathematicae Debrecen 60 (3-4) (2002), 661-677. [5] Z. Popovic, S. Bogdanovic, T. Petkovic, and M. Ciric, Generalized directable automata, in: Words, languages and combinatorics, III, Proceedings of the Third International Colloquium in Kyoto, Japan, (M. Ito and T. Imaoka, eds.), World Scientific, 2003, pp. 378-395.

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 261â€“269, 2011 http://ijamc.psit.in

New approach for convergence of the series solution to a class of Hammerstein integral equations M. A. Abdoua , I. L. El-Kallab , A. M. Al-Bugamic a

Department of Mathematics Faculty of Education, Alexandria University, Egypt. b Physics and Engineering Mathematics Department, Faculty of Engineering, Mansoura University, PO 35516, Mansoura, Egypt. Email: al kalla@man.edu.eg c Department of Mathematics, Faculty of Applied Sciences, Taif University, KSA.

Abstract: The proof of convergence of the series solution to a class of nonlinear two-dimensional Hammerstein integral equation (NTHIE), including the necessary and sufficient conditions that guarantee a unique solution, is introduced. Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM) are used to solve the NTHIE. It was found that, when using the traditional Adomian polynomials (1.4), ADM and HAM are exactly the same. But when using the proposed accelerated Adomian polynomials formula (1.5), ADM converges faster than HAM. The proposed accelerated Adomian polynomials formula is used directly to prove the convergence of the series solution. Convergence approach is reliable enough to estimate the maximum absolute truncated error. Key words: Hammerestein integral equation; Cauchy-Schwarz inequality; Fixed point theorem; Adomian Method; Homotopy Analysis Method.

1

Introduction

Integral equations provide an important tool for modeling a numerous phenomena and processes and also for solving boundary value problems for both ordinary and partial differential equations. Their historical development is closely related to the solution of boundary value problems in potential theory. Progress in the theory of integral equations also had a great impact on the development of functional analysis. Reciprocally, the main results of the theory of compact operators have taken the leading part to the foundation of the existence theory for integral equations of the second kind [1]-[4]. Therefore, many different methods are used to obtain the solution of the linear and nonlinear integral equations. During the last years, significant progress has been made in numerical analysis of one-dimensional version. However, the numerical methods for two-dimensional integral equations seem to have been discussed in only a few places. Brunner and Kauthen [5] introduced collocation and iterated collocation methods for solving the two-dimensional Volterra integral equation (T-DVIE). In [6] authors proposed a class of explicit Runge-Kutta-type methods of order 3 for solving nonlinear T-DVIE. In [7] authors studied the approximate solution of T-DVIEs by the two-dimensional differential transform method. Abdou, in [8]-[11], used different methods to obtain the solution of Fredholm-Volterra integral equation of the first and second kinds in which the

Corresponding author: I. L. El-Kalla

262

Fredholm integral term is considered in position while the Volterra integral term is considered in time. In this paper, we use ADM and HAM for solving NTHIE Z

b

Z

d

k (x, y, t, s) γ (t, s, u(t, s)) dtds,

µu (x, t) = f (x, y) + λ

(1.1)

c

a

of the second kind where, the free term f (x, y) ∈ J = [a, b] × [c, d], k ∈ J × J is the given kernel and γ (x, y, u(x, y)) is the nonlinear term containing the unknown function u which represents the solution of the NTHIE (1.1). In general µ defines the kind of the integral equation, µ = 0 for the first kind, µ = const 6= 0 for the second kind and µ = µ(x, y) for the third kind. Also, λ is a constant, may be complex, that has a physical meaning.

1.1

Adomian Decomposition Method (ADM)

ADM has been known as a powerful device for solving many functional equations as algebraic equations, ordinary and partial differential equations and integral equations [12]-[15]. In many papers for example [16]-[18] ADM were used to solve some classes of integral equations in one dimension. In this work, the two dimensions NTHIE (1.1) of the second kind (µ = const 6= 0) will be solved using ADM. Without loss of generality (1.1) can be rewritten in the form Z

b

Z

d

u(x, y) = f (x, y) + λ

k(x, y, t, s)γ(t, s, u(t, s))dtds. a

(1.2)

c ∞ P

ADM assume the solution in the series form u(x, y) =

un (x, y) and the nonlinear term γ(t, s, u)

n=0

is decomposed into an infinite series of Adomian polynomials ∞ X

γ(t, s, u) =

An ,

(1.3)

n=0

which can be determined using the traditional formula An =

∞ X 1 dn [γ(t, s, λi ui )]λ=0 . n! dλn i=0

(1.4)

Another formula of Adomian polynomials, called accelerated Adomian polynomials, was deduced by El-Kalla in [19]-[20] in the recursive form An = γ(Sn ) −

n−1 X

Ai ,

(1.5)

i=0

where the partial sum Sn =

n P

ui (x, y) and A0 = γ(u0 ). Application of ADM to Eq. (1.2) yields

i=0

u(x, y) =

∞ X

un (x, y),

(1.6)

i=0

where, Z

b

Z

d

k(x, y, t, s)Ai−1 dtds, i ≥ 1

u0 (x, y) = f (x, y), ui (x, y) = λ a

c

(1.7)

263

1.2

Homotopy Analysis Method (HAM)

Since 1992, Liao in [21] employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely the homotopy analysis method and then modified it step by step [22, 23]. This method has been successfully applied to solve different types of nonlinear problems [24, 25]. HAM is in principle based on Taylor series with respect to an embedding parameter. More importantly, different from all perturbation and traditional non-perturbation methods, HAM provides us a simple way to ensure the convergence of solution series, and therefore, the HAM is valid even for strongly nonlinear problems. In this subsection NTHIE (1.2) will be solved using the HAM. Z bZ d k(x, y, t, s)γ(t, s, u(t, s))dtds = 0, (1.8) N [u] = u(x, y) − f (x, y) − λ a

c

where, N is an operator and u = (x, y) is the unknown function. Let u0 (x, y) denote an initial guess of the exact solution u(x, y), h 6= 0 an auxiliary parameter, H(x, y) an auxiliary function and L an auxiliary linear operator with the property L[g(x, y)] = 0 when g(x, y) = 0. Using r ∈ [0, 1], as an embedding parameter, we construct such a homotopy b (1 − r)L[φ(x, y; r) − u0 (x, y)] − rhH(x, y)N [φ(x, y; r)] = H[φ(x, y; r); u0 (x, y), H(x, y), h, r], (1.9) b is a second auxiliary function . It should be emphasized that we have a great freedom to where H choose the initial guess u0 (x, y), the auxiliary linear operator L, the non-zero auxiliary parameter h and the auxiliary function H(x, y). Assume the homotopy (1.9) to be zero, i.e. b H[φ(x, y; r); u0 (x, y), H(x, y), h, r] = 0. We have the so called zero-order deformation equation (1 − r)L[φ(x, y; r) − u0 (x, y) = rhH(x, y)N [φ(x, y; r)].

(1.10)

When r = 0, the zero-order deformation (1.10) becomes φ(x, y; 0) = u0 (x, y),

(1.11)

also, when r = 1, h 6= 0 and H(x, y) 6= 0, the zero–order deformation (1.10) is equivalent to φ(x, y; 1) = u(x, y).

(1.12)

By Taylor’s theorem φ(x, y; r) can be represents in a power series form of r as follows φ(x, y; r) = u0 (x, y) +

∞ X

um (x, y)rm

(1.13)

m=1

where, 1 ∂ m φ(x, y; r) |r=0 . (1.14) m! ∂rm The nonzero auxiliary parameter h, and the auxiliary function H(x, y)are properly chosen, so that the power series (1.13) of φ(x, y; r) converges at r = 1. Under these assumptions we have the series solution ∞ X u(x, y) = φ(x, y; 1) = u0 (x, y) + um (x, y), (1.15) um (x, y) =

m=1

therefore, we can define the vector → − u n (x, y) = φ(x, y; 1) = {u0 (x, y), u1 (x, y), ..., un (x, y)}.

(1.16)

264

According to the definition (1.13), the governing equation of um (x, y) can be derived from the zero-order deformation equation (1.10). Differentiating (1.10) m times with respective to r, then dividing by m! and setting r = 0,we have the so called mth -order deformation equation − L[um (x, y) − ηm um−1 (x, y)] = hH(x, y)<m (→ u m−1 (x, y)), um (0, 0) = 0,

(1.17)

where, ∂ m−1 N [φ(x, y; r)] 1 |r=0 , (m − 1)! ∂rm−1

− <m (→ u m−1 (x, y)) =

(1.18)

and ηm = 0 (for m ≤ 1) or = 1 (for m > 1). Note that the high-order deformation of Eq. (1.17) − is governing by the linear operator L, and the term <m (→ u m−1 (x, y)) can be expressed simply by (1.18). To obtain a simple iteration formula for um (x, y), choose Lu = u as an auxiliary linear operator, the zero–order approximation u0 (x, y) = f (x, y) is taken and the nonzero auxiliary parameter h and the auxiliary function H(x, y) can be taken as h = −1, H(x, y) = 1. Application of HAM on (1.2) yields u0 (x, y) um (x, y)

= f (x, y) Z bZ = λ a

d

k(x, y, t, s)<m−1 (φp )dtds, m ≥ 1,

c

and the corresponding homotopy series solution is given by u(x, y) =

∞ X

um (x, y)

(1.19)

m=0

2

Existence of a unique solution

To prove the existence of a unique solution of (1.1), we use the principle of contraction mapping on a complete metric space. Rewrite (1.1) in the operator form 1 f (x, y) + W u(x, y), (µ 6= 0), µ

W u(x, y) =

(2.1)

where, λ W u(x, y) = µ

b

Z

Z

d

k(x, y, t, s)γ(t, s, u(t, s))dtds, a

(2.2)

c

and assuming the following conditions: ˚, N ˚ is a constant 1- The kernel k(x, y, t, s) satisfies the continuity condition |k(x, y, t, s)| ≤ N 2- The given function f (x, y) is continuous in the Banach space C[J] and satisfy Z

b

Z

! 21

d

2

|f (x, y)| dxdy a

=δ

c

where δ is a constant. 3- For constants A > A1 and A > P, the function γ(x, y, u(x, y)) satisfies: Z

b

Z

i−{

d

2

1

|γ(x, y, u(x, y))| dxdy} 2 a

≤ A1 ku(x, y)k

c

ii − |γ(x, y, u1 (x, y)) − γ(x, y, u2 (x, y))|

≤ M (x, y) |u1 (x, y) − u2 (x, y)|

265

where kM (x, y)k = P

21

R R

b d 2 4- The unknown function u(x, y) ∈ C[J] with the norm ku(x, y)k = a c |u(x, y)| dxdy

Lemma 1. Under the conditions (1) – (3-i), the operator W , defined by (1.2), maps the space C[J] into itself. Proof. Using equations (1.2), (1.3) and condition 2 with Cauchy-Schwarz inequality, we have

Z bZ d

1 2 W u(x, y) ≤ δ + λ {|k(x, y, t, s)|}{ |γ(x, y, u(x, y))| dxdy} 2 . |µ| µ

c a Using conditions (1) and (3-i), the above inequality takes the form

˚ W u(x, y) ≤ δ + σ ku(x, y)k , σ = λ N

µ A. |µ|

(2.3)

The inequality (1.4) shows that the operator W maps the space C[J] into itself. Also, inequality (1.4) shows that the operators W and W are bounded where, kW u(x, y)k = σ ku(x, y)k

(2.4)

Lemma 2. If the conditions 1) and 3-ii) are satisfied, then the operator W contractive in C[J]. Proof. For any two functions u1 (x, y), u2 (x, y) ∈ C[J], formulas (1.2) and (1.3) lead to

Z b Z d λ (W u1 − W u2 )(x, y) ≤ |k(x, y, t, s)| |γ(t, s, u (t, s)) − γ(t, s, u (t, s))| dtds . 1 2

µ a c Using the conditions 1) and 3-ii) and applying Cauchy-Schwarz inequality we have (W u1 − W u2 )(x, y) ≤ σ ku1 (x, y) − u2 (x, y)k .

(2.5)

Under the condition σ < 1, the operator W contractive and the proof is complete. Theorem 2.1. If the conditions (1)–(3) are satisfied, then the integral equation (1.1) has a unique ˚A. solution in the space C[J] whenever |µ| > |λ| N

˚ ˚A and the proof Proof. From Lemma1 and Lemma 2 since σ = µλ N A < 1 this leads to |µ| > |λ| N is complete.

3

Convergence Analysis

Convergence of the Adomian series solution was studied for different problems and by many authors. In [26, 27], convergence was investigated when the method applied to a general functional equations and to specific type of equations in [28, 29]. In convergence analysis, Adomian’s polynomials play a very important role however, these polynomials cannot utilize all the information concerning the obtained successive terms of the series solution, which could affect directly the accuracy as well as the convergence region and the convergence rate. In the present analysis we suggest an alternative approach for proving the convergence. This approach depends mainly on El-Kalla accelerated Adomian polynomial formula (1.5). As a result to this approach, the rate of convergence for the series solution is accelerated and the maximum absolute truncated error of the series solution is estimated. Define a mapping F : E → E where, E = (C[J], k.k) is the Banach space of all continuous functions on J with the norm ku(x, y)k = max |u(x, y)| . ∀x,y∈J

266

Theorem 3.1. The series (1.6) converges to a unique continuous solution of Eq. (1.2) if f (x, y) and k(x, y, t, s) are continuous and bounded functions and γ(t, s, u) satisfies Lipschitz condition. Proof. Define a sequence {Sn } of partial sum such that Sn = u0 + u1 + ... + un we are going to prove that {Sn } is a Cauchy sequence in Banach space E. Let Sn and Sm are two arbitrary distinct partial sums in the sequence {Sn } and n > m we have kSn − Sm k = max |Sn − Sm | . x,y∈J

Using El-Kalla formula (5) we have kSn − Sm k

n

X

= max

ui (x, y)

x,y∈J

i=m+1

=

max

x,y∈J

n−1 X i=m

Z

b

Z

d

k(x, y, t, s)Ai dtds

λ a

c

Z Z

b d

= |λ| max

k(x, y, t, s)[γ(Sn−1 ) − γ(Sm−1 )]dtds

x,y∈J a

c ≤

|λ| N (b − a)(d − c) max |γ(Sn−1 ) − γ(Sm−1 )| . x,y∈J

Since γ(u) satisfies Lipschitz condition so ∃ a constant L such that |γ(u) − γ(h)| ≤ L |u − h|. and we can write kSn − Sm k ≤ α kSn−1 − Sm−1 k , where, α = |λ| N L(b − a)(d − c). Substituting n = m + 1 we have kSm+1 − Sm k ≤ α kSm − Sm−1 k ≤ α2 kSm−1 − Sm−2 k ≤ · · · ≤ αm kS1 − S0 k . Using the triangle inequality we have kSn − Sm k

≤

kSm+1 − Sm k + kSm+2 − Sm+1 k + · · · + kSn − Sn−1 k m ≤ α + αm+1 + · · · + αn−1 kS1 − S0 k ≤ αm 1 + α + α2 + · · · + αn−m−1 kS1 − S0 k 1 − αn−m m ≤ α ku1 (x, y)k . 1−α

If 0 < α < 1, i.e. 1 − αn−m ≤ 1 then kSn − Sm k ≤

αm max |u1 (x, y)| , 1 − α ∀x,y∈J

(3.1)

but max |u1 (x, y)| < ∞ (since f (x, y) is bounded) then kSn − Sm k → 0 as m → ∞, from which ∀x,t∈J P∞ we conclude that {Sn } is a Cauchy sequence in E so, the series i=0 ui (x, y) converges and the proof is complete.

