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International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 2 Issue 4 Dec 2012 1-10 Š TJPRC Pvt. Ltd.,

ON THE NUMERICAL SOLUTION FOR THE LINEAR FRACTIONAL KLIEN-GORDON EQUATION USING LEGENDRE PSEUDOSPECTRAL METHOD 1 1

N. H. SWEILAM, 2M. M. KHADER & 3A. M. S. MAHDY

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt 2 3

Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

ABSTRACT Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the linear fractional Klien-Gordon equation is considered. The fractional derivative is described in the Caputo sense. The method is based on Legendre approximations. The properties of Legendre polynomials are utilized to reduce the proposed problem to a system of ordinary differential equations, which solved using the finite difference method. Numerical solutions are presented and the results are compared with the exact solution.

KEYWORDS: Fractional Klien-Gordon Equation, Caputo Fractional Derivative, Finite Difference Method, Legendre Polynomials

INTRODUCTION Ordinary and partial fractional differential equations (FDEs) have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering (Bagley & Torvik (1984)). Consequently, considerable attention has been given to the solutions of FDEs of physical interest. Most FDEs do not have exact solutions, so approximate and numerical techniques (He (1998), Sweilam et al. (2011), Sweilam et al. (2012a-e)), must be used. Recently, several numerical methods to solve FDEs have been given such as variational iteration method (Inc (2008)), homotopy perturbation method (El-Sayed et al. (2012a, b), Sweilam et al. (2007)), Adomian decomposition method (El-Sayed (2003), Jafari & Daftardar (2006)), homotopy analysis method (Hashim et al. (2009)) and collocation method (Khader (2011), Khader (2012), Khader & Hendy (2012b), Sweilam & Khader (2010), Sweilam et al. (2012e)). The linear or nonlinear PDEs arising in physics and engineering play an important role in mathematical modelling. The hyperbolic PDEs model the vibrations of structures. So, searching the numerical and exact solutions to these linear or nonlinear models gains importance. For this reason, many methods were developed for solving differential equations in literature. The Klein-Gordon equation plays a significant role in mathematical physics and many scientific applications such as solid-state physics, nonlinear optics, and quantum field theory (Wazwaz (2006)). The equation has attracted much attention in studying solitons (Sassaman & Biswas (2011a, b)) and condensed matter physics, in investigating the interaction of solitons in a collisionless plasma, the recurrence of initial states, and in examining the nonlinear wave equations (El-Sayed (2003)). Wazwaz has obtained the various exact travelling wave solutions such as compactons, solitons and periodic solutions by using the tanh method (Wazwaz (2006)). The study of numerical solutions of the Klein-Gordon equation has been investigated considerably in the last few years. In the previous studies, the most papers have carried out different spatial discretization of the equation (Golmankhaneh et al. (2011), Yusufoglu (2008)).


2

N. H. Sweilam, M. M. Khader & A. M. S. Mahdy

Where, the numerical solution using radial basis functions is given in (Dehghan & Shokri (2009)), collocation and finite difference-collocation methods for the solution of proposed problem is introduced in (Lakestani & Dehghan (2010)), finally, the tension spline approach for the numerical solution of nonlinear Klein-Gordon equation is implemented in (Rashidini & Mohammadi (2010)). (Khader et al. (2011)) introduced a new approximate formula of the fractional derivative and used it to solve numerically the fractional diffusion equation. Also, (Khader and Hendy (2012a)) used this formula to solve numerically the fractional delay differential equations. We present some necessary definitions and mathematical preliminaries of the fractional calculus theory that will be required in the present paper.

DEFINITION The Caputo fractional derivative operator

D α f ( x) =

D α of order α is defined in the following form

x 1 f ( m ) (t ) dt , Γ ( m − α ) ∫0 ( x − t ) α − m +1

α > 0,

x > 0,

where m − 1 < α ≤ m, m ∈ N. Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operation

D α (λ f ( x ) + µ g ( x)) = λ D α f ( x) + µ D α g ( x), where λ and µ are constants. For the Caputo's derivative we have (Podlubny (1999))

D α C = 0,

C is a constant , (1)

 0,  α n D x =  Γ(n + 1) n −α  Γ(n + 1 − α ) x , 

for

n ∈ Ν 0 and n < α ;

for

n ∈ Ν 0 and n ≥ α . (2)

