International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 3 Sep 2012 70-75 © TJPRC Pvt. Ltd.,
SOLVING HELMHOLTZ EQUATION BY THE HOMOTOPY PERTURBATION TRANSFORM METHOD R.RAJARAMAN & G.HARIHARAN Department of Mathematics, School of Humanities & Sciences, SASTRA University, Thanjavur,Tamilnadu, India
ABSTRACT In this paper, Homotopy Perturbation transform Method (HPTM) is used for analytic treatment of the Helmholtz equation. This method is the combined form of Homotopy perturbation method and Laplace transform method. The Nonlinear terms can be easily decomposed by use of He’s polynomials. This method can provide analytical solutions to the problems by just utilizing the initial conditions. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. The proposed method solves nonlinear problems without using Adomain polynomials which is the advantage of this method over Adomain Decomposition method. The results reveal that the HPTM is very effective, simple, convenient, flexible and accurate.
KEYWORDS: Homotopy perturbation method, Lapalce transform method, He’s polynomials, Helmholtz equation. INTRODUCTION Many engineering problems related to steady-state oscillations (mechanical, acoustical, thermal, electromagnetic) lead to the two-dimensional Helmholtz equation. For
λ <0,
the equation
describes mass transfere processes with volume chemical reactions of the first order. The two-dimensional Helmholtz equation has the following form
∂ 2ω ∂ 2ω + + λω = −Φ ( X , Y ) ∂X 2 ∂Y 2
(1)
Many phenomena in real world are essentially nonlinear and are described by nonlinear equations. It is still difficult to obtain accurate solutions of nonlinear problems and often more difficult to get an analytical approximation than a numerical one of a given nonlinear problem. Recently various iterative methods are employed for the numerical and analytical solutions of Linear and Nonlinear partial differential equations. Also some iterative methods are applied for solving Helmholtz equations [1, 2, 3]. In this paper the Homotopy perturbation transform method [15] is applied to solve telegraph equations. . In recent years this method has been successfully employed to solve many types of nonlinear homogeneous or nonhomogeneous partial differential equations. The HPTM has certain advantages over
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Solving Helmholtz Equation by the Homotopy Perturbation Transform Method
routine numerical methods. Numerical methods use discretization which gives rise to rounding off errors causes loss of accuracy and requires large computer memory and time. This computational method yields analytical solutions and is effective and accurate than standard numerical methods. The HTPM method does not involve discretization of the variables and hence free from rounding off errors and does not require large computer memory or time. Recently various methods are proposed to solve nonlinear partial differential equations such as Adomain docomposition method (ADM) iteration method (VIM)
[4,5,6] .
Variational
[7,8,9] and Differential transform method [13,14] .Most of these methods have
their inbuilt deficiencies like the calculation of Adomain polynomials, the Lagrange multiplier, divergent results and huge computational work. He developed Homotopy perturbation method (HPM)
[10,11,12]
by merging standard homotopy and perturbation for solving various physical problems. The Laplace transform is totally incapable of handling nonlinear equations because of the difficulties that are caused by nonlinear terms. To overcome the deficiencies. Homotopy perturbation method is combined with Laplace transform method to produce highly effective technique to deal with these nonlinearities.The suggested HPTM provides the solution in a rapid convergence series which may lead the solution in closed form. Also very accurate results are obtained in a wide range via one or two iteration steps.