3.1

Error Estimate

Upon the final result of theorem 2, we can estimate the maximum absolute truncated error of the series solution in the next theorem. Theorem 3.2. The maximum absolute truncation error of the series (1.6) to problem (1.2) is Pm m+1 estimated to be: max |u (x, t) − i=0 ui (x, y)| ≤ α1−α max |γ (u0 )|. ∀x,y∈J

∀x,y∈J

267

Proof. From Theorem 2 inequality (3.1) we have αm max |u1 (x, t)| . 1 − α ∀x,y∈J

kSn − Sm k ≤ As n → ∞ then Sn → u (x, y) and we have

max |u1 (x, y)| ≤ |λ1 | N L(b − a)(c − d) max |γ (u0 )| ≤ α max |γ (u0 )| ,

∀x,y∈J

∀x,y∈J

∀x,y∈J

so, ku (x, t) − Sm k ≤

αm+1 max |γ (u0 )| . 1 − α ∀x,y∈J

Finally the maximum absolute truncation error in the interval J is

m

αm+1 X

max |γ (u0 )| . max u (x, y) − ui (x, y) ≤

1 − α ∀x,y∈J ∀x,y∈J

(3.2)

i=0

4

Numerical Experiments and discussions

Example1. Consider the NTHIE Z

1

Z

u(x, y) = f (x, y) + 0

1

(exy s2 )(u(t, s))k dtds,

(4.1)

0

Using Maple 10, this example will be solved by ADM and HAM when k = 1 (as a linear case) and when k = 2 (as a nonlinear case). In linear case, the free term f (x, y) = x2 y 2 − 0.06666666667e(xy) while in the nonlinear case f (x, y) = x2 y 2 −0.02857142857e(xy) . The exact solution in both cases is u(x, y) = x2 y 2 . Using 10 term approximation when k = 1, table 1 presents a comparison between errors resulted from ADM (ErrorADM ) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1. Table 1 (Linear case, k=1)

x 0.0 0.2 0.4 0.6 0.8 1.0

y 0.0 0.2 0.4 0.6 0.8 1.0

ErrorADM 6.36100E − 08 6.62060E − 08 7.46500E − 08 9.12000E − 08 1.20600E − 07 1.72900E − 07

ErrorHAM 6.36100E − 08 6.62060E − 08 7.46500E − 08 9.12000E − 08 1.20600E − 07 1.72900E − 07

In the nonlinear case (k = 2) we solve using the traditional Adomian polynomials (1.4) and using the accelerated Adomian polynomials (5). Table 2 presents a comparison between errors resulted from ADM with traditional formula (1.4) (ErrorADM 1 ), with accelerated formula (1.5) (ErrorADM 2 ) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1. Table 2 (Nonlinear case, k=2)

x 0.0 0.2 0.4 0.6 0.8 1.0

y 0.0 0.2 0.4 0.6 0.8 1.0

ErrorADM 1 2.20000E − 10 2.28978E − 10 2.58172E − 10 3.15332E − 10 4.17335E − 10 5.98022E − 10

ErrorADM 2 2.00000E − 11 2.08162E − 11 2.34702E − 11 2.86665E − 11 3.79296E − 11 5.43656E − 11

ErrorHAM 2.20000E − 10 2.28978E − 10 2.58172E − 10 3.15332E − 10 4.17335E − 10 5.98022E − 10

268

Example2. Consider the NTHIE Z

1

Z

u(x, y) = f (x, y) +

1

cos(xy)(u(t, s))k dtds

(4.2)

0

0

Also, Maple 10 is used to solve this example by ADM and HAM when k = 1 (as a linear case) and when k = 2 (as a nonlinear case). In linear case, the free term f (x, y) = sin(xy) − 0.2118455042xy while in the nonlinear case f (x, y) = sin(xy)−0.08246639025xy. The exact solution in both cases is u(x, y) = sin(xy). Using 10 term approximation when k = 1, table 3 presents a comparison between errors resulted from ADM (ErrorADM ) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1. Table 3 (Linear case, k=1)

x 0.0 0.2 0.4 0.6 0.8 1.0

y 0.0 0.2 0.4 0.6 0.8 1.0

ErrorADM 0.00000E + 00 4.00000E − 12 1.60000E − 11 3.60000E − 11 6.40000E − 11 1.00000E − 10

ErrorHAM 0.00000E + 00 4.00000E − 12 1.60000E − 11 3.60000E − 11 6.40000E − 11 1.00000E − 10

In the nonlinear case (k = 2) we solve using the traditional Adomian polynomials (1.4) and using the accelerated Adomian polynomials (1.5). Table 2 presents a comparison between errors resulted from ADM with traditional formula (1.4) (ErrorADM 1 ), with accelerated formula (1.5) (ErrorADM 2 ) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1. Table 4 (Nonlinear case, k=2)

x 0.0 0.2 0.4 0.6 0.8 1.0

y 0.0 0.2 0.4 0.6 0.8 1.0

ErrorADM 1 0.00000E + 00 4.00000E − 13 1.60000E − 12 3.60000E − 12 6.40000E − 12 1.00000E − 11

ErrorADM 2 0.00000E + 00 0.00000E + 00 0.00000E + 00 0.00000E + 00 0.00000E + 00 0.00000E + 00

ErrorHAM 0.00000E + 00 4.00000E − 13 1.60000E − 12 3.60000E − 12 6.40000E − 12 1.00000E − 11

It is clear that, in linear case both ADM and HAM give the same approximate solution. In the nonlinear case, It was found that, both HAM and ADM with traditional formula (1.4) are exactly the same, but ADM with accelerated formula (1.5) converges faster than HAM.

5

Conclusion

The necessary and sufficient conditions that guarantee a unique solution to NTHIE is introduced. ADM and HAM are exactly the same for solving linear two-dimensional Hammerstein integral equation. In nonlinear case, both HAM and ADM with traditional formula (1.4) are exactly the same, but ADM with accelerated formula (1.5) converges faster than HAM. Moreover, accelerated Adomian polynomials formula (1.5) is used directly in convergence analysis. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution.

References [1] K.E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, 1976. [2] F.G. Tricomi, Integral Equations, New York (1985). [3] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Philadelphia, New York, 1985. [4] M.A. Golberg, Numerical Solution of Integral Equations, Plenum press, New York, 1990.

269 [5] H. Brunner, J. P. Kauthen, The numerical solution of two-dimensional Volterra integral equations by collocation and iterated collocation, IMA J. Numer. Anal. 9 (1989) 47-59. [6] B. A. Beltyukov, L.N. Kuznechikhina, A Runge-Kutta method for the solution of two-dimensional nonlinear Volterra integral equations, Differential Equations, 12 (1976) 1169-1173. [7] P. Darania and A. Ebadian, Numerical solutions of the nonlinear two-dimensional Volterra integral equations, NJOM, Volume 36 (2007), 163-174. [8] M. A. Abdou, Fredholm-Volterra integral equation of the first kind and the contact problem, Apple. Math. Comput, 125, (2002), 177-193. [9] M .A. Abdou, Fredholm-Volterra integral equation and generalized potential kernel, Apple. Math. and Computing, 13, (2002), 81-94. [10] M. A. Abdou, On asymptotic method for Fredholm-Volterra integral equation of the second kind in contact problems, J .Comp. Appl . Math. 154, (2003), 431-446. [11] M. A. Abdou, Fredholm-Volterra integral equation with singular kernel, Apple. Math. Comput, 137, (2003), 231-243. [12] G. Adomian, Solving Frontier problems of Physics: The Decomposition method, Kluwer, 1995. [13] I. L. El-Kalla, New results on the analytic summation of Adomian series for some classes of differential and integral equations, Appl. Math. Comp., 217, (2010), 3756–3763. [14] A.M. El-Sayed, I. L. El-Kalla, E. A. Ziada, Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations, Applied Numerical Mathematics, 60, (2010), 788–797. [15] I. L. El-Kalla, Error estimate of the series solution to a class of nonlinear fractional differential equations, Commun Nonlinear Sci Numer Simulat, 16, (2011), 1408–1413. [16] I. L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Appl. Math. Lett., 21, (2008), 372-376. [17] A. Wazwaz, The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterra integro–differential equations, Appl. Math. and Comp., 216, (2010), 1304-1309. [18] A. Wazwaz, S. Khuri, Two methods for solving integral equations, Appl. Math. and Comp, Volume 77, (1996), 79-89. [19] I. L. El-Kalla, Convergence of Adomian’s Method Applied to A Class of Volterra Type Integro-Differential Equations, Inter. J. of Differential Equations and Appl., 10, No.2, (2005), 225-234. [20] I. L. El-Kalla, Error analysis of Adomian series solution to a class of nonlinear differential equations, Applied math. E-Notes, 7(2007), 214-221. [21] S.J. Liao, The proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. [22] S.J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003. [23] S.J. Liao, On the homotopy analysis method for nonlinear problems. Appl. Math. Comput., 147, (2004), 499513. [24] S.J. Liao Notes on the homotopy analysis method: some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation, (2008), 4-13. [25] Shijun Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 15, (2010), 2003-2016. [26] Y.cherruault, G.adomian, Decomposition method: new ideas for proving convergence of decomposition methods, Comput. Math. Appl., Vol. 29, No. 7, (1995), 103-108. [27] M.M. Hosseini, H. Nasabzadeh, On the convergence of Adomian decomposition method, Appl. Math. Comput, 182, (2006), 536–543. [28] K.Abbaoui and Y.Cherruault, Convergence of Adomian’s method applied to differential equations, Comput. Math. Appl., Vol. 28, No. 5, (1994), 103-109. [29] R. Rajaram, M. Najafi, Analytical treatment and convergence of the Adomian decomposition method for a system of coupled damped wave equations, Appl. Math. Comput, 212, (2009), 72–81.

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 270–273, 2011 http://ijamc.psit.in

Application of homotopy perturbation method for high order nonlinear partial differential equations Esmail Alibeigi, Mostafa Eslami, Ahmad Neirameh1 Department of Mathematics Islamic Azad University of Gonbad Kavos, Iran Email: 1 neirameh m@yahoo.com

Abstract: In the paper, we extend the homotopy perturbation method to solve nonlinear fifth order KdV equation. As a result, we successfully obtain some available approximate solutions of them. Numerical solutions obtained by the homotopy perturbation method are compared with the exact solutions. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear partial differential equations. Key words: Homotopy perturbation method; fifth order KdV equation.

1

Introduction

The application of the homotopy perturbation method (HPM) in nonlinear problems has been devoted by scientists and engineers, because this method is to continuously deform a simple problem which is easy to solve into the under study problem which is difficult to solve. The homotopy perturbation method [5], proposed first by He in 1998 and was further developed and improved by He [1-8]. It can be said that He’s homotopy perturbation method is a universal one, is able to solve various kinds of nonlinear functional equations. In this method the solution is considered as the summation of an infinite series which usually converges rapidly to the exact solutions. This method continuously deforms a simple problem, easy to solve, into the difficult problems under study. For the purpose of applications illustration of the methodology of the proposed method, using homotopy perturbation method, we consider the following nonlinear differential equation, A(u) − f (r) = 0,

r ∈ Ω,

(1.1)

B(u, ∂u/∂n) = 0,

r ∈ Γ,

(1.2)

where A is a general differential operator, f (r) is a known analytic function, B is a boundary condition and Γ is the boundary of the domain Ω. The operator Acan be generally divided into two operators, L and N , where L is a linear, while N is a nonlinear operator. Equation (1.1) can be, therefore, written as follows: L(u) + N (u) − f (r) = 0

Corresponding author: A. Neirameh

(1.3)

271

Using the homotopy technique, we construct a homotopy U (r, p) : Ω × [0, 1] → R which satisfies: H(U, p) = (1 − p)[L(U ) − L(u0 ))] + p[A(U ) − f (r)] = 0,

p ∈ [0, 1],

r ∈ Ω,

(1.4)

or H(U, p) = L(U ) − L(u0 ) + pL(u0 ) + p[N (U ) − f (r)] = 0

(1.5)

Wherep ∈ [0, 1], is called homotopy parameter, and u0 is an initial approximation for the solution of Eq.(1.1), which satisfies the boundary conditions. Obviously from Esq. (1.4) and (1.5) we will have. H(U, 0) = L(U ) − L(u0 ) = 0,

(1.6)

H(U, 1) = A(U ) − f (r) = 0,

(1.7)

we can assume that the solution of (1.4) or (1.5) can be expressed as a series inp, as follows: U = U0 + pU1 + p2 U2 + . . .

(1.8)

Setting p = 1, results in the approximate solution of Eq. (1.1) u = lim U = U0 + U1 + U2 + . . .

(1.9)

p→1

Example : fifth order KdV equation [10] ∂u ∂u ∂u ∂ 2 u ∂3u ∂5u + 45u2 − 15 − 15u + = 0. ∂t ∂x ∂x ∂x2 ∂x3 ∂x5

(1.10)

√ u(x, 0) = a0 − 2c2 sec h2 ( c2 x).

(1.11)

With initial condition,

With the exact solution √ u = a0 − 2c2 sec h2 ( c2 (x − (45a20 − 60a0 c2 + 16c22 )t)),

c2 > 0.

Where a0 , c2 are arbitrary constants. To solve Eq. (1.10) by homotopy perturbation method, we construct the following homotopy (1 − p)(

∂U ∂u0 ∂U ∂U ∂U ∂ 2 U ∂3U ∂5U − ) + p( + 45U 2 − 15 − 15U 3 + ) = 0, 2 ∂t ∂t ∂t ∂x ∂x ∂x ∂x ∂x5

or ∂U ∂u0 ∂u0 ∂U ∂U ∂ 2 U ∂3U ∂5U − + p( + 45U 2 − 15 − 15U + ) = 0, ∂t ∂t ∂t ∂x ∂x ∂x2 ∂x3 ∂x5 Suppose the solution of Eq. (1.12) has the following form

(1.12)

U = U0 + pU1 + p2 U2 + . . .

(1.13)

Substituting (1.13) into (1.12) and equating the coefficients of the terms with the identical powers of p leads to p0 : p1 : p2 : .. .

∂U0 ∂t ∂U1 ∂t ∂U2 ∂t

0 − ∂u = 0, ∂t 2 3 5 0 0 ∂ U0 + 45U02 ∂U − 15 ∂U − 15U0 ∂∂xU30 + ∂∂xU50 = 0, ∂x ∂x ∂x2 2 2 3 3 0 ∂ U1 0 1 1 ∂ U0 + 90U1 U0 ∂U + 45U02 ∂U − 15 ∂U − 15 ∂U − 15U1 ∂∂xU30 − 15U0 ∂∂xU31 + ∂x ∂x ∂x ∂x2 ∂x ∂x2

pj : .. .