We use the ceiling function

α  to denote the smallest integer greater than or equal to α and

Ν 0 = {0,1,2,...} . Recall that for α ∈ Ν , the Caputo differential operator coincides with the usual differential operator of integer order. For more details on fractional derivatives definitions and its properties see (Das (2008), Podlubny (1999)). Our main goal in this paper is concerned with the application of Legendre pseudospectral method to obtain the numerical solution of the linear fractional Klien-Gordon equation (LFKGE) of the form

u tt ( x, t ) − d ( x) D α u ( x, t ) = f ( x, t ), the parameter

α

0 < x < 1,

refers to the Caputo fractional order of spatial derivatives with 1 < α

(3)

≤ 2. The function f ( x, t )

is the

source term. We also assume the following initial conditions

u ( x,0) = g1 ( x),

u t ( x,0) = g 2 ( x),

x ∈ (0,1),

(4)

and the Dirichlet boundary conditions

u (0, t ) = h1 (t ), u (1,0) = h2 (t ), where

g1 , g 2 , h1 and h2 are known functions and u ( x, t ) is the unknown function.

Note that at

α = 2,

Eq.(3) is the classical linear Klien-Gordon equation

(5)


3

On the Numerical Solution for the Linear Fractional Klien-Gordon Equation Using Legendre Pseudospectral Method

u tt ( x, t ) − d ( x)u xx ( x, t ) = f ( x, t ). Our idea is to apply the Legendre collocation method to discretize (3) to get a linear system of ODEs thus greatly simplifying the problem, and use finite difference method (Meerschaert & Tadjeran (2006), Smith (1965)) to solve the resulting system. Legendre polynomials are well known family of orthogonal polynomials on the interval

[−1,1] that have many

applications (Bell (1968), Khader et al. (2011), Khader & Hendy (2012a)). They are widely used because of their good properties in the approximation of functions. However, with our best knowledge, very little work was done to adapt these polynomials to the solution of FDEs. The organization of this paper is as follows. In the next section, the approximation of fractional derivative

D α y (x) is obtained. Section 3 summarizes the application of Legendre collocation method to the solution of (3). As a result a system of ordinary differential equations is formed and the solution of the considered problem is introduced. In section 4, some numerical results are given to clarify the proposed method. Also a conclusion is given in section 5.

AN APPROXIMATE FORMULA OF THE FRACTIONAL DERIVATIVE The well known Legendre polynomials are defined on the interval

[−1,1] and can be determined with the aid

of the following recurrence formula (Bell (1968))

L k +1 ( z ) = where

2k + 1 k z Lk ( z) − L k −1 ( z ), k +1 k +1

L0 ( z ) = 1 and L1 ( z ) = z . In order to use these polynomials on the interval [0,1] we define the so

called shifted Legendre polynomials by introducing the change of variable Let the shifted Legendre polynomials

Pk +1 ( x ) = where

k = 1,2,...,

z = 2 x − 1.

Lk (2 x − 1) be denoted by Pk (x) . Then Pk (x) can be obtained as follows

(2 k + 1)(2 x − 1) k Pk ( x ) − Pk −1 ( x ), ( k + 1) k +1

(6)

k = 1,2,...,

P0 ( x) = 1 and P1 ( x) = 2 x − 1 . The analytic form of the shifted Legendre polynomials Pk (x)

of degree

k

is

given by k

Pk ( x ) =

∑ ( − 1)

( k + i )! xi. ( k − i )( i! ) 2

k +i

i=0

Note that

(7)

Pk (0) = (−1) k and Pk (1) = 1 . The orthogonality condition is ∫

The function

1 0

P i ( x ) P j ( x ) dx

1   2i + 1 ,  =  0,  

for

i =

j;

for

i ≠

j

(8)

y (x) , which is square integrable in [0,1] , may be expressed in terms of shifted Legendre polynomials as ∞

y ( x) = ∑ yi Pi ( x), i=0 1

where the coefficients

y i are given by y i = (2i + 1) ∫ y ( x ) Pi ( x)dx,

In practice, only the first

0

i = 1,2,....

(m + 1) -terms of shifted Legendre polynomials are considered. Then we have


4

N. H. Sweilam, M. M. Khader & A. M. S. Mahdy

m

ym ( x) =

∑ y P ( x ). i

(9)

i

i=0

The main approximate formula of the fractional derivative is given in the following theorem.