HOMOTOPY PERTURBATION TRANSFORM METHOD (HPTM) To illustrate the basic idea of the method, we consider a general non-homogeneous partial differential equation with initial conditions of the form
Du ( x, t ) + Ru ( x, t ) + Nu ( x, t ) = g ( x, t )
(2)
u ( x,0) = h( x), ut ( x,0) = f ( x)
(3)
D= where D is the second order linear differential operator
∂2 ∂t 2 , R is the linear differential
operator of less order than D, N represents the general non-linear differential operator and g(x,t) is the source term. Taking the Laplace transform denoted by L on both sides of Eq. (1):
L ( Du ( x, t )) + L ( Ru ( x, t )) + L ( Nu ( x, t )) = L ( g ( x, t ))
(4)
Using the differentiation property of Laplace transform, we have
h( x ) f ( x ) 1 1 1 + 2 − 2 L( Ru( x, t )) + 2 L( g ( x, t )) − 2 L( Nu ( x, t )) L (u ( x, t )) = s s s s s (5)
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R.Rajaraman & G.Hariharan
Operating with Laplace inverse on both sides of Eq. (3) gives
u ( x, t ) = G( x, t ) − L−1 (
1 L( Ru( x, t + Nu ( x, t ))) s2
(6)
Where G(x,t) represent the term arising from the source term and the prescribed initial conditions. Now we apply Homotopy perturbation method ∝
∑p
n
u ( x, t ) = n=0
un ( x, t )
(7)
And the nonlinear term can be decomposed as ∝
Nu ( x, t ) = ∑ p n H n (u ), n =0
(8)
For some He’s polynomials that are given by
H n (u0 .......un ) =
1 ∂n n ! ∂p n
∝ i ( N ∑ ( p ui )) i =0
p =0
, n=0, 1, 2, 3,…
(9)
Substituting Eqs.(6) and (5) in Eqs.(2) and (3), we get ∝
∑ p u ( x, t ) = G ( x , t ) − p ( L
−1
n
n =0
n
1 2 s
∝ ∝ L R ∑ p nu n ( x, t ) + ∑ p n H n (u ) ) n =0 n =0
(10) Which is the coupling of the Laplace transform and the homotopy perturbation method using He’s polynomials. Comparing the coefficient of like powers of p, the following approximations are obtained.
p 0 : u0 ( x, t ) = G ( x, t ),
1 L [ Ru0 ( x, t ) + H 0 (u ) ] , 2 s 1 p 2 : u2 ( x, t ) = − 2 L [ Ru1 ( x, t ) + H1 (u ) ] s 1 3 p : u3 ( x, t ) = − 2 L [ Ru2 ( x, t ) + H 2 (u )] s
(11)
p1 : u1 ( x, t ) = −
….
(12)
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Solving Helmholtz Equation by the Homotopy Perturbation Transform Method
APPLICATIONS In this section, the exact solutions of some Helmholtz equations are found out using Homotopy perturbation method. Example 3.1 Consider a special case of Helmholtz equation
∂ 2 u ( x, y ) ∂ 2 u ( x, y ) + − u ( x, y ) = 0 ∂x 2 ∂y 2 (13) with initial conditions u(0,y)=y, ux(0,y)=y+sinhy
(14)
We apply Homotopy perturbation method in x-direction. Therefore we set the second order linear differential operator D=∂2/∂x2 By applying Homotopy perturbation transform method, we get ∝ n n n n −1 1 p u ( x , t ) = x sinh y + y ( 1 + x ) − pL L 2 ∑ ( p un ( x, t ))xx + ( p un ( x, t )) yy − p un ( x, t ) ∑0 n s 0 ∝
(15) u0(x,y)=xsinhy+y(1+x)
(16)
u1(x,y)=xsinhy+y(1+x+x2/2!+x3/3!)
(17)
……… The exact solution is un(x,y)= xsinhy+y(1+x+x2/2!+x3/3!+x4/4!+x5/5!+………………..) =xsinhy+yex
(18)
Example 3.2 We consider a special case of Helmholtz equation The approximation can also be obtained by y-direction.
∂ 2 u ( x, y ) ∂ 2 u ( x, y ) + + 5u ( x, y ) = 0 ∂x 2 ∂y 2 (19) with initial conditions
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R.Rajaraman & G.Hariharan
u(x,y)=0, ux(x,y)=3cosh(2y) ∝
∑ p u ( x, t ) = 3x cosh(2 y) − pL
−1
n
n
0
(20)
1 ∝ n n n 2 L ∑ ( p un ( x, t )) xx + ( p un ( x, t )) yy + 5 p un ( x, t ) s 0
(21) u0(x,y)=3xcosh(2y)
(23)
u1(x,y)=cosh(2y)(3x-(3x)3/3!
(24)
………… The exact solution is un(x,y)= cosh(2y)(3x-(3x)3/3!+(3x)5/5!-..........) =cosh(2y) sin3x
(25)
The approximation can also be obtained by y-direction.
CONCLUSIONS In this paper exact solutions for different types of the Helmholtz equation have been established. The Homotopy perturbation transform method (HPTM) is successfully used to develop these solutions. This work shows that HPTM has significant advantages over the existing techniques. It avoids the need for calculating the Adomain polynomials which can be difficult in some cases. The reliability of the method and reduction in the size of computational domain give this method wider applicability. The results show that HPTM is a powerful mathematical tool for finding the exact and approximate solutions of the nonlinear equation.
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Solving Helmholtz Equation by the Homotopy Perturbation Transform Method
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