∂Uj ∂t

+ 45

Pj−1 Pj−i−1 i=0

k=0

Ui Uk

∂Uj−k−i−1 ∂x

− 15

2 ∂Uk ∂ Uj−1−k k=0 ∂x ∂x2

Pj−1

− 15

Pj−1

k=0

Uk

∂ 5 U1 ∂x5

∂ 3 Uj−1−k ∂x3

+

= 0, ∂ 5 Uj ∂x5

= 0,

272

We take √ U0 = u0 = a0 − 2c2 sec h2 ( c2 x)

(1.14)

We have the following recurrent equations for j = 1, 2, 3 . . . . t

Z Uj = −

(45 0

j−1 j−i−1 X X i=0

k=0

j−1

Ui Uk

j−1

X ∂Uk ∂ 2 Uj−1−k X ∂Uj−k−i−1 ∂ 3 Uj−1−k ∂ 5 Uj − 15 − 15 Uk + ) dt = 0 2 3 ∂x ∂x ∂x ∂x ∂x5 k=0 k=0 (1.15)

With the aid of the initial approximation given by Eq. (1.14) and the iteration formula (1.15) we get the other of component as follows p p √ 3 √ 2 √ 2 √ 5 5 ( c2 x) − 480ta U1 = 720ta 0 c2 sec h ( c2 x) tanh( c2 x) p p0 c2 sec h√( c2 x) tanh √ √ √ 3 4 7 c72 sec h4 ( c2 x) tanh( c2 x) −2160t p c2 sec h√( c2 x) tanh√ ( c2 x) +1200t p √ √ 2 2 3 5 −180pc2 sec h ( c2 x) tanh( c2 x)ta0 +720p c2 sec h4 ( c2 x) tanh( c2 x)ta0 √ √ √ √ 5 c72 sec h6 ( c2 x) tanh( c2 x)t−1440 c72psec h2 ( c2 x) tanh ( c2 x)t −720 p √ √ √ √ +1920 c72 sec h2 ( c2 x) tanh3 ( c2 x)t −544 c72 sec h2 ( c2 x) tanh( c2 x)t, √ √ √ U2 = 353792t2 c62 sec h2 ( c2 x) + 162000t2 a30 c32 sec h2 ( c2 x) tanh4 ( c2 x) √ √ √ +238080t2 a0 c52 sec h2 ( c2 x) − 43480800t2 c62 sec h6 ( c2 x) tanh6 ( c2 x) √ √ √ √ −74606400t2 c62 sec h4 ( c2 x) tanh8 ( c2 x) − 130394880t2 c62 sec h2 ( c2 x) tanh6 ( c2 x) 6 √ 2 2 4 2 √ 2 6 4 √ +85680t a0 c2 sec h ( c2 x) + 156470400t c2 sec h ( c2 x) tanh ( c2 x) √ √ √ √ +5702400t2 c62 sec h8 ( c2 x) tanh2 ( c2 x) − 9504000t2 c62 sec h8 ( c2 x) tanh4 ( c2 x) √ √ √ √ 8 +119750400t2 c62 sec h2 ( c2 x) tanh ( c2 x) − 17540640t2 c62 sec h6 ( c2 x) tanh2 ( c2 x) √ √ √ −129600t2 c52 sec h8 ( c2 x)a0 − 10830336t2 c62 sec h2 ( c2 x) tanh2 ( c2 x) √ √ √ +61036800t2 c62 sec h2 ( c2 x) tanh4 ( c2 x) + 97200t2 c42 sec h6 ( c2 x)a20 √ √ −172800t2 c42 sec h4 ( c2 x)a20 − 544320t2 c52 sec h4 ( c2 x)a0 √ √ √ +21600t2 a30 c32 sec h2 ( c2 x) + 4050t2 a40 c22 sec h2 ( c2 x) + 432000t2 c52 sec h6 ( c2 x)a0 √ √ √ +23326080t2 c62 sec h4 ( c2 x) tanh2 ( c2 x) − 32400t2 c32 sec h4 ( c2 x)a30 √ √ √ √ −39916800t2 c62 sec h2 ( c2 x) tanh10 ( c2 x) − 712800t2 c62 sec h10 ( c2 x) tanh2 ( c2 x) √ √ √ √ +55663200t2 c62 sec h6 ( c2 x) tanh4 ( c2 x) − 104843520t2 c62 sec h4 ( c2 x) tanh4 ( c2 x) √ 2 √ 4 √ 2 5 2 5 2 √ −5068800t a0 c2 sec h( c2 tanh ( c2 x) + 19353600t a0 c2 sec h ( c2 x) tanh ( c2 x) √ √ √ √ −25401600t2 a0 c52 sec h( c2 tanh6 ( c2 x) − 1587600t2 c42 sec h2 ( c2 x) tanh6 ( c2 x)a20 √ √ √ √ 8 6 +10886400t2 a0 c52 sec h( c2 tanh ( c2 x) + 18273600t2 c52 sec h4 ( c2 x) tanh ( c2 x)a0 √ √ √ √ −26481600t2 a0 c52 sec h4 ( c2 x) tanh4 ( c2 x) + 9720000t2 a0 c52 sec h4 ( c2 x) tanh2 ( c2 x) √ √ √ √ −2268000t2 c42 sec h4 ( c2 x) tanh4 ( c2 x)a20 + 1836000t2 c42 sec h4 ( c2 x) tanh2 ( c2 x)a20 √ √ √ √ +8553600t2 c52 sec h6 ( c2 x) tanh4 ( c2 x)a0 − 5875200t2 c52 sec h6 ( c2 x) tanh2 ( c2 x)a0 √ √ √ √ −162000t2 a30 c32 sec h2 ( c2 x) tanh2 ( c2 x) − 12150t2 a40 c22 sec h2 ( c2 x) tanh2 ( c2 x) √ 2 √ 2 2 4 2 √ 2 √ 2 2 4 −1164240t a0 c2 sec h ( c2 x) tanh ( c2 x) + 2646000t a0 c2 sec h ( c2 x) tanh4 ( c2 x) 2 √ 2 √ 2 3 4 √ 3 2 4 6 √ 2 +162000t c2 sec h ( c2 x)a0 tanh ( c2 x) − 680400t c2 sec h ( c2 x)a0 tanh ( c2 x) √ √ √ +1166400t2 c52 sec h8 ( c2 x)a0 tanh2 ( c2 x) − 868800t2 c62 sec h4 ( c2 x) √ √ √ +64800t2 c62 sec h10 ( c2 x) + 796320t2 c62 sec h6 ( c2 x) − 345600t2 c62 sec h8 ( c2 x), .. . Approximate solution of (1.10) can be obtained by setting p = 1 u = lim U = U0 + U1 + U2 + . . . . p→1

Suppose u∗ =

P3

j=0

Uj , the results are presented in Table 1.

273 Table 1: The numerical results, whena0 = c2 = 0.01 for solutions of Eq. (1.10) for initial condition (1.11).

x 0.1 0.1 0.15 0.2 0.35 0.45 0.5 0.45 0.4 0.8 0.85 0.9 0.95 1

2

t 0.1 0.15 0.1 0.3 0.25 0.45 0.3 0.7 0.8 0.8 0.95 0.9 0.9 1

u∗ (x, t) -0.00999800053325245730 -0.00999800073320080290 -0.00999550127471400001 -0.00999200453139060139 -0.00997552348861796656 -0.00995956269009242584 -0.00995008919529557727 -0.00995956717739295321 -0.00996804687387015724 -0.00987256954075552987 -0.00985622515843887668 -0.00983890285245324707 -0.00982061427034014788 -0.00980136528619636983

|u∗ − u| 1.0 × 10−19 1.0 × 10−19 1 × 10−20 1 × 10−20 4 × 10−20 4 × 10−20 3 × 10−20 1 × 10−20 4 × 10−20 3 × 10−20 8 × 10−20 7 × 10−20 2 × 10−20 3 × 10−20

Conclusion

In this article, He’s homotopy perturbation method has been successfully applied to find the solution of nonlinear fifth order KdV equation are presented in table 1 , for differential results of x and t to show the stability of the method for this two problems. The approximate solutions obtained by the homotopy perturbation method are compared with exact solutions. Revealing that the obtained solutions are more accurate then variatioal method. This method is introduced to overcome the difficulty arising in calculating Adomain polynomial in Adomian method. In our work, we use the maple package to carry the computations.

References [1] J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation 151 (2004) 287-292. [2] J. H. He, Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons and Fractals 26 (2005) 695–700. [3] J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (2006) 87-88. [4] J. H. He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals 26 (1.3) (2005) 827-833. [5] J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) 257–262. [6] J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics 35 (1.1) (2000) 37–43. [7] J. H. He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation 156 (2004) 527-539. [8] J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73-79. [9] Engui Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos, Solitons and Fractals 16 (2003) 819-839.

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 274â€“282, 2011 http://ijamc.psit.in

Role of cloud droplets on the removal of gaseous pollutants from the atmosphere: A nonlinear model Shyam Sundar1 and Ram Naresh2 1

Department of Mathematics, P.S. Institute of Technology, Bhaunti, Kanpur-208020, India Email: ssmishra 75@yahoo.co.in

2

Department of Mathematics, H.B.Technological Institute Kanpur-208002, India Email: ramntripathi@yahoo.com

Abstract: In this paper, a four-dimensional nonlinear mathematical model is proposed to study the effect of density of cloud droplets on the removal of gaseous pollutants by rain in the atmosphere. The atmosphere, during rain, is assumed to consist of four nonlinearly interacting phases i.e. the phase of cloud droplets, the phase of raindrops, the phase of gaseous pollutants and the absorbed phase of gaseous pollutants in the raindrops. It is further assumed that these phases undergo ecological type growth and nonlinear interactions. The proposed model is analyzed by stability theory of differential equations and computer simulations. It is shown that the cumulative concentration of pollutants decreases due to rain and its equilibrium level depends upon the density of cloud droplets, the rate of formation of raindrops, emission rate of pollutants, the rate of falling absorbed phase on the ground, etc. It is noted here that if due to unfavorable conditions, cloud droplets are not formed, gaseous pollutants would not be removed due to non-occurrence of rain. Key words: Gaseous pollutants; precipitation; cloud droplets; stability; computer simulation.

1

Introduction

It is well known that the intensity of rainfall depends upon the density of droplets in clouds, more dense the cloud, more intense is the rain fall. During rain gaseous pollutants are removed by absorption/impaction by raindrops falling on the ground. Some investigations have been made in India regarding removal of air pollutants from the atmosphere and it has been found that the atmosphere becomes cleaner during and after rain [1 - 5]. In particular, Sharma et al. [5] measured the concentration of particulate matters in Kanpur city, India and found considerable decrease in their concentrations during monsoon season. Some experimental observations regarding rain washout have also been made for industrial cities elsewhere in the world [6 - 11]. For example, Flossmann [9] presented the interactions of clouds and pollution and found that clouds take up pollution mass that is removed when cloud precipitates in the form of rain. Several investigations have been conducted to study the phenomenon of pollutants removal by precipitation scavenging due to rain, snow or fog using mathematical models [12 - 17]. In particular,

Corresponding author: Shyam Sundar

275

Hales [12] has proposed a theory of gas scavenging by rain. Hales et al. [13] have also proposed a model for predicting the rain washout of gaseous pollutants from the atmosphere. Kumar [14] has given an Eulerian model to describe the simultaneous process of trace gas removal from the atmosphere and absorption of these gases in raindrops by considering the precipitation scavenging of these gases present below the clouds. It is pointed out here that the above mentioned models are linear but in real situations during precipitation, the densities of cloud droplets and raindrops change, which affect the interaction process. [18 - 21]. In this direction, Naresh et al. [20] proposed a nonlinear model for removal of pollutants from the atmosphere of a city. They have shown that, under appropriate conditions, gaseous pollutants and particulate matters, emitted with a constant rate, can be washed out from the atmosphere. This model, however, does not incorporate the role of cloud droplets in the modeling process. In view of above, in this paper, we propose a nonlinear model for the removal of gaseous pollutants from the atmosphere by precipitation incorporating the phase of cloud droplets. The proposed model is analyzed to see the effect of cloud density on the equilibrium level of gaseous pollutants in the atmosphere. Numerical simulation of the model is also carried out to support the analytical results.

2

Mathematical Model

To model the phenomena, let Cd (t) be the density of cloud droplets, Cr (t) be the density of raindrops in the atmosphere, C(t) be the cumulative concentration of gaseous pollutants in the atmosphere and Ca (t) be the cumulative concentration of gaseous pollutants in the absorbed phase. It is considered that Q is the cumulative emission rate of gaseous pollutants with natural depletion rate δC. It is also assumed that the absorption of gaseous pollutants by raindrops is proportional to the cumulative concentration of the pollutants and the density of raindrops (i.e. αCCr ). It is considered that the gaseous pollutants with concentration Ca in the absorbed phase may be removed due to falling of raindrops with rate kCa . It is also assumed that the removal of absorbed gaseous pollutants due to falling of rain drops on the ground is proportional to the density of rain drops as well as their concentration in absorbed phase (i.e. νCa Cr ). Thus, the dynamics of the removal processes is governed by the following system of nonlinear differential equations involving bilinear interactions of various phases, dCd dt dCr dt dC dt dCa dt

=

λ − λ0 Cd + π r Cr C

=

µ Cd − r0 Cr − r Cr C

=

Q − δ C − α CCr

=

α CCr − k Ca − ν Ca Cr

(2.1)

with Cd (0) ≥ 0, Cr (0) ≥ 0, C(0) ≥ 0, Ca (0) ≥ 0 In the first equation of model (2.1), λ is the rate of formation of water droplets in the cloud with the natural depletion rate coefficient λ0 . In the second equation of model (2.1), µ is the growth rate coefficient of Cr due to cloud droplets, r0 is the natural depletion rate coefficient of raindrops from the atmosphere. It is assumed that if the pollutant species are hot, raindrops may vaporize to enhance the growth of clouds. Thus, the depletion of raindrops is taken to be in direct proportion to the number density of raindrops as well as the concentration of gaseous pollutants (i.e. r Cr C) and a part of it (i.e. π r Cr C with 0 ≤ π ≤ 1) may re-enter into the atmosphere enhancing the growth of clouds. In the third and fourth equations of model (2.1), the constants δ and k are natural removal rate coefficients of C and Ca respectively, α and ν are removal rate

276

coefficients of C and Ca respectively due to interactions with Cr . All the constants considered here are taken to be non-negative. It is remarked here that if Îť0 is very large for a given concentration C due to atmospheric d conditions then dC dt may become negative. In such a case no cloud droplets formation takes place, then precipitation does not occur and pollutants would not be removed. In the following, we analyze the nonlinear model (2.1) by using the stability theory of differential equations under the situations Q(t) = 0 (instantaneous emission of pollutants) and Q(t) = Q (continuous emission of pollutants with a constant rate). We state the region of attraction of the model (2.1) as follows, to describe the bounds of dependent variables. The set Q Îť (2.2) , 0 â‰¤ C + Ca â‰¤ â„Ś = (Cd , Cr , C, Ca ) : 0 â‰¤ Cd + Cr â‰¤ Îťm Î´m attracts all solutions initiating in the interior of the positive octant, where Îťm = min {Îť0 âˆ’ Âľ, r0 } and Î´m = min {Î´, k}.

3 3.1

Stability Analysis Case 1: When Q(t) = 0 (Instantaneous emission)

The concentration of pollutants in the atmosphere is C (0) >0. In this case the model has only one non-negative equilibrium E0 ÎťÎť0 , rÂľ0 ÎťÎť0 , 0, 0 in Cd âˆ’ Cr âˆ’ C âˆ’ Ca space. After computing variational matrix of the model (2.1) corresponding toE0 , we have found that all the eigen values are negative proving that E0 is locally asymptotically stable. Theorem 3.1. IfCd (0) > 0, then E0 is globally asymptotically stable in the positive octant. Proof. From the first two equations of model (2.1), we have dCd dCr + â‰¤ Îť âˆ’ (Îť0 âˆ’ Âľ)Cd âˆ’ r0 Cr dt dt From this we get lim sup {Cd (t) + Cr (t)} â‰¤ tâ†’âˆž

Îť Îťm ,

(3.1)

where Îťm = min {Îť0 âˆ’ Âľ, r0 }

Again from the third and fourth equations of model (2.1), we get dC dCa + = âˆ’Î´ C âˆ’ k Ca âˆ’ Î˝ Cr Ca dt dt â‰¤ âˆ’Î´ C âˆ’ k Ca â‰¤ âˆ’Î´m (C + Ca )

(3.2)

lim sup C(t) = lim sup Ca (t) = 0

(3.3)

where, Î´m = min{Î´, k} Thus, we get tâ†’âˆž

tâ†’âˆž

Hence the proof. This theorem implies that in the case of instantaneous emission, gaseous pollutants are removed completely from the atmosphere by rain and the time taken for removal will depend upon the rates of cloud droplets and raindrops formation.