THEOREM Let

y (x) be approximated by shifted Legendre polynomials as (9) and also suppose α > 0 then m

D α ( y m ( x)) =

i

∑α ∑α y w α x ( ) i ,k

i

k −α

,

(10)

i=   k =  

where

wi(,αk )

is given by

w i(,αk ) =

( − 1) ( i + k ) ( i + k )! . ( i − k )! ( k )! Γ ( k + 1 − α )

(11)

PROOF Since the Caputo's fractional differentiation is a linear operation we have D

α

m

( y m ( x )) =

yiD

α

(12)

( P i ( x )).

i=0

Employing Eqs.(1)-(2) in Eq.(7) we have

i = 0,1,..., α  − 1, α > 0.

D α Pi ( x) = 0,

(13)

Therefore, for i = α , α  + 1, ..., m and by using Eqs.(1)-(2) and (7) we get

D α Pi ( x ) =

i

∑ k=0

( − 1) i + k ( i + k )! α k D (x ) = ( i − k )! ( k ! ) 2

i

∑α

k=

( − 1) i + k ( i + k )! x k −α , ( i − k )! ( k ! ) Γ ( k + 1 − α )

(14)

a combination of Eqs.(12), (13) and (14) leads to the desired result.

PROCEDURE SOLUTION OF THE LINEAR FRACTIONAL KLIEN-GORDON EQUATION Consider the LFKGE of type given in Eq.(3). In order to use Legendre collocation method, we first approximate u ( x, t ) as m

um ( x, t ) = ∑ui (t ) Pi ( x).

(15)

i =0

From Eqs.(3), (15) and Theorem 1 we have m

∑ i=0

We now collocate Eq.(16) at m

m i d 2 u i (t ) Pi ( x ) − d ( x ) ∑ ∑ u i ( t ) w i(,αk ) x k − α = f ( x , t ). 2 dt i = α  k = α 

(m + 1 − α  )

points m

x p , p = 0,1,..., m − α 

∑u&&i (t ) Pi ( x p ) − d ( x p ) ∑ i=0

i

∑α u (t ) w α i

( ) i ,k

as

x kp−α = f ( x p , t ).

i = α  k =  

For suitable collocation points we use roots of shifted Legendre polynomial Also, by substituting Eq.(15) in the boundary conditions (5) we can obtain

(16)

Pm +1− α  ( x) .

α  equations as follows

(17)


On the Numerical Solution for the Linear Fractional Klien-Gordon Equation Using Legendre Pseudospectral Method

m

∑ ( − 1)

m

i

∑u

u i ( t ) = h1 ( t ),

i= 0

α 

Eq.(17), together with

i

5

(18)

( t ) = h 2 ( t ).

i= 0

equations of the boundary conditions (18), give

(m + 1)

of ordinary differential equations

which can be solved, for the unknowns

ui , i = 0,1,..., m,

using the finite difference method, as described in the following

section.

NUMERICAL RESULTS In this section, we implement the proposed method to solve LFKGE (3) with

u tt ( x, t ) − d ( x) D 1.8 u ( x, t ) = f ( x, t ), with the coefficient function: the source function:

α = 1.8 , of the form

0 < x < 1, t > 0,

d ( x ) = Γ (1.2) x1.8 ,

f ( x, t ) = x 2 (4 x − 1)e − t ,

under initial conditions and Dirichlet conditions

u ( x,0) = x 2 (1 − x), ut ( x,0) = x 2 ( x − 1),

u (0, t ) = u (1, t ) = 0.

The exact solution to this problem is

u ( x, t ) = x 2 (1 − x )e − t , which can be verified by applying the fractional

differential formula (2). We apply the proposed method with

m = 3 , and approximate the solution as follows 3

u3 ( x, t ) = ∑ui (t ) Pi ( x).

(19)

i =0

Using Eq.(17) we have 3

3

i

k −1.8 = f ( x p , t ), ∑u&&i (t ) Pi ( x p ) − d ( x p )∑∑u i (t )wi(1.8) ,k x p i=0

where

xp

p = 0,1,

(20)

i=2 k = 2

are roots of shifted Legendre polynomial

P2 ( x) , i.e. x0 = 0.211324, x1 = 0.788675.