277

3.2

Case 2: Q(t) = Q (Constant emission)

In this case also, the model (2.1) has only one equilibrium namely E ∗ (Cd∗ , Cr∗ , C ∗ , Ca∗ ), where Cd∗ , Cr∗ , C ∗ and Ca∗ are positive solutions of the following equations, Cd =

λ + π r Cr f (Cr ) λ0

(3.4)

µ Cd − r0 Cr − r Cr C = 0 Q C= = f (Cr ) δ + α Cr α Cr f (Cr ) Ca = k + ν Cr ∗ To show the existence of E , we write eq.(3.5) as F (Cr ) =

(3.5) (3.6) (3.7)

µ {λ + π r Cr f (Cr )} − r0 Cr − r Cr f (Cr ) λ0

(3.8)

From eq.(3.8) we note that, F (0) =

µλ >0 λ0

(3.9)

Again, using the maximum value of Cr , we get λ r0 µ λm rλ µπ λ λ =− 1− + 1− f F λm λm r0 λ0 λm λ0 λm Since µ < λ0 , λm < r0 and 0 ≤ π ≤ 1, therefore λ F <0 λm

(3.10)

Hence there exist a root Cr∗ in 0 < Cr < λλm . For this root to be unique we have F 0 (Cr ) < 0. Here, µπ µπ F 0 (Cr ) = − r0 + r Cr f 0 (Cr ) 1 − + r f (Cr ) 1 − <0 (3.11) λ0 λ0 Thus, F (Cr ) = 0 has a unique positive root (sayCr∗ ) without any condition. Using Cr∗ we can calculate Cd∗ , C ∗ and Ca∗ from eqs.(3.4), (3.6) and (3.7) respectively. From eqs.(3.6) and (3.7), we note that Cand Ca both decrease with increase in the density of raindrops and they may even tend to zero for its larger values. We also note here that the removal of gaseous pollutants would depend upon different removal parameters. In the following, we check the characteristics of various phases with respect to relevant parameters analytically. 3.2.1

Variation of Cr with µ

From eq. (3.5), we have µ {λ + π r Cr f (Cr )} − r0 Cr − r Cr f (Cr ) = 0 λ0 Differentiating with respect to ‘µ’ we get, dCr =h dµ r0 1 − This implies,

dCr dµ

π rµ λ0 r0

λ λ0

+

πr λ0 Cr f (Cr )

+ r Cr f 0 (Cr ) 1 −

> 0. Thus , Cr increases as µ increases.

πµ λ0

i + r f (Cr )

278

3.2.2

Variation of C with µ

From eq. (3.6), we note that

dC dCr

< 0. Now, dC dC dCr = <0 dµ dCr dµ

r since dC dµ > 0 dCa dC d Therefore C decreases as µ increases. Similarly, we can also show that dC dλ1 < 0, dα < 0, dα > dCa dC 0, dλ < 0, dλ < 0, etc. Thus, it may be concluded that the removal rate of gaseous pollutants increases as the density of cloud droplets increases. We also note that, d 1. If the coefficient λ0 is so large that dC dt becomes negative, the cloud droplets formation will not take place and rain will not occur.

2. If the coefficient α is so large that the atmosphere. 3. For large k and ν, not exist.

dCa dt

dC dt

< 0, all the gaseous pollutants will be removed from

< 0 and the formation of absorbed phase is very transient and it may

To see the stability behavior of E∗ , we state the following theorems. Theorem 3.2. Let the following inequalities π r µ C∗ <

λ0 (r0 + r C ∗ ) 6

1 (r0 + r C ∗ )(δ + α Cr∗ ) 9 2 π αµ Cr∗ < λ0 (δ + α Cr∗ ) 3 hold, then E∗ is locally stable. (See appendix A for proof ) rα Cr∗ C ∗ <

(3.12) (3.13) (3.14)

Theorem 3.3. Let the following inequalities π r µ C∗ <

λ0 r0 6

(3.15)

r0 δ 9

(3.16)

rα Cr∗ C ∗ <

2 λ0 δ (3.17) 3 hold inΩ, then E ∗ is globally asymptotically stable with respect to all solutions initiating in the interior of the positive octant. (See appendix B for proof ) π αµ Cr∗ <

The above theorems imply that under certain conditions, the gaseous pollutants from the atmosphere would be removed and the removal rate increases as the density of cloud droplets increases. Remarks: 1. If π = 0, r = 0, then the inequalities (3.12) – (3.17) are satisfied automatically. This implies that these parameters have destabilizing effect on the system dynamics. It shows that, in absence of these parameters, the gaseous pollutants would be washed out completely from the atmosphere by precipitation. 2. If µ = 0, α = 0, then the inequalities (3.12) – (3.17) are again satisfied automatically. It shows that the gaseous pollutants would be removed completely from the atmosphere due to gravity.

279

4

Figure 1: Global stability in Cd − Cr plane

Figure 2: Variation of Cr with time 0 t0 for different values of λ

Figure 3: Variation of Cwith time 0 t0 for different values of λ

Figure 4: Variation of Ca with time 0 t0 for different values of λ

Numerical Simulation

To see the effect of various removal parameters on the system dynamics, we conduct numerical simulation of the model system by considering the following set of parameter values, λ = 5, λ0 = 0.4, π = 0.001, µ = 0.3, r0 = 0.06, r = 0.08, Q = 20, δ = 0.15, α = 0.65, k = 0.30, ν = 0.55 The equilibrium E ∗ is calculated as, Cd∗ = 12.506089, Cr∗ = 21.931986, C ∗ = 1.388330, Ca∗ = 1.600938 Eigen values corresponding to E ∗ are obtained as, −12.362592, −0.060690, −14.516168, −0.399998 Since all the eigen values corresponding to E ∗ are negative, therefore E ∗ is locally asymptotically stable. The global stability behavior of E ∗ in Cd − Cr plane is shown in the fig.1. In figs.2-4, the variation of density of raindrops, cumulative concentrations of gaseous pollutants Cand its absorbed phase Ca with time 0 t0 is shown for different values of growth rate of cloud droplets λ (i.e. at

280

Figure 5: Variation of Cwith time 0 t0 for different values of µ

Figure 6: Variation of Ca with time 0 t0 for different values of µ

λ = 4, 5, 6) respectively. From these figures, it is seen that the density of raindrops increases while the cumulative concentrations of gaseous pollutants C and that in absorbed phase Ca decrease as growth rate of cloud droplets increases. Thus, if the density of cloud droplets is higher, the removal of gaseous pollutants as well as the pollutants in absorbed phase is quite significant due to enhanced rainfall. In figs.5-6, the variation of cumulative concentrations of gaseous pollutants and the pollutants in absorbed phase with time 0 t0 is shown for different values of growth rate of raindrops µ (i.e. atµ = 0.1, 0.2, 0.3) at λ = 5 respectively. It is seen that the cumulative concentrations of gaseous pollutants and that in absorbed phase decrease with time as the growth rate of raindrops increases.

5

Conclusion

A nonlinear mathematical model is proposed to see the effect of density of cloud droplets on the removal of gaseous pollutants from the atmosphere by precipitation. The model is analyzed using stability theory of differential equations and numerical simulations. The model analysis shows that the density of raindrops increases while the cumulative concentrations of gaseous pollutants and pollutants in absorbed phase decrease as growth rate of cloud droplets increases. It is also shown that the magnitude of gaseous pollutants removed by rainfall depends upon the intensity of rain caused by cloud formation. It is noted that the gaseous pollutants are removed from the atmosphere, but the remaining equilibrium amount would depend upon the rate of emission of gaseous pollutants, growth rate of cloud droplets and raindrops, the rate of falling raindrops on the ground and other removal parameters. Acknowledgements: The financial support for this research from University Grants Commission, New Delhi, India through project F. No. 39-33/2010 (SR) is gratefully acknowledged.

References [1] J. Pandey, M. Agrawal, N. Khanam, D. Narayanan and D.N. Rao, Air pollution concentrations in Varanasi, India, Atmos. Environ. 26 B (1992) 91-98. [2] A. G. Pillai, M.S. Naik, G. Momin, P. Rao, K. Ali, H. Rodhe and L. Granat, Studies of wet deposition and dustfall at Pune, India, Water, Air and Soil Pollution 130 (1-4) (2001) 475-480. [3] P.S. Prakash Rao, L.T. Khemani, G.A. Momin, P.D. Safai and A.G. Pillai, Measurements of wet and dry deposition at an urban location in India, Atmos. Environ. 26B (1992) 73-78. [4] K. Ravindra, S. Mor, J. S. Kamyotra and C. P. Kaushik, Variation in spatial pattern of criteria air pollutants before and during initial rain of monsoon, Environmental Monitoring and Assessment 87 (2.2) (2003) 145 – 153.

281 [5] V.P. Sharma, H.C. Arora and R.K. Gupta, Atmospheric pollution studies at Kanpur- suspended particulate matter, Atmos. Environ. 17 (1983) 1307-1314. [6] J.D. Blando and B.J. Turpin, Secondary organic aerosol formation in cloud and fog droplets: a literature evaluation of plausibility, Atmos. Environ. 34 (2000) 1623-1632. [7] T. D. Davies, Precipitation scavenging of SO2 in an industrial area, Atmos. Environ., 10 (1976) 879 – 890. [8] T. D. Davies, Sulphur dioxide precipitation scavenging, Atmos. Environ. 17 (1983) 797-805. [9] A. I. Flossmann, Clouds and Pollution, Pure and App. Chem. 70 (3.5) (1998) 1345 – 1352. [10] G. A. Loosmore and R. T. Cederwall, Precipitation scavenging of atmospheric aerosols for emergency response applications: testing an updated model with new real time data, Atmos. Environ. 38(3.5) ( 2004) 93-1003. [11] K.F. Moore, D.E. Sherman, J.E Reilly and J.L. Collett, Drop size dependent chemical composition in cloud and fog, part 1, observations, Atmos. Environ. 38(10) (2004) 1389-1402. [12] J.M. Hales, Fundamentals of the theory of gas scavenging by rain, Atmos. Environ. 6 (1972) 635-650. [13] . J.M. Hales, M.A Wolf and M.T. Dana, A linear model for predicting the washout of pollutant gases from industrial plume, AICHE Journal 19 (1973) 292-297. [14] S. Kumar, An Eulerian model for scavenging of pollutants by rain drops, Atmos. Environ. 19 (1985) 769-778. [15] R. Naresh, An analytical approach to study the problem of air pollutants removal in a two patch environment, Ultra Science, (Int. J. Physical Sc.) 16(2.1) M, (2004) 83-96. [16] J.B. Shukla, R. Nallaswany, S. Verma and J.H. Seinfeld, Reversible absorption of a pollutant from an area source in a stagnant fog layer, Atmos. Environ. 16 (1982) 1035-1037. [17] W.G.N. Slinn, Some approximations for the wet and dry removal of particles and gases from the atmosphere, Water, Air and Soil Pollution, 7 (1977) 513-543. [18] R. Naresh, Qualitative analysis of a nonlinear model for removal of air pollutants, Int. J. Nonlinear Sciences and Numerical Simulation 4 (2003) 379-385. [19] R. Naresh and S. Sundar, Mathematical modelling and analysis of the removal of gaseous pollutants by precipitation using general nonlinear interaction, IJAMC. 2(2.2) (2010) 45 – 56. [20] R. Naresh, S. Sundar and J. B. Shukla, Modelingthe removal of gaseous pollutants and particulate matters from the atmosphere of a city, Nonlinear Analysis: RWA 8(2.1) (2007) 337-344. [21] J.B.Shukla, A.K. Misra, R. Naresh and P.Chandra, How artificial rain can be produced ? A mathematical model, Nonlinear Analysis: RWA 11 (2010) 2659-2668.

Appendix A Proof of Theorem 2 Using the following positive definite function in the linearized system of (2.1), 2 + k C2 + k C2 + k C2 ) (A1) V = 21 (k0 Cd1 1 r1 2 1 3 a1 where Cd1 , Cr 1 , C1 , Ca1 are small perturbations from E∗ , as follows ∗ ∗ ∗ Cd = Cd + Cd1 , Cr = Cr + Cr1 ,C = C + C1 , Ca = Ca∗ + Ca1 Differentiating (A1) with respect to0 t0 we get, in the linearized system corresponding to E ∗ 2 − k (r + r C ∗ )C 2 − k (δ + α C ∗ )C 2 − k (k + ν C ∗ )C 2 V˙ = −k1 λ0 Cd1 2 0 3 4 r r r1 1 a1 +(k1 π r C ∗ + k2 µ) Cd1 Cr1 + k1 π r Cr∗ Cd1 C1 − (k2 r Cr∗ + k3 α C ∗ ) Cr1 C1 +k4 (α C ∗ − ν Ca∗ )Cr1 Ca1 + k4 α Cr∗ C1 Ca1 ˙ Now V will be negative definite under the following conditions, (A2) (k1 π r C ∗ + k2 µ)2 < 32 k1 k2 λ0 (r0 + r C ∗ ) (A3) k1 (π r Cr∗ )2 < 23 k3 λ0 (δ + α Cr∗ ) (A4) (k2 r Cr∗ + k3 α C ∗ )2 < 94 k2 k3 (r0 + r C ∗ )(δ + α Cr∗ ) (A5) k4 (α C ∗ − ν Ca∗ )2 < 32 k2 (r0 + r C ∗ )(k + ν Cr∗ ) (A6) k4 (α Cr∗ )2 < 32 k3 (δ + α Cr∗ )(k + ν Cr∗ ) Assuming k1 = 1, and we write the equation (A2) as (π r C ∗ − k2 µ)2 + 4k2 π r µ C ∗ < π r C∗ , above equation µ π r µ C ∗ < λ60 (r0 + r C ∗ )

Choosing k2 = (A7)

2 k2 λ0 (r0 + r C ∗ ) 3

reduces to

Similarly, from equation (A4), choosing k3 =

π r 2 Cr∗ αµ

it reduces to

282 (A8) rα Cr∗ C ∗ < 19 (r0 + r C ∗ )(δ + α Cr∗ ) From equation (A3), usingk3 , we get, (A9) π αµ Cr∗ < 23 λ0 (δ + α Cr∗ ) From equations (A5) and (A6), usingk n 2 and k3 , we get

o (δ+α Cr∗ ) r (r +r C ∗ )C ∗ + ν Cr∗ ) πµr min (α0C ∗ −ν C ∗ )2 , 3 C∗ α a r Under conditions (A7) – (A9), V˙ will be negative definite showing that V is a Liapunov’s function and hence the theorem. (A10) k4 <

2 (k 3

Appendix B Proof of Theorem 3 . Using the following positive definite function, (B1) U = 21 [m1 (Cd − Cd∗ )2 + m2 (Cr − Cr∗ )2 + m3 (C − C ∗ )2 + m4 (Ca − Ca∗ )2 ] Differentiating with respect to‘t’ we get, U˙ = −m1 λ0 (Cd − Cd∗ )2 − m2 (r0 + rC)(Cr − Cr∗ )2 − m3 (δ + αCr )(C − C ∗ )2 − m4 (k + ν Cr )(Ca − Ca∗ )2 +(m1 π r C ∗ + m2 µ)(Cd − Cd∗ )(Cr − Cr∗ ) + m1 π r Cr (Cd − Cd∗ )(C − C ∗ ) − (m2 r Cr∗ + m3 α C ∗ )(Cr − Cr∗ )(C − C ∗ ) +m4 (α C ∗ − ν Ca∗ )(Cr − Cr∗ )(Ca − Ca∗ ) + (m4 α Cr )(C − C ∗ )(Ca − Ca∗ ) Now U˙ will be negative definite under the following conditions, (B2) (m1 π r C ∗ + m2 µ)2 < 32 m1 m2 λ0 (r0 + r C) (B3) m1 (π r Cr )2 < 23 m3 λ0 (δ + α Cr ) (B4) (m2 r Cr∗ + m3 α C ∗ )2 < 94 m2 m3 (r0 + r C)(δ + α Cr ) (B5) m4 (α C ∗ − ν Ca∗ )2 < 32 m2 (r0 + r C)(k + ν Cr ) (B6) m4 (α Cr )2 < 23 m3 (δ + α Cr )(k + ν Cr ) Maximizing LHS and minimizing RHS, choosing m1 = 1and ( ) 2 πrk δ r Cr∗ λm 2 r0 C ∗ min m4 < , 3 µ (α C ∗ − ν Ca∗ )2 α3 λ Equations (B2) – (B6) reduce to (B7) π r µ C ∗ < λ06r0 (B8) rα Cr∗ C ∗ < r09δ (B9) π αµ Cr∗ < 23 λ0 δ Under conditions (B7) – (B9), U˙ will be negative definite showing that U is a Liapunov’s function and hence the theorem.

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 283â€“289, 2011 http://ijamc.psit.in

Effect of ion and electron streaming on the formation of ion-acoustic solitons in weakly relativistic magnetized ion-beam plasmas S. N. Barman1 and A. Talukdar2 1,2

Department of Mathematics, Arya Vidyapeeth College, Guwahati, Assam, India, 781016. E-mail: snbarman.ghy@gmail.com

Abstract: Existence of both compressive and rarefactive solitons is established in weakly relativistic but magnetized plasma with cold ions and ion-beams in presence of electron inertia. It is observed that rarefactive (compressive) solitons exist for smaller (higher) difference of electron and ion initial streaming speeds for different ion to ion-beam mass ratio and wave speed. Interestingly, the amplitude of the rarefactive solitons is seen to increase with the increase of the difference of initial streaming speeds to certain limit before turning out to be compressive after that limit. Key words: KdV equation; soliton; relativistic plasma; nonlinear phenomena

1

Introduction

Nonlinear waves may play important roles in rarefied space plasmas in the Earthâ€™s auroral zone [1,2,3], the physics of solar atmosphere [4], and other astrophysical plasmas[5,6]. Plasmas are an intrinsically nonlinear medium that can support a great variety of diverse waves. It has been well established both theoretically and experimentally that the behavior of ion-acoustic solitary waves in collisionless plasmas can be described by the Korteweg-de Vries (KdV) equation in the small amplitude and long-wavelength region. During the last three decades or so, many workers have investigated the existence of solitary waves both theoretically [7,8,9,10,11] and experimentally [12,13,14,15,16] in different physical situations of plasma compound. In recent space observations, it has been investigated that the high-speed streaming ions as well as the electrons play a major role in the physical mechanism of the nonlinear wave structures. When we assume that the ion and electron energies depend only on the kinetic energy, velocities of plasma particles in the solar atmosphere and the magnetosphere have to attain relativistic speeds [17,18]. Thus, by considering such relativistic effects on ion and electron velocities, one can take the relativistic motion of such particles into consideration in the study of nonlinear plasma waves. When the velocity of the plasma particles approaches near to that of light, the nonlinear waves which occur in the space, exhibit a peculiar feature due to the effect of the high speed ions [19,20]. Starting from the works of Das and Paul [21], many workers like Nejoh [20], Das et al.[21], Pakira et al.[23], Kalita et al.[24], El-Labany and Shaaban [25], Singh et al.[26], Gill et al.[,27], Lee and Choi [28] have considered relativistic effects and investigated the existence of ion-acoustic solitary waves under various physical situations in plasmas.

Corresponding author: S. N. Barman

284

In this paper, the investigation of ion-acoustic solitary waves in weakly relativistic magnetized plasmas with cold ions and ion-beams together with isothermal electrons is carried out. The external magnetic field B0 zˆ is taken along the z-direction, while the propagation of the wave is assumed to take place in a ξ-direction in the (x, z) plane, inclined an angle θ to the direction of the magnetic field. We assume that the ion-beam has a constant drift velocity Ud along the x-direction perpendicular to the direction of the magnetic field. The layout of this paper is as follows: In section 2, we present the basic equations of motions for a relativistic three components plasma namely cold ions, ion-beams and electrons together with the Poisson equation. Using perturbation method, KdV equation is derived from this basic set of equations. In section 3, condition for the existence of solitons and solitary wave solutions are discussed. The last section 4 is devoted to the concluding discussion.

2

Equations of motion and derivation of KdV equation

We consider a model of weakly relativistic magnetized plasma consisting of cold ions and ion-beams together with the isothermal electrons. The equations of motion governing the state of this plasma in a ξ- direction (inclined at an angle θ to the direction of the magnetic field) are as follows: For the cold ions, ∂ ∂ns + (ns vsξ ) = 0, (2.1) ∂t ∂ξ ∂ ∂ e ∂φ + vsξ − B0 vsy , βs vsx = − sin θ (2.2) ∂t ∂ξ mi0 ∂ξ e ∂ ∂ + vsξ βs vsy = − B0 vsx , (2.3) ∂t ∂ξ mi0 ∂ e ∂φ ∂ + vsξ βs vsz = − cos θ , (2.4) ∂t ∂ξ mi0 ∂ξ For the ion-beams, ∂nb ∂ ∂nb + Ud sin θ + (nb vbξ ) = 0, ∂t ∂ξ ∂ξ ∂ ∂ e ∂φ + (vbξ + Ud sin θ) βb vbx = − − B0 vby , sin θ ∂t ∂ξ mb0 ∂ξ ∂ e ∂ + (vbξ + Ud sin θ) vbx B0 , βb vby = − ∂t ∂ξ mb0 ∂ ∂ e ∂φ + (vbξ + Ud sin θ) βb vbz = − cos θ , ∂t ∂ξ mb0 ∂ξ

(2.5) (2.6) (2.7) (2.8)

For the electrons, ∂ne ∂ + (ne veξ ) = 0, ∂t ∂ξ ∂ 1 ∂φ Te ∂ne ∂ + veξ βe vez = e cos θ − cos θ , ∂t ∂ξ me0 ∂ξ ne ∂ξ

(2.9) (2.10)

With the Poisson equation, ∂2φ = 4π e(ne − ns − nb ), ∂ξ 2 where, βr =

2 vrξ 1− 2 c

!−1/2 ≈1+

2 vrξ , r = s, b, e. 2c2

(2.11)

285

We have normalized the physical quantities appearing in the set of equations (2.1) – (2.11) as: densities by the equilibrium plasma density n0 , velocities by cs συ , potential φ by Te /e, distances 1/2 −1 by the Debye length λD = Te /4π n0 e2 and time by the ion plasma period ωpi = cs συ /λD = −1/2 2 4π n0 e /mi0 . We denote the ratio cold-ion rest mass (mi0 ) to ion-beam rest mass (mb0 ) by α = mi0 /mb0 , electron rest mass (me0 ) to ion rest mass by Q = me0 /mi0 , the ion-acoustic 1/2 speed by cs = (Te /mi0 ) , the cyclotron frequency by ωBi = eB0 /mi0 , the ratio of plasma frequency to cyclotron frequency by µ = ωpi /ωBi , and the direction of the wave propagation by συ (= cos θ). For fast ion-acoustic waves the ratio of plasma frequency to wave frequency is largeµ >> 1 i.e. ωpi >> ωBi ( ωpi >> ωBi suggests weak magnetic resistance to the plasma waves and so it is termed as “fast ion-acoustic” ignoring presence of strong magnetic field). In this situation, we choose µ = bε−3/2 , where b is greater than zero and depends upon the distribution of cold ions and ion-beams. To derive the KdV equation, we introduce the stretched coordinates as η = ξ − M t,

τ =εt

(2.12)

where M is the phase velocity of the ion-acoustic wave in (ξ, t)space, and ε is a small dimensionless expansion parameter. We expand the flow variables asymptotically about the equilibrium state in terms of this parameter as follows: ns = (1 − Nb ) + ε ns1 + ε2 ns2 + ....., nb = Nb + ε nb1 + ε2 nb2 + ....., 2 ne = 1 + ε ne1 + ε ne2 + ....., 2 φ = εφ1 + ε φ2 + ....., 2 vez = vez0 + ε vez1 + ε vez2 + ....., (2.13) 2 veξ = veξ0 + ε veξ1 + ε veξ2 + ....., vjξ = vjξ0 + ε vjξ1 + ε2 vjξ2 + ....., 2 vjx = vjx0 + ε vjx1 + ε vjx2 + ....., 2 vjy = ε vjy1 + ε vjy2 + ....., 2 vjz = vjz0 + ε vjz1 + ε vjz2 + ....., j = s, b. With the use of the transformation (2.12) and the expansions (2.13) in the normalized set of equations (2.1) – (2.11) subject to the boundary conditions vjx1 = 0, vjy1 = 0, vjz1 = 0, vjξ1 = 0, (j = s, b), vez1 = 0, and φ1 = 0 as |η| → ∞

(2.14)

We get, from the ε-order equations, the following quantities: vsξ1 =

φ1 αφ1 2 , vbξ1 = β (M −U sin θ−v 2 , veξ1 βs1 (M −vsξ0 )συ b1 d bξ0 )συ (1−Nb )φ1 α Nb φ1 2 , nb1 = β (M −U sin θ−v 2 2 , ne1 βs1 (M −vsξ0 )2 συ b1 d bξ0 ) συ

ns1 = ne1 − ns1 − nb1 = 0.

= =

(M −veξ0 )φ1 1−Qβe1 (M −veξ0 )2 , φ1 1−Qβe1 (M −veξ0 )2 ,

3v 2

where, βr1 = 1 + 2crξ0 2 , r = s, b, e. Using the values of ne1 , ns1 , nb1 in the last equation of above, we obtain the expression for the phase velocity M as 1 1 − Nb α Nb − − = 0. 2 2 2 1 − Qβe1 (M − veξ0 ) βs1 (M − vsξ0 ) συ βb1 (M − Ud sin θ − vbξ0 )2 συ2

(2.15)

Again from ε2 - order equations obtained from the equations (2.1) – (2.11), with the use of the first order quantities, we can find the following equations: 2(1 − Nb ) (1 − Nb ) ∂φ1 ∂ns2 ∂φ2 − + βs1 συ2 (M − vsξ0 )3 ∂τ ∂η βs1 (M − vsξ0 )2 συ2 ∂η +

2(1 − Nb )c2 βs1 (M − vsξ0 ) + (1 − Nb )[βs1 c2 − 3vsξ0 (M − vsξ0 )] ∂φ1 φ1 = 0, 3 c2 βs1 συ4 (M − vsξ0 )4 ∂η

(2.16)

286

∂φ1 ∂nb2 α Nb ∂φ2 2α Nb − + βb1 συ2 (M − Ud sin θ − vbξ0 )3 ∂τ ∂η βb1 (M − Ud sin θ − vbξ0 )2 συ2 ∂η +

2α2 Nb c2 βb1 (M − Ud sin θ − vbξ0 ) + α2 Nb [βb1 c2 − 3vbξ0 (M − Ud sin θ − vbξ0 )] ∂φ1 = 0, φ1 3 4 c2 βb1 συ (M − Ud sin θ − vbξ0 )4 ∂η

(2.17)

2Qβe1 (M − veξ0 ) ∂φ1 ∂ne2 1 ∂φ2 + − [1 − Qβe1 (M − veξ0 )2 ]2 ∂η ∂η 1 − Qβe1 (M − veξ0 )2 ∂η +

3Qβe1 c2 (M − veξ0 )2 − 3Q veξ0 (M − veξ0 )3 − c2 = 0, [1 − Qβe1 (M − veξ0 )2 ]3

∂ 2 φ1 = ne2 − ns2 − nb2 ∂η 2

(2.18)

(2.19)

Combining the equations (2.16) – (2.19) and using (2.15), we obtain the KdV equation ∂φ1 ∂φ1 ∂ 3 φ1 + Aφ1 +B = 0, ∂τ ∂η ∂η 3

(2.20)

where A = A1 /A2 ,

B = B1 /B2 ,

3 A1 = βb1 (M − Ud sin θ − vbξ0 )4 [1 − Qβe1 (M − veξ0 )2 ]3 {2(1 − Nb )c2 βs1 (M − vsξ0 ) 3 + (1 − Nb )[βs1 c2 − 3vsξ0 (M − vsξ0 )]} + βs1 (M − vsξ0 )4 [1 − Qβe1 (M − veξ0 )2 ]3 {2Nb × α2 c2 βb1 (M − Ud sin θ − vbξ0 ) + α2 Nb [βb1 c2 − 3vbξ0 (M − Ud sin θ − vbξ0 )]} 3 3 4 + c2 βs1 βb1 συ (M − vsξ0 )4 (M − Ud sin θ − vbξ0 )4 [3Qβe1 c2 (M − veξ0 )2 − 3Q veξ0 (M − veξ0 )3 − c2 ], 2 2 2 A2 = 2c2 βs1 βb1 συ (M − vsξ0 )(M − Ud sin θ − vbξ0 )[1 − Qβe1 (M − veξ0 )2 ]{(1 − Nb )βb1 × (M − Ud sin θ − vbξ0 )3 [1 − Qβe1 (M − veξ0 )2 ]2 + α Nb βs1 (M − vsξ0 )3 × [1 − Qβe1 (M − veξ0 )2 ]2 + 2Qβe1 βs1 βb1 συ2 (M − veξ0 )(M − vsξ0 )3 (M − Ud sin θ − vbξ0 )3 }, B1 = βs1 βb1 συ2 (M − vsξ0 )3 (M − Ud sin θ − vbξ0 )3 [1 − Qβe1 (M − veξ0 )2 ]2 , B2 = 2{(1 − Nb )βb1 (M − Ud sin θ − vbξ0 )3 [1 − Qβe1 (M − veξ0 )2 ]2 + α Nb βs1 (M − vsξ0 )3 × [1 − Qβe1 (M − veξ0 )2 ]2 + 2Qβe1 βs1 βb1 συ2 (M − veξ0 )(M − vsξ0 )3 (M − Ud sin θ − vbξ0 )3 }.

3

Condition for existence of solitons and solitary wave solution

The expression given in equation (2.15) shows that, irrespective of the magnitude of the streaming velocities vjξ0 (j = s, b, e), the phase velocity M has the relativistic impact through βs1 , βb1 and βe1 on the plasma particles and thus affects the same on the dynamics of the soliton propagation. The soliton solution of the KdV equation (2.20) is possible if A and B(> 0) are non-zero and finite for which we have that Qβe1 (M − veξ0 )2 6= 1, Ud sin θ + vbξ0 6= M. (3.1) To find solitary wave solution of the KdV equation (2.20), we introduce the variableχ = η − V τ , where V is the soliton speed in the linear χ- space. Using the boundary conditions φ1 =

∂ 2 φ1 ∂φ1 = = 0 as |χ| → ∞, ∂χ ∂χ2

(3.2)

equation (2.20) can be integrated to give φ1 =

3V sec h2 A

V 4B

1/2 χ.

(3.3)

287

Figure 1: Amplitudes of both compressive and rarefactive ion-acoustic solitons versus direction of wave propagation συ for |veξ0 − vsξ0 | = 12.5(1), 13.0(2), 13.5(3), 15.5(4), 39.5(5) when V = 0.1, α = 0.1, Nb = 0.01, vsξ0 = 30.

Figure 2: Amplitudes of both compressive and rarefactive ion-acoustic solitons versus wave velocity V for |veξ0 − vsξ0 | = 20.5(1), 22.5(2), 23.5(3), 26.5(4), 29.5(5), 39.5(6) when συ = 0.8, α = 0.1, Nb = 0.01, vsξ0 = 30.

Figure 3: Soliton amplitudes versus initial streaming difference of ion and electron for fixedV = 0.1, α = 0.1, Nb = 0.01, vsξ0 = 30and for different values ofσυ = 0.9(1), 0.8(2), 0.7(3), 0.6(4).

Figure 4: Soliton amplitudes versus ion to beamion mass ratio α for fixed V = 0.05, Nb = 0.1, |veξ0 − vsξ0 | = 29.75, vsξ0 = 30and for higher values of συ =0.9(1), 0.95(2), 0.97(3).

The amplitude and the width of the solitary waves are given respectively by 3V ,∆ = φ0 = A

4

4B V

1/2 .

Discussion

In this model of plasma, both compressive and rarefactive solitons are established to exist which are predicted to propagate depending on a definite range of difference of electron and ion streaming speeds in various ξ-directions making an angle θ to the direction of the magnetic field B0 zˆ. Computational work reveals the existence range of solitons as 9 < |veξ0 − vsξ0 | < 42.5. It is observed that rarefactive (compressive) solitons exist for the smaller (higher) difference of electron and ion (or ion-beam) initial streaming speeds for different values of α, V, Nb . For small α = 0.1 that is for heavy concentration of ion-beam mass in the plasma and for small ion-beam density Nb = 0.01 and wave velocityV = 0.1, the amplitude of the rarefactive solitons are found to increase with

288

Figure 5: Soliton widths versus wave velocity for fixedα = 0.1, Nb = 0.01,|veξ0 − vsξ0 | = 29.75, vsξ0 = 30 and for συ = 0.6, 0.7, 0.8, 0.9.