By using Eqs.(18) and (20) we can obtain the following system of ODEs

u&&0 (t ) + k1 u&&1 (t ) + k 2 u&&3 (t ) − R1 u 2 (t ) − R2 u 3 (t ) = f 0 (t ),

(21)

u&&0 (t ) + k11 u&&1 (t ) + k 22 u&&3 (t ) − R11 u 2 (t ) − R22 u 3 (t ) = f1 (t ),

(22)

u0 (t ) − u1 (t ) + u2 (t ) − u3 (t ) = 0,

(23)

u0 (t ) + u1 (t ) + u2 (t ) + u3 (t ) = 0,

(24)

where

k1 = P1 ( x0 ),

k2 = P3 ( x0 ),

k11 = P1 ( x1 ),

k22 = P3 ( x1 ),

(1.8) 0.2 R1 = d ( x0 ) w2,2 x0 ,

(1.8) 0.2 (1.8) 1.2 R2 = d ( x0 )[ w3,2 x0 + w3,3 x0 ],

(1.8) 0.2 R11 = d ( x1 ) w2,2 x1 ,

(1.8) 0.2 (1.8) 1.2 R22 = d ( x1 )[ w3,2 x1 + w3,3 x1 ].

Now, to use FDM for solving the system (21)-(24), we use the notations ti

= iτ

to be the integration time 0 ≤ t i ≤ T ,


6

N. H. Sweilam, M. M. Khader & A. M. S. Mahdy

τ = T/N ,

uin = ui (tn ), f i n = f i (tn ).

for i = 0,1,..., N for an arbitrary integer number N . Define

Then the system (21)-(24), is discretized in time and takes the following form

u 0n +1 − 2 u 0n + u 0n −1

τ

+ k1

2

u 0n +1 − 2u 0n + u 0n −1

τ2

u 1n +1 − 2 u 1n + u 1n −1

τ

+ k 11

+ k2

2

u1n +1 − 2u1n + u1n −1

τ2

+ k 22

u 3n +1 − 2 u 3n + u 3n −1

τ2 u 3n +1 − 2u 3n + u 3n −1

τ2

(25)

− R1 u 2n +1 − R 2 u 3n +1 = f 0n +1 ,

− R11 u 2n +1 − R 22 u 3n +1 = f 1n +1 ,

(26)

u0n+1 − u1n +1 + u 2n +1 − u3n+1 = 0,

(27)

u0n+1 + u1n+1 + u2n+1 + u3n+1 = 0.

(28)

We can write the above system (25)-(28) in the following matrix form 1  1 1  1   2  2 0  0  

k1

−τ

k 11

−τ

2 2

−1

1

1

1

2k1

R1 R 11

0

2k2

2 k 11

0

2 k 22

0

0

0

0

0

0

R 2  u 0    k 22 − τ 2 R 22   u 1   u  −1  2   u 3  1     n k1  u 0  1     u 1   1 k 11  u  −  0 0 2    0  u 3  0       k2 − τ

2

n +1

(29)

=

k 2  u 0  k 22   u 1 0  u 2  0  u 3  

0 0 0 0

       

n −1

τ 2 f 0  2  τ f1 +  0   0  

       

n +1

.

We use the notation for the above system

AU n +1 = BU n − CU n −1 + F n +1 , or , U n +1 = A −1 BU n − A −1CU n −1 + A −1 F n +1 , where

(30)

U n = (u 0n ,u1n ,u 2n ,u 3n ) T and F n = (τ 2 f 0n , τ 2 f 1n ,0,0) T . The obtained numerical results by means of the proposed method are shown in Table 1 and figures 1-2. In

Table 1, the absolute errors between the exact solution with the final time solution at

T =2

uex

and the approximate solution

uapprox

at

m=3

and

m=5

are given. Also, in figures 1 and 2, comparison between the exact solution and the approximate

T = 1 with time step τ = 0.0025, m = 3

and

m = 5 , respectively, are presented.