συ but in the contracted range of existence as the difference of electron and ion (or ion-beam) initial streaming speeds increases to a critical value (figure 1) before change over to compressive solitons. But after the critical value of this difference, between 13.5 and 15.5 (figure 1) the formation of soliton turns out to be compressive character attaining relatively smaller amplitudes rather in the entire range. It is seen that the compressive solitons tend to disappear in the fully expanded range as the difference of the initial streaming speeds increases. Both compressive and rarefactive solitons (figure 2) of high amplitudes exist near the magnetic field for small α = 0.1 and ion-beam density Nb = 0.01 quite with linear growth relative to wave velocity V when vsξ0 = 30 as the difference of initial streaming speeds of electrons and ions increases. The amplitude of the rarefactive solitons increases as the difference of initial streaming speeds increases to certain limit but moment it crosses that limit, the solitons turn out to be compressive being consistent with the characters of figure 1. Unlike rarefactive solitons, the amplitude of the compressive solitons increasingly diminishes with the increase of |veξ0 − vsξ0 | and the wave velocity V . Compressive and rarefactive solitons (figure 3) are found to exist for fixed V = 0.1, α = 0.1 and Nb = 0.01 in all directions συ = 0.9, 0.8, 0.7, 0.6 as the difference of initial streaming speeds increases. But at the greater (for compressive) and smaller (for rarefactive) difference of the initial streaming speeds, the small amplitudes remain almost constant. Further, it exhibits indication of disappearance of the non-linearity range in every direction of propagation showing ample scope of study of modified KdV solitons. Besides, this range has shifted to the smaller value of |veξ0 − vsξ0 | as the direction of wave propagation deviates from that of the magnetic field. Interestingly, much concentration of ions in the plasma compound is observed to generate both compressive and rarefactive high amplitude solitons and that too near the magnetic field (figure 4). The widths of the solitons admitting sharp fall for small V and attaining high values gradually diminishes at the increase of the wave velocity V (figure 5) in all directions συ when α = 0.1, Nb = 0.01 and vsξ0 = 30. In all our numerical calculations, we consider Q = 1/1836 with reference to the lightest ion.

References [1] M. Temerin, K Cerny, W. Lotko , F.S. Moser, Observation of double layers and solitary waves in the auroral plasma, Phys. Rev. Lett. 48 (1982)1175-1179. [2] R. Bostrom , G. Gustafsson, B. Holback , G. Holmgren , H. Koskinen, Characteristics of solitary waves and weakly double layers in the magnetospheric plasma, Phys. Rev. Lett. 61(1988) 82-85. [3] L.P. Block, C.G. Falhammar, The role of magnetic-field-aligned electric fields in auroral acceleration, J. Geophys. Res. 95(1990) 5877-5888. [4] H. Alfven, The plasma universe, Phys. Today 39(1986) 22-27. [5] H. Alfven, Double layers and circuits in astrophysics, IEEE Trans. Plasma Sci. PS-14 (1986) 779-793.

289 [6] D.A. Gurnett, L.A. Frank, Observed relationship between electric fields and auroral particle precipitation, J. Geophys. Res. 78) 1973) 145-170. [7] P.K. Shukla, M.Y. Yu, Exact solitary ionacoustic waves in a magnetoplasma, J. Math. Phys. 19(1978) 25062508. [8] M.Y.Yu , P.K. Shukla, S. Bujarbarua, Fully nonlinear ion-acoustic solitary waves in a magnetized plasma, Phys. Fluids 23(1980) 2146-2147. [9] M. Y. Ivanov, Analysis of ion-acoustic solitons in a low-pressure magnetized plasm, Sov. J. Plasma Phys. 7 (1981) 640-642. [10] L.C. Lee, J.R. Kan, Nonlinear ion-acoustic waves and solitons in a magnetized plasma, Phys. Fluids 24 (1981) 430-433. [11] B.C. Kalita, M.K. Kalita, J.Chutia, Drifting effect of electrons on fully nonlinear ion-acoustic waves in a magnetoplasma, J. Phys. A: Math. Gen. 19(1986) 3559-3563. [12] H. Ikezi, R.J. Taylor, D.R. Baker, Formation and interaction of ion-acoustic solitons, Phys. Rev. Lett. 25(1970) 11-14. [13] H. Ikezi, Experiments on ion-acoustic solitary waves, Phys. Fluids 16(1973)1668-1675. [14] E. Okutsu, M. Nakamura, Y. Nakamura, T. Itoh, Amplification of ion-acoustic soitons by an ion beam, Plasma Phys. 20 (1978)561-565. [15] G.O. Ludwig, J.L. Ferreira, Y. Nakamura, Observations of ion-acoustic rarefaction solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett. 52(1984) 275-278. [16] J.L. Cooney, M. T. Gavin, K.E. Lonngren, Experiments on Korteweg-de Vries solitons in a positive ion-negative ion plasma, Phys. Fluids B3(1991)2758-2766. [17] F.L. Scarf , F.V. Coroniti , C.F. Kennel , E.J.Smith , J.A. Slavin , B.T. Tsurutani , S.J.Bame, W.C. Feldman, Plasma wave spectra near slow mode shocks in distant magnetotail, Geophys. Res. Lett. 11(1984)1050-1053. [18] F.L. Scarf , F.V. Coroniti , C.F. Kennel , R.W. Fredricks , D.A. Gurnett , E.J.Smith, ISEE-3 wave measurements in the distant geomagnetic tail and boundary layer, Geophys. Res. Lett. 11(1984) 335-338. [19] C.J. Mckinstrie, D.F. DuBois, Relativistic solitary wave solutions of the Beat-wave equations, Phys. Rev. Lett. 57(1986) 2022-2025. [20] Y. Nejoh, The effect of the ion temperature on the ion-acoustic solitary waves in a collsionless relativistic plasma, J. Plasma Phys. 37(1987) 487-495. [21] G.C. Das, S.N. Paul, Ion-acoustic solitarywaves in relativistic plasmas, Phys. Fluids 28 (1985) 823-825. [22] G.C. Das , B. Karmakar, S.N. Paul, Propagation of solitary waves in relativistic plasmas, IEEE Trans. Plasma Sci. 16(1988) 22-26. [23] G. Pakira , A. Roychowdhury , S.N. Paul, Higher order corrections to the ion-acoustic waves in a relativistic plasma(isothermal case), J.Plasma Phys. 40(1988) 359-364. [24] B.C. Kalita , S.N. Barman, G. Goswami, Weakly relativistic solitons in a cold plasma with electron inertia, Phys. Plasmas 3 (1996) 145-148. [25] S.K. El-Labany , S.M. Shaaban, Contribution of higher order nonlinearity to nonlinear ion-acoustic waves in a weakly relativistic warm plasma, J. Plasma Phys. 53(1995) 245-252. [26] K. Singh , V. Kumar, H.K. Malik, Electron inertia effect on small amplitude solitons in a weakly relativistic two-fluid plasma, Phys. Plasmas 12(2005) 052103-12. [27] T.S. Gill , A. Singh , H. Kaur , N.S. Saini , P. Bala, Ion-acoustic solitons in weakly relativistic plasma containing electron-positron and ion, Phys.Lett. A, 361(2007)364-367. [28] N.C. Lee, C.R. Choi, Ion-acoustic solitary waves in a relativistic plasma, Phys. Plasmas 14(2007) 022307-15.

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 290–299, 2011 http://ijamc.psit.in

Uniformly convergent scheme for Convection Diffusion problem K. Sharath babu1 and N. Srinivasacharyulu2 Department of Mathematics, National Institute of Technology, warangal-507004. Email: 1 sharathsiddipet@gmail.com, 2 nitw nsc@yahoo.co.in

Abstract: In this paper a study of uniformly convergent method proposed by Ilin Allen-South well scheme was made. A condition was contemplated for uniform convergence in the specified domain. This developed scheme is uniformly convergent for any choice of the diffusion parameter. The search provides a first- order uniformly convergent method with discrete maximum norm. It was observed that the error increases as step size h gets smaller for mid range values of perturbation parameter. Then an analysis carried out by [16] was employed to check the validity of solution with respect to physical aspect and it was in agreement with the analytical solution. The uniformly convergent method gives better results than the finite difference methods. The computed and plotted solutions of this method are in good agreement with the exact solution. Key words: Boundary layer; Peclet number; Uniformly convergence; Perturbation parameter.

1

Introduction

Consider the elliptic operator whose second order derivative is multiplied by a parameter ε that is close to zero. These derivatives model diffusion while first-order derivatives are associated with the convective or transport process. In classical problems ε is not close to zero. This kind of problem that was studied in the paper [17]. To summarize when a standard numerical method is applied to a convection-diffusion problem, if there is too little diffusion then the computed solution is often oscillatory, while if there is superfluous diffusion term, the computed layers are smeared. There is a lot of work in literature dealing with the numerical solution of singularly perturbed problems, showing the interest in this nature of problems in Kellog et al [10], Kadalbajoo et al [9], Bender [4], Robert E.O’s Malley, Jr [8], Mortan [13] and Miller et al [12]. We can see that the solution of this problem has a convective nature on most of the domain of the problem, and the diffusive part of the differential operator is influential only in the certain narrow sub-domain. In this region the gradient of the solution is large. This nature is described by stating that the solution has a boundary layer. The interesting fact that elliptic nature of the differential operator is disguised on most of the domain, it means that numerical methods designed for elliptic problems will not work satisfactorily. In general they usually exhibit a certain degree of instability.

Corresponding author: K. Sharath babu

291

2

Motivation and History

The numerical solution of convection-diffusion problems dates back to the 1950s, but only in the 1970s it did acquire a research momentum that has continued to this day. In the literature this field is still very active and as we shall see more effort can be put in. Perhaps the most common source of convection-diffusion problem is the Navier Stokes equation having nonlinear terms with large Reynolds number. Morton [13] pointed out that this is by no means the only place where they arise. He listed ten examples involving convection diffusion equations that include the driftdiffusion equations of semiconductor device modeling and the BlackSholes equation from financial modeling. He also observed that accurate modeling of the interaction between convective and diffusive processes is the most ubiquitous and challenging task in the numerical approximation of partial differential equations. In this paper, the diffusion coefficient ε is a small positive parameter and coefficient of convection a(x) is continuously differentiable function. Consider the convection diffusion problem Lu(x) = −εu00 (x) + a(x)u0 (x) + b(x)u(x) f or0 < x < 1 with

u(0) = u(1) = 0

(2.1)

Where 0 < ε 1, a(x) > α > 0 and b(x) ≥ 0 on [0, 1] , Here assume that a(x) ≤ 1 The above problem is solved by the method proposed by the Il’in Allen uniformly convergent method. The convergence criterion is realized through computation, based on explanation given by Roos et al [16], for lower values of the diffusion coefficient. The reciprocal of the diffusion coefficient is called the Piclet number. For a finite Piclet number the solution patterns matches with the exact solution.

3

Construction of a Uniformly Convergent Method

We describe a way of construction of uniformly convergent difference scheme. We start with the standard derivation of an exact scheme for the convection-diffusion problem (2.1). Introduce the formal adjoint operator L∗ of L and for the sake of convenience select b = 0 in (2.1) Let gi be local Greens function of L∗ with respective to the argument xi ; i.e., [ 00 0 L∗ gi = − ε gi −a gi = 0 in (xi−1 , xi ) ( xi , xi+1 ) (3.1) Let us impose boundary conditions gi (xi−1 ) = gi (xi+1 ) = 0

(3.2)

And impose additional conditions 0

0

+ g ε (gi (x− i ) − i ( xi ) ) = 1

Equation (2.1) is multiplied by g i , integrated with respective to x between the limits xi−1 and xi+1 to get Z xi+1 Z xi+1 g (Lu) i dx = f gi dx xi−1

Z

xi+1

xi−1

00

xi−1 0

(−εu (x) + a u (x)) gi dx =

Z

xi+1

f gi dx xi−1

(3.3)

292

Now L.H.S of (3.3) : Z xi Z 00 0 = (−εu (x) + a u (x))gi dx + xi−1

xi+1

00

0

(−ε u (x) + a u (x)gi dx

xi xi

0

0

x = (−εu + au) gi (x) | +(−εu + au) gi (x) |xi+1 i xi−1

Z

xi

0

0

(−ε u +a u) gi dx −

−

Z

xi−1

xi+1

0

0

(−ε u +a u) gi dx

xi

0

0

= [−ε u (x− ) + a u(xi )] g i (xi ) − [−ε u (xi−1 ) + a u(xi−1 )] g i (xi−1 ) i 0

0

g + [−ε u (xi+1 ) + a u (xi+1 )] g i ( xi+1 ) − (−ε u (x+ i ) + a u(xi )) i (xi )] Z xi Z xi+1 Z xi Z xi+1 0 0 0 0 0 0 − (a u)gi dx − (a u) gi dx + (ε u ) gi dx + (ε u ) gi dx xi−1

xi

xi−1

0

0

0

xi

0

+ g xi xi+1 g g g = −ε u (x− i ) i (xi ) + ε u (xi ) i (xi ) + [εu(x) i (x)]xi−1 + [εu(x) i (x)]xi Z xi Z xi+1 00 0 00 0 + (−εgi − agi )udx + (−εgi − a gi )udx xi−1

xi

0

Since u is continuous on (xi−1, xi+1), we have 0

0

0

0

g + g − g + = [εu(xi ) gi (x− i ) − ε u(xi−1 ) i (xi−1 )] + [εu(xi+1 ) i (xi+1 ) − εu (xi ) i (xi )] Z xi+1 0 0 = −ε gi (xi−1 ) ui−1 + ui + ε gi (xi+1 ) ui+1 = f gi dx

(3.4)

xi−1 0

The difference scheme of equation (3.1) is exact. We can able to evaluate each gi exactly The solution of the equation (3.1) is given by −ax −ε ) e ε on(xi−1 , xi+1 ) a

(3.5)

−ax −ε ) e ε on(xi−1 , xi+1 ) a

(3.6)

gi (x− ) = c1 + c2 ( 0

0

gi (x+ ) = c1 + c2 ( 0

0

Here there are 4 unknowns c1 , c2 ,c1 , c2 requiring 4 equations gi (xi−1 ) = 0

(3.7)

gi (xi+1 ) = 0

(3.8)

0

0

+ ε ( gi ( x− i ) − gi ( xi ) ) = 1

(3.9)

and, from continuity of gi at x=xi + gi (x− i ) = gi (xi ).

(3.10)

On imposing boundary conditions (3.7) and (3.8) on (3.5), (3.6) it can be seen gi (xi−1 ) = c1 + c2 ( 0

0

−axi−1 −ε ) e ε =0 a

(3.11)

−axi+1 −ε )e ε =0 a

(3.12)

gi (xi+1 ) = c1 + c2 (

293

On differentiation of equations (3.5), (3.6) −axi 0 0 0 ε −a −axi ε −a gi (x− ) e ε , gi (x+ )e ε i ) = c2 (− )( i ) = c2 (− )( a ε a ε

Then the equation (3.9) can be written in the following form ε(c2 e Using the fact

−axi ε

+ gi (x− i ) = gi (xi )

0

−c2 e

−axi ε

e

a xi ε

axi+1 ε

(3.13)

] = 0

(3.14)

at x= xi in (3.11) ,(3.12) it follows

−ε e ⇒ c1 + c2 ( a )

On assumption that αi =

1 axi e ε ε

0

) = 1 ⇒ c2 − c2 =

ah ε

, ρi =

=e

−axi ε

a(xi + h) ε

0

0

e − [c1 + c2 ( −ε a )

−axi ε

, above equations may be rewritten as = eαi +ρi

,

e

axi−1 ε

= eαi −ρi

Hence on transformation of the equations (3.11) to (3.14) in to the equations (3.15) to (3.18) −ε −αi +ρi )e = 0 a

(3.15)

−ε − (αi +ρi ) )e = 0 a

(3.16)

c1 + c2 ( 0

0

c1 + c2 (

1 αi e ε 0 0 ε (c1 − c1 ) + (c2 − c2 )(− ) e−αi = 0 a On insertion of (3.17) into the equation (3.18) 0

c2 − c2 =

0

(c 1 − c 1 ) +

(3.17) (3.18)

1 αi −ε −αi e ( )e = 0 ε a

1 (3.19) a Subtracting the equation (3.16) from the equation (3.15) , then by using equations (3.17) & (3.19) it may be obtained 0

(c 1 − c 1 ) =

0

0

(c 1 − c 1 ) + (c2 e−αi +ρi − c2 e−αi −ρi )(

−ε ) = 0 a

1 1 −ε + (c2 e−αi + ρi −(c2 − eαi )(e−(αi +ρi ) ( ) = 0 a ε a 1 1 αi −αi −ρi −ε −αi + ρi −(αi +ρi ) + (c2 e − c2 e + e e )( ) = 0 a ε a From (3.20) it follows eαi (1 − e−ρi ) c2 = ε (eρi − e−ρi )

(3.20)

(3.21)

0

To find c2 the value of c2 is substituted in (3.17) , to get 0

c2 =

eαi (1 − eρi ) ε (eρi − e−ρi )

(3.22)

294

Again employing the value of c2 in (3.15) the value of c1 can be obtained as 1 eρi − 1 a (eρi − e−ρi )

c1 =

(3.23)

0

Next the value of c1 is used in (3.19) to obtain c1 1 e−ρi − 1 a (eρi − e−ρi )

0

c1=

(3.24)

Now on imposition of equations (3.21)- (3.24) , on (3.5) , (3.6) they may be rewritten as gi ( x− ) =

1 eαi eρ i − 1 + a (eρi − e−ρi ) ε

−ax (1 − e−ρi ) −ε ( ) e ε ρ −ρ i i (e − e ) a

(3.25)

gi ( x+ ) =

eαi 1 e−ρi −1 + ρ −ρ i i a (e − e ) ε

−ax (1 − eρi ) −ε ( ) e ε ρ −ρ i i (e − e ) a

(3.26)

The derivatives of equations (3.25) , (3.26) are 0

g i (x− ) = 0

g i (x+ ) =

axi 1 −ax e ε e ε ε

(1 − e−ρi ) (eρi − e−ρi )

(3.27)

axi 1 −ax e ε e ε ε

(1 − eρi ) (eρi − e−ρi )

(3.28)

Now from (3.27) , (3.28) and (3.9) it follows. 1 ah (1 − e−ρi ) eε i.e. ε (eρi − e−ρi )

0

gi (x− i−1 ) = 0

gi (x− i−1 ) = 0

gi (x+ i+1 ) =

1 (eρi − 1) ε (eρi − e−ρi )

(3.29)

1 (e−ρi − 1) ε (eρi − e−ρi )

(3.30)

Now by inserting values of gi + and gi −− from (3.29) , (3.30) in (3.2) & (3.3) it may be obtained Z x Z xi Z xi+1 i+1 axi ah − gi+ dx ] where ρi = f , αi = gi dx = f [ gi dx + ε ε xi−1 xi−1 xi R xi =

1 eρi −1 xi−1 [ a (eρi −e−ρi )

+

−ρ eαi ( 1− e i ) ε (eρi −e−ρi )

e ( −ε a )

−ax ε

] dx +

xi+1

−ax 1 e−ρi − 1 eαi ( 1 − eρi ) −ε e ε ] dx ) + ( ρ −ρ ρ −ρ a (e i − e i ) ε (e i − e i ) a xi ah (eρi − 1 ) (1 − e−ρi ) ε αi −ax i e ε eε )] + e + [ (1 − ρ 2 ρ −ρ −ρ a (e i − e i ) (e i − e i ) −ρi − ah (e − 1) (1 − eρi ) ε αi −ax i ε e e − 1) ] + [ (e ε ρ −ρ 2 ρ −ρ i i i i a (e − e (e − e ) )

Z

[

=

h a

h a

ε αi −ax (1 − e−ρi ) (1 − eρi ) + (1 − eρi )(e−ρi − 1 ) h (eρi + e−ρi − 2) i e ε ( + [ )] e ρ −ρ 2 a a (e i − e i ) (eρi − e−ρi ) h = a (e ρ2i

ρi

(e 2

− e

− e −ρi 2

) (e

−ρi 2 ρi 2

)2 + e

−ρi 2

= )

h (eρi − 1 ) a (eρi + 1)

295

Finally, as follows Z xi+1 it can be represented h ( eρi − 1 ) gi dx = f This gives the final scheme as f a (eρi + 1) xi−1 1 − e−ρi h ( eρi − 1 ) (eρi − 1 ) ui−1 + ui − ρ ui+1 = f − ρi −ρ −ρ (e − e i ) (e i − e i ) a (eρi + 1)

(3.31)

ah . ε The equation (3.31) is the Il’in-Allen scheme. This method is tested for a linear problem by applying various perturbation parameter values with in the defined range. I t is observed from the numerical results that Il’in-Allen scheme is converging uniformly in the entire domain. In the boundary layer region , it is appreciable thing that the scheme is uniformly converging one. For testing the algorithm outlined above the twopoint boundary value problem here ρi =

00

0

−ε u (x) + u (x) = 2x withu(0) = u(1) = 0

(3.32)

Is considered with ka(x)k ≤ 1 The analytical solution of (3.32) is u(x) =

(1 + 2ε) (e

1 ε

− 1)

−

(1 + 2ε ) (e

1 ε

− 1)

x

e ε + x2 + 2 ε x , 0 < ε << 1

(3.33)

The computational method is executed with various choices of the diffusion co-efficient by applying forward difference method, upwind method, central difference method and the Il’in-Allen scheme. The results obtained are presented in the table.

4

Error Analysis:

The present scheme is first-order uniformly convergent in the discrete maximum norm, i.e., M ax |u(xi ) − ui | ≤ Ch i

The region of solution u is divided into two parts, (2.1) smooth region with bounded derivatives 2) boundary layer region with chaotic behavior where

in u = v + z , where v is a boundary layer function and the bound on the smooth function z j has a factor ε1−j The calculation of |z (xi ) − zi | is now considered. The corresponding consistency error |τi | is estimated with the help of Taylor series, proposed by H.G. Roos et al [16] which give the inequality Z xi+1

00

|τ i | ≤ C (ε z 3 (t) + a z (t) ) dt xi−1

≤ Ch + C ε−1

Z

xi+1

exp(−a0 xi−1

1−t )dt ε

a0 h 1 − xi ≤ Ch + C sinh( ) exp ( − ao ). ε ε An application of the discrete comparison principle indicates the increase of power of ε i i.e., |z (xi ) − zi | ≤ Ch + C sinh( a0ε h ) exp ( − ao 1−x ) for i= 1,2 ,3,. . . . . . n ε for ε ≤ h that can be easily obtained | z( xi ) − zi | ≤ Ch. In the second case h ≤ ε , using the inequality 1 − e−t ≤ ct f or t > 0 the desired desired estimate can be put as | z(xi ) − zi | ≤ Ch

296 Table 1: Case1 : ε = 0.05

x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Forwardscheme 0 0.001000 0.002200 0.003600 0.005200 0.007000 0.009000 0.011200 0.013600 0.016200 0.019000 0.058000 0.123999 0.204997 0.305981 0.426848 0.566617 0.716180 0.790694 0.780071 0.762193 0.735194 0.696747 0.643937 0.573125 0.479760 0.358154 0.20119 0

Backwardscheme 0 0.001200 0.002600 0.004200 0.006000 0.008000 0.010200 0.012600 0.015200 0.018000 0.02100 0.062000 0.122998 0.203985 0.304899 0.425363 0.563036 0.703420 0.756763 0.745514 0.728374 0.704127 0.671311 0.628171 0.572603 0.502081 0.413575 0.167335 0

CentralScheme 0 0.001100 0.002400 0.003900 0.005600 0.007500 0.009600 0.011900 0.014400 0.017100 0.020000 0.060000 0.119999 0.19995 0.299960 0.419700 0.557771 0.703422 0.776690 0.768388 0.754197 0.732763 0.702433 0.661185 0.606549 0.535505 0.444363 0.182736 0

Allen-Il’in scheme 0 0.001103 0.002407 0.005613 0.005613 0.007517 0.009620 0.011923 0.014427 0.017130 0.020033 0.060067 0.120100 0.200129 0.300127 0.419895 0.557961 0.703453 0.756044 0.767636 0.753337 0.731799 0.701373 0.660049 0.605369 0.534331 0.44321 0.182179 0

Exact solution 0 0.0010999 0.0023999 0.0038999 0.005599 0.0074999 0.0095999 0.0118999 0.0143999 0.0170999 0.019999 0.06509985 0.127098 0.209091 0.2999500 0.41963099 0.5572733 0.6998527 0.75113119 0.737271 0.7463138 0.7166433 0.706286 0.6866133 0.646286 0.592832 0.4342072 0.17849617 0

2

h Similarly | v( xi ) − vi | ≤ C h+ε ≤ Ch as proposed by Kellog et al [10]. This shows that Il’in-Allen scheme is uniformly convergent of first order. In the above scheme the absolute value of a(x) the convection coefficient is less than or equal to unity, the scheme converges faster to the exact solution.

5

Result Analysis

We have solved the problem by using forward difference scheme, upwind scheme, central difference scheme and Il’in- Allen scheme by selecting the step width h = 0.01 and varying the perturbation parameter or diffusion coefficient . We have selected ε = 0.05, 0.001, 0.0001, 0.00001. 1. for ε =0.05 all the schemes behaves similarly in the smooth region as well as in the boundary layer region. 2. for ε = 0.001 forward scheme is not matching with the exact solution , upwind scheme converging to exact solution well and the central difference scheme converges in the smooth region and oscillates in the boundary layer. where as Il’in scheme converges uniformly in the entire region. 3. for ε = 0.0001, 0.00001 forward scheme diverges , central scheme oscillates . Upwind scheme has given good numeric results in the specified domain. But at the boundary i.e near to the point x=1 the upwind scheme is not matching with the exact solution. The solution of the upwind scheme is not uniformly convergent in the discrete maximum norm due to its behavior in the layer, where as the proposed scheme is uniformly convergent of first order even for lower values of ε through out the domain.

297 Table 2: Case2 : ε = 0.001 = 10−3

x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Forwardscheme 0 -0.12447 -0.99928 -1.01274 -1.01058 -1.00993 -1.00080 -0.00858 -0.99616 -0.99354 -0.99072 -0.97362 -0.92442 -0.85522 -0.76602 -0.65682 -0.52762 -0.37842 -0.20922 -0.19120 -0.17298 -0.15456 -0.13594 -0.11712 -0.00981 -0.07888 -0.05946 -0.02002 0

Backwardscheme 0 0.000220 0.000640 0.001260 0.002080 0.003100 0.004320 0.005740 0.007360 0.009180 0.011200 0.042400 0.093600 0.164800 0.256000 0.367200 0.498400 0.649600 0.820800 0.839020 0.857440 0.876060 0.894880 0.913899 0.933114 0.952469 0.971384 0.91816 0

CentralScheme 0 0.000120 0.000440 0.000960 0.001680 0.002600 0.003720 0.005040 0.006560 0.008280 0.010200 0.040400 0.090600 0.160800 0.251000 0.361200 0.491404 0.641805 0.823617 0.812195 0.874828 0.826879 0.945302 0.814667 1.058120 0.740940 1.265210 1.683413 0

Allen-Il’in scheme 0 0.00020 0.000600 0.001200 0.002 0.0030 0.04200 0.005600 0.007200 0.009000 0.01100 0.042000 0.093000 0.164000 0.255000 0.366001 0.497001 0.648001 0.819001 0.837201 0.855601 0.874201 0.893001 0.912001 0.931201 0.950601 0.970201 0.990001 0

Exact solution 0 0.00012 0.00044 0.00096 0.00168 0.002600 0.0037199 0.005040 0.006560 0.00828 0.01020 0.04040 0.0906 0.160800 0.251000 0.3611999 0.49140 0.641600 0.81180 0.829920 0.848240 0.866760 0.885480 0.904400 0.92352 0.9428399 0.9623599 0.9820345 0

Table 3: Case3 : ε = 0.0001 = 10−4

x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Forwardscheme 0 -1.020404 -1.009895 -1.009597 -1.008994 -1.008192 -1.00719 -1.005988 -1.004586 -1.002984 -1.001182 -0.972162 -0.923142 -0.854122 -0.765102 -0.656082 -0.527062 -0.378042 -0.209022 -0.191020 -0.172818 -0.154416 -0.135814 -0.117012 -0.098010 -0.078808 -0.059406 -0.020002 0

Backwardscheme 0 0.000202 0.000604 0.001206 0.002008 0.003010 0.004212 0.005614 0.007216 0.009018 0.011020 0.042040 0.093060 0.164080 0.255100 0.366120 0.497140 0.648100 0.819180 0.837382 0.855784 0.874386 0.893188 0.912190 0.931216 0.950616 0.970216 1.008987 0

CentralScheme 0 -0.03588 0.00187 -0.03661 0.00466 -0.03666 0.00839 -0.03605 -0.01306 -0.03479 0.01869 0.06165 0.13098 0.22981 0.36280 0.53694 0.76261 1.05533 1.43826 0.13857 1.52845 0.11939 1.62392 0.09635 1.72504 0.06906 1.83223 0.03709 0

Allen-Il’in scheme 0 0.000200 0.000600 0.001200 0.002000 0.003000 0.004200 0.007200 0.007200 0.009000 0.011000 0.042000 0.093000 0.164000 0.255000 0.366000 0.497000 0.648000 0.819000 0.837200 0.855600 0.874200 0.893000 0.912000 0.931200 0.950600 0.970200 0.990000 0

Exact solution 0 0.000102 0.000404 0.000906 0.001608 0.0025100 0.0036199 0.0049140 0.006416 0.00818 0.01002 0.040040 0.09006 0.16008 0.250100 0.360120 0.490140 0.640160 0.81018 0.828282 0.846584 0.865086 0.883788 0.902690 0.921792 0.941094 0.9605959 0.980298 0

298 Table 4: Case4 : ε = 0.00001 = 10−5

Forwardscheme 0 -1.011037 -1.009825 -1.009426 -1.008825 -1.008025 -1.007025 -1.005825 -1.004424 -1.002824 -1.001024 -0.972021 -0.923018 -0.854015 -0.765012 -0.656009 -0.527007 -0.378005 -0.209002 -0.191002 -0.172802 -0.154402 -0.135801 -0.117001 -0.098001 -0.078801 -0.059401 -0.039800 0

Backwardscheme 0 0.000200 0.000600 0.001201 0.002001 0.003001 0.004201 0.005601 0.007202 0.009002 0.011002 0.042004 0.093006 0.164008 0.255010 0.366011 0.497013 0.648014 0.819016 0.837216 0.855616 0.874216 0.89016 0.912016 0.931216 0.950616 0.970216 1.008987 0

CentralScheme 0 -0.818348 0.003681 -0.820841 0.008188 -0.822561 0.013522 -0.082350 0.019682 -0.823680 0.026669 0.074019 0.142077 0.230871 0.340432 0.470792 0.621983 0.794038 0.986993 -0.168016 1.028095 -0.135937 1.070035 -0.103095 1.112814 -0.069491 1.156430 -0.035126 0

Allen-Il’in scheme 0 0.00200 0.000600 0.001200 0.002000 0.003000 0.004200 0.005600 0.007200 0.009000 0.01100 0.042000 0.193000 0.264000 0.355000 0.466000 0.697000 0.74800 0.819000 0.837200 0.855600 0.874200 0.893000 0.912000 0.931200 0.950600 0.970200 0.99000 0

Exact solution 0 0.000100 0.0121022 0.0144024 0.0169026 0.0196028 0.022503 0.0256032 0.0289034 0.0324036 0.0361028 0.0841058 0.1521078 0.2401097 0.3481117 0.4761138 0.6241158 0.7921178 0.81001 0.827004 0.848282 0.8650680 0.873788 0.900691 0.920006 0.940002 0.9600231 0.980098 0

4. For finite value of the Peclet number Il’in-Allen scheme behaves well with the exact solution in the region [0,1] . 5. The standard finite difference scheme of upwind and central scheme on equally spaced mesh does not converge uniformly. Because, the point wise error is not necessarily reduced by successive uniform improvement of the mesh in contrast to solving unperturbed problems. The standard central difference scheme is of order O(h2 ) .It is numerically unstable in the boundary layer region and gives oscillatory solutions unless the mesh width is small comparatively with the diffusion coefficient but it is practically not possible as diffusion coefficient is very small. 6. For any value of x in [0,1] , a(x)=1 Il’in- Allen scheme converges uniformly. This has been thoroughly verified through computation.