Table 1: The Absolute Error between the Exact Solution and the Approximate Solution at at m = 3 , m = 5 and T = 2

x

u ex − u approx

at m = 3

u ex − u approx

at m = 5

0.0

4.483787e-03

2.726496e-04

0.1

4.479660e-03

3.455890e-04

0.2

4.201329e-03

3.809670e-04

0.3

3.695172e-03

3.809103e-04

0.4

3.007566e-03

3.514280e-04

0.5

2.184889e-03

3.009263e-04

0.6

1.273510e-03

2.387121e-04

0.7

0.319831e-03

1.735125e-04

0.8

0.629793e-03

1.119821e-04

0.9

1.528978e-03

0.572150e-04

1.0

2.331347e-03

0.072566e-04


7

On the Numerical Solution for the Linear Fractional Klien-Gordon Equation Using Legendre Pseudospectral Method

Figure 1: Comparison between the Exact Solution and the Approximate Solution at

T = 1 with Ď&#x201E; = 0.0025 and

m=3

Figure 2: Comparison between the Exact Solution and the Approximate Solution at T = 1 with

Ď&#x201E; = 0.0025

and

m=5 CONCLUSIONS In this paper, we considered the numerical solutions for linear fractional Klein-Gordon equation using the application of Legendre pseudo-spectral method. The properties of the Legendre polynomials are used to reduce the proposed problem to the solution of system of ordinary differential equations which solved by using FDM. The fractional derivative is considered in the Caputo sense. The solutions obtained using the suggested method are in excellent agreement with the exact solution and show that this approach can be solved the problem effectively. Although we only considered a model problem in this paper, the main idea and the used techniques in this work are also applicable to many other problems. It is evident that the overall errors can be made smaller by adding new terms from the series (15). Comparisons are made between the approximate solution and the exact solution to illustrate the validity and the great potential of the technique. All computations in this paper are done using Matlab 8.

REFERENCES 1.

Bagley, R. L., & Torvik, P. J. (1984). On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51, 294-298


8

N. H. Sweilam, M. M. Khader & A. M. S. Mahdy

2.

Bell, W. W. (1968). Special Functions for Scientists and Engineers, Great Britain, Butler & Tanner Ltd, Frome and London

3.

Das, S. (2008). Functional Fractional Calculus for System Identification and Controls, Springer, New York.

4.

Dehghan, M. & Shokri, A. (2009). Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, Journal of Computational and Applied Mathematics, 230, 400-410

5.

El-Sayed, S. M. (2003). The decomposition method for studying the Klein-Gordon equation, Chaos, Solitons & Fractals, 18(5), 1026-1030

6.

El-Sayed, A. M. A., Elsaid, A. & Hammad, D. (2012). A reliable treatment of homotopy perturbation method for solving the nonlinear Klein-Gordon equation of arbitrary (fractional) orders, Journal of Applied Mathematics, 2012, 1-13

7.

El-Sayed, A. M. A., Elsaid, A., El-Kalla, I. L. & Hammad, D. (2012). A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains, Applied Mathematics & Computation, 218, 8329-8340

8.

Golmankhaneh, A. K., Golmankhaneh, A. K. & Baleanu, D. (2011). On nonlinear fractional Klein-Gordon equation, Signal Processing, 91, 446-451

9.

Hashim, I., Abdulaziz, O. & Momani, S. (2009). Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 14, 674-684

10. He, J. H. (1998). Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167(1-2), 57-68 11. Inc, M. (2008). The approximate and exact solutions of the space-and time-fractional Burger's equations with initial conditions by VIM, J. Math. Anal. Appl., 345, 476-484 12. Jafari, H. & Daftardar-Gejji, V. (2006). Solving linear and nonlinear fractional diffusion and wave equations by ADM, Appl. Math. & Comput., 180, 488-497 13. Khader, M. M. (2011). On the numerical solutions for the fractional diffusion equation, Communications in Nonlinear Science & Numerical Simulations, 16, 2535-2542 14. Khader, M. M. (2012). Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials. Arab Journal of Mathematical Sciences, 18, 61-71 15. Khader, M. M., Sweilam, N. H., & Mahdy, A. M. S. (2011). An efficient numerical method for solving the fractional diffusion equation. J. of Applied Mathematics & Bioinformatics, 1, 1-12 16. Khader, M. M., & Hendy, A. S. (2012). The approximate and exact solutions of the fractional order delay differential equations using Legendre pseudospectral method. International Journal of Pure & Applied Mathematics, 74(3), 287-297 17. Khader, M. M. & Hendy, A. S. (2012). A numerical technique for solving fractional variational problems. Accepted in Mathematical Methods in Applied Sciences 18. Lakestani, M. & Dehghan, M. (2010). Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation. Computer Physics Communications, 181, 1392-1401


On the Numerical Solution for the Linear Fractional Klien-Gordon Equation Using Legendre Pseudospectral Method

9

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