References [1] V.B. Andreev and N.V. Kopteva, Investigation of difference Schemes with an approximation of the first derivative by a central difference relation, Zh. Vychisl. Mat.i Mat. Fiz. 36 (1996), 101–117. [2] Arthur E.P. Veldman, Ka-Wing Lam, Symmetry-preserving upwind discretization of convection on non-uniform grids. Applied Numerical Mathematics 58 (2008). [3] A. Brandt and I. Yavneh, Inadequacy of first-order upwind difference schemes for some recirculating flow, J. Comput. Phys. 93 (1991), 128-143. [4] C.M. Bender , S.A.Orszag, Advanced Mathematical Methods for Scientists and Engineers , McGraw-Hill , New York, (1979). [5] J. C. Butcher, Numerical Methods for Ordinary Differential Equations , Second edition ,John wiley & Sons, Ltd. [6] D. Gilbarg, N.S.Trudinger, Elliptic partial differential equation of second order, springer, Berlin, (1983).

299

Figure 1 Figure 2

Figure 3

Figure 4

[7] A. M. Il’in, A difference scheme for a differential equation with a Small Parameter multiplying the highest derivative, Mat. Zametki, 6 (1969),237–248. [8] E.O. Robert, Introduction to singular Perturbationproblems, Malley,Jr, Academic press. [9] M. K. Kadalbajoo, Y.N. Reddy, Asymptotic and numerical analysis of singular Perturbation problems: a survey, Appl. Math.Comp. 30 (1989), 223-259. [10] R.B. Kellog , A. Tsan, Analysis of some difference approximations for a singularly Perturbed problem without turning points. Math. Comp., 32 (1978), 1025–1039. [11] Martin Stynes, Steady-state convection-diffusion problems, Acta Numerica (2005), 445–508. [12] J. Miller, E. O’Riordan, G. Shishkin, Fitted Numerical Methods for Singularly Perturbed problems, World Scientific, Singapore, (1996). [13] K. W. Morton (1996), Numerical solution of Convection-Diffusion problems, Applied Mathematics and Mathematical Computation, Vol. 12, Chapman & Hall, London , 1995. [14] Mikhail Shashkov, Conservative finite difference methods on General grids, CRS Press(Tokyo), (2005). [15] Dennis G. Roddeman, Some aspects of artificial diffusion in flow analysis, TNO Building and Construction Research , Netherlands. [16] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection –Diffusion and Flow Problems Springer,Berlin, (1996). [17] N. Srinivasacharyulu, K. Sharath babu, Computational method to solve Steady-state convection-diffusion problem, Int. J. of Math, Computer Sciences and Information Technology, 1 (2008), 245–254. [18] M. Stynes and L. Tobiska, A finite difference analysis of a streamline Diffusion method on a Shishkin meshes, Numer. Algorithms, 18 (1998), 337–360.

c Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(4),pp 300â€“306, 2011 http://ijamc.psit.in

Thermal radiation effects on hydro-magnetic flow due to an exponentially stretching sheet P. Bala Anki Reddy1 and N. Bhaskar Reddy2 1

Department of Mathematics, NBKRIST, Vidyanagar, Nellore, A.P., India-524413.

2

Department of Mathematics, S.V. University, Tirupati-517502, A.P. (INDIA). Email: pbarsvu@gmail.com, nbrsvu@yahoo.co.in

Abstract: A steady laminar two-dimensional boundary layer flow of a viscous incompressible radiating fluid over an exponentially stretching sheet, in the presence of transverse magnetic field is studied. The non-linear partial differential equations describing the problem under consideration are transformed into a system of ordinary differential equations using similarity transformations. The resultant system is solved by applying Runge-Kutta fourth order method along with shooting technique. The flow phenomenon has been characterized by the thermo physical parameters such as magnetic parameter (M), radiation parameter (R) and Eckert number (E). The effects of these parameters on the fluid velocity, temperature, wall skin friction coefficient and the heat transfer coefficient have been computed and the results are presented graphically and discussed quantitatively. Key words: Thermal radiation; MHD; Boundary layer flow; exponentially stretching sheet.

1

Introduction

The study of viscous incompressible flow over a stretching surface has become increasing important in the recent years due to its numerous industrial applications such as the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film, condensation process of metallic plate in a cooling bath and glass, and also in polymer industries. Since the pioneering work of Sakiadis [1], various aspects of the stretching flow problem have been investigated by many authors like Cortell [2], Xu and Liao [3], Hayat et al [4] and Hayat and Sajid [5]. On the other hand, Gupta and Gupta [6] stressed that realistically, stretching surface is not necessarily continuous. Due to the fact that the rate of cooling influences the quality of the product with desired characteristics, Ali [7] has investigated the thermal boundary layer flow by considering the nonlinear stretching surface. Further, a new dimension has been added to this investigation by Elbashbeshy [8] who examined the flow and heat transfer characteristics by considering an exponentially stretching continuous surface. He considered an exponential similarity variable and exponential stretching velocity distribution on the coordinate considered in the direction of stretching.

Corresponding author: P. Bala Anki Reddy

301

For some industrial applications such as glass production and furnace design, electrical power generation, Astrophysical flows, solar power technology, which operates at high temperatures, radiation effects, can be significant. An extensive literature that deals with flows in the presence of radiation is now available. Cortell [9] has solved a problem on the effect of radiation on Blasius flow by using fourth –order RungeKutta approach. Later, Sajid and Hayat [10] considered the influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet by solving the problem analytically via homotopy analysis method (HAM). Recently, El-Aziz [11] studied effects on the flow and heat transfer an unsteady stretching sheet. Bidin and Nazar [12] studied the boundary layer flow over an exponential stretching sheet with thermal radiation, using Keller-box method. There has been a renewed interest is studying magnetohydrodynamic flows and heat transfer due to the effect of magnetic fields on the boundary layer flow control and on the performance of many systems involving electrically conductive flows. In addition, this type of flow finds applications in many engineering problems such as MHD generators, Plasma studies, Nuclear reactors, and Geothermal energy extractions. Raptis et al. [13] studied the effect of thermal radiation on the magnetohydrodynamic flow of a viscous fluid past semi-infinite stationary plate and Hayat et al. [14] extended the analysis for a second grade fluid. Later Aliakbar et al. [15] analyzed the influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets. In this paper an attempt is made to investigate the effects of thermal radiation on the study laminar two dimensional boundary layer flow of a viscous incompressible electrically conductive and radiating fluid over an exponentially stretching sheet. The governing boundary layer equations are solved using Runge–Kutta fourth order along with shooting technique.

2

Mathematical Formulation

A two dimensional boundary layer flow of a viscous incompressible electrically conductive and radiative fluid bounded by a stretching surface is considered. The x-axis is taken along the stretching surface in the direction of the motion and y-axis perpendicular to it. The fluid is assumed to be gray, absorbing-emitting but non scattering. A uniform magnetic fluid is applied in the direction perpendicular to be stretching surface. The transverse applied magnetic field and magnetic Reynolds number are assumed to be very small, so that the induced magnetic field is negligible. Then under the above assumptions, in the absence of an input electric field, the governing boundary layer equations are: ∂u ∂v + =0 (2.1) ∂x ∂y ∂u ∂u ∂ 2 u σB02 +v =υ 2 − u ∂x ∂y ∂y ρ 2 ∂T ∂T ∂2T ∂qr ∂u ρcp u +v =k 2 − +µ ∂x ∂y ∂y ∂y ∂y u

(2.2) (2.3)

where u and v are the velocities in the x-and y-directions respectively, υ-the kinematic viscosity,σthe electrical conductivity,B0 -the magnetic induction, ρ-the fluid density,cp -the specific heat at constant pressure, T- the temperature,k- the thermal conductivity, qr - the radiative heat flux and µ- the dynamic viscosity. The second and third terms on the right hand side of Equation (2.3) represent the radiative heat flux and the viscous dissipative heat. The boundary conditions for the velocity and temperature fields are: x

u (0) = U0 e L u → 0,

T → 0 as y → ∞ (4)

,

v (0) = 0,

2x

T (0) = T∞ + T0 e L ,

302

in which U0 - the reference velocity, T0 and T∞ - the temperatures at far away from the plate and L- the constant. By employed Rosseland approximation (Sajid and Hayat 2008), the radiative heat flux is given by 4σ ∗ ∂T 4 (2.4) qr = − ∗ 3k ∂y where σ ∗ is the Stefan- Boltzmann constant and k ∗ - the mean absorption coefficient. We should be noted that the by using the Rosseland approximation, the present analysis is limited to optically thick fluids. If the temperature differences with in the flow field are sufficiently small, then Equation (2.4) can be linearized by expanding T 4 into the Taylor series aboutT∞ , which after neglecting higher order forms takes the form 3 4 T4 ∼ T − 3T∞ = 4T∞

(2.5)

Invoking Equations (2.3),(2.4) and (2.5), it can be written as

2 2 3 ∂T ∂T 16σ ∗ T∞ ∂ T ∂u ρcp u +v + µ = k+ ∂x ∂y 3k ∗ ∂y 2 ∂y

(2.6)

Introducing the following non-dimensional quantities x

u = U0 e L f 0(η) , r v=−

o υU0 x n e 2L f (η) + ηf 0(η) , 2L 2x

T = T∞ + T0 e L , r η=

U0 x e 2L y, 2υL

M=

σB02 2L x , ρU0 e L

Pr =

µcp , k

R=

3 4σ ∗ T∞ , k∗ k

E=

U02 , T0 cp

(2.7)

Equation (2.1) is automatically satisfied, and Equations (2.2) and (2.6) reduce to f 000 − 2(f 0 )2 + f f 00 − M f 0 = 0 4 1 + R θ00 Pr f θ0 − 4f 0 θ + E(f 00 )2 = 0 3

(2.8) (2.9)

WhereM ,R,Prand E are the magnetic parameter, radiation parameter, Prandtl number and Eckert number, respectively and primes denote the differentiation with respect to η. The corresponding boundary conditions are f (0) = 0,

f 0 (0) = 1,

θ(0) = 1,

f 0 → 0, θ → 0 as η → ∞ (11) For the type of boundary layer flow, the skin-friction coefficient and heat transfer coefficient are important physical parameters. Knowing the velocity field, the skin-friction at the stretching surface can be obtained, which in non-dimensional form is given by cf = −2 (Re)

− 12

00

f (0)

(2.10)

Knowing the temperature field, the rate of heat transfer coefficient at the stretching surface can be obtained, which in non-dimensional form, in terms of the Nusselt number, is given by 1

N u = − (Re) 2 θ0 (0) Where Re =

U0 L υ is

the Reynolds number.

(2.11)

303

3

Figure 1: Profiles for f ’(η), f (η) and θ(η)

Figure 2: Effect of M on the Veloci

Figure 3: Effect of M on the Temperature

Figure 4: Effect of Pr on the Temperature

Solution of the problem

The governing boundary layer Equations (2.8) and (2.9) subjects to boundary conditions (11) are solved numerically by using Runge-Kutta fourth order technique along with shooting method. First of all higher order non-linear differential Equations (2.8) and (2.9) are converted into simultaneous linear differential equations of first order and they are further transformed into initial value problem by applying the shooting technique (Jain et al. [16]). The resultant initial value problem is solved by employing Runge-Kutta fourth order technique. The step size ∆η = 0.05 is used o obtain the numerical solution with five decimal place accuracy as the criterion of convergence. From the process of numerical computation, the skin-friction coefficient and the Nusselt number which are respectively proportional to f 00 (0) and −θ0 (0), are also sorted out and their numerical values are presented in a tabular form.

4

Results and Discussion

In order to get a physical insight of the problem, a representative set of numerical results we shown graphically in Figs.1-12 to illustrate the influence of physical parameters viz., the magnetic parameter M , Prandtl number Pr, Radiation parameter R and Eckert number E on the velocity f 0 (η), f (η) and temperatureθ (η). Fig.1 depicts the profiles of velocity, f (η) and θ (η) profile for M =1, Pr = 1, R = 1 and E = 0.2. It is observed that the profiles of the velocity f 0 (η) and f (η) are inversely proportional

304

Figure 5: Effect of R on the Temperature

Figure 6: Effect of E on the Temperature

Figure 7: Effects of R and E on the Temperature

Figure 8: Effects of Pr and E on the Temperature

to each other. The velocity profile is unique for all values of M ,Pr, Rand E due to the decoupled Equations (2.8) and (2.9). Figs.(2.2) and (2.3) illustrate the velocity and temperature profiles for different values of the magnetic parameterM . It is observed that the velocity decreases as the magnetic parameter increases (Fig.2). This is because that the application of transverse magnetic field will result a resistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. Also, the boundary layer thickness decreases with an increase in the magnetic parameter. From Fig.(2.3), it is noticed that an increase in the magnetic parameter results in an increase in the temperature. The effect of the Prandtl number Pr on the temperature field is shown in Fig.4. The Prandtl number defines the ratio of momentum diffusivity to thermal diffusivity. It is noticed that as Pr increases, the temperature decreases. This is because, physically, if Pr increases, the thermal diffusivity decreases and these phenomena lead to the decreasing of energy ability that reduces to thermal boundary layer. The influence of the thermal radiation parameter R on the temperature is shown in Fig.5. The radiation parameter R defines the relative contribution of conduction heat transfer to thermal radiation transfer. It is observed that as R increases, the temperature profiles and thermal boundary layer thickness also increase. For different values of the viscous dissipation parameter i.e., the Eckert number E on the temperature is shown in Fig.6. The Eckert number E express the relationship between the kinetic energy in the flow and the enthalpy. It embodies the conservation of kinetic energy into internal

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Figure 9: Effects of M and Pr on the Temperature

Figure 10: Effects of M and R on the Temperature

Table 1: Nu for various values of M, R, E and Pr

M 0 0.5 1.0 0.5 0.5 0.5 0.5

R 0.5 0.5 0.5 1.0 0.5 0.5 0.5

E 0.5 0.5 0.5 0.5 0.9 0.5 0.5

Pr 1.0 1.0 1.0 1.0 1.0 2.0 3.0

Nu 1.2381 1.1904 1.1485 0,9605 1.1265 1.8155 2.2924

energy by work done against the viscous fluid stress. The positive Eckert number implies cooling of the sheet i.e., loss of heat from the sheet to the fluid. It is found that the temperature profiles and thermal boundary layer thickness increase slightly with an increase in E. For further observations, comparison is made between the various physical parameters involved in the problem and shown in Figs.7-10. The effects of R and E with fixed M = 1 and Pr = 1 are shown in Fig.7. It is seen that as Eor R increases, the temperature profiles also increase and the effects of R are more pronounced than the effects of E. The effects of Eand Pr, with fixed M = 1 and R = 1 are illustrated in Fig. 8. It is observed that their effects are opposite in nature, in which the increase in Eand the decrease in Pr lead to the increase in the temperature profiles. The effects of M and Pr, with fixed E = 0.5 and R = 1 are shown in Fig. 9. It is found that their effects are opposite in nature, in which the increase in M and the decrease in Pr lead to the increase in the temperature profiles. The effects of R and M with fixed E = 0.5 and Pr = 1 are presented in Fig.10. It is noticed that as M and R increase, the temperature profiles also increase and the effects of R are more pronounced than the effects of M. From equations (2.8) and (2.10), it is clear that the variations in the Prandtl number Pr, radiation parameter R and Eckert number E do not effect the wall skin-friction coefficient due to the decoupled equations. However, since the magnetic parameter M is coupled with the momentum equation, it has significant effect on the wall skin-friction coefficient. The wall skin-friction coefficient has unique values 1.28213 and 1.62918, for non- magnetic (M=0) and magnetic (M=1) cases respectively. It is interesting to note that the value of the wall skin-friction coefficient in nonmagnetic case is in good agreement with that of Bidin and Nazar [12], whose solved the problem using the Keller-box method. The effects of various governing parameters on the Nusselt number N u are shown in Table1. It is observed that the Nusselt number increases as Pr increases, where as it decreases as M or R or

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E increases.

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