International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email:
[email protected],
[email protected] Volume 1, Issue 4, November – December 2012 ISSN 2278-6856
Homotopy Perturbation Transform method for solving Klein-Gordan equations R.Rajaraman1, G.Hariharan2 and K.Kannan3 1,2,3
Department of Mathematics, School of Humanities & Sciences SASTRA University, Thanjavur-613 401, Tamilnadu, India
Abstract: In this work Homotopy Perturbation transform Method (HPTM) is used to solve some of the Klein-Gordan equations .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 HPTM provides the solution in a rapid convergent series which may lead the solution in a closed form. The results reveal that the results reveal that the HPTM is very effective, simple, convenient, flexible and accurate. Outcomes prove that HPTM is in very good agreement with ADM, VIM and HPM.
Keywords: Homotopy perturbation method, Lapalce transform method, He’s polynomials, Klein-Gordan equations
1. INTRODUCTION Nonlinear phenomena have important effects on applied mathematics, physics and issues related to engineering. Many such physical phenomena are modeled in terms of nonlinear partial differential equations. The importance of obtaining the exact or approximate solutions of nonlinear partial differential equations in physics and mathematics is still a significant problem that needs new methods to discover exact or approximate solutions. Recently various iterative methods are employed for the numerical and analytical solutions of Linear and Nonlinear partial differential equations. In this paper the Homotopy perturbation transform method 1, 2,3, 4 is applied to solve Klein-Gordan equations. We consider the general form of Klein-Gordan equation
which gives rise to rounding off error 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) 5,6,7 .
Variational
iteration
method
(VIM) 8,9,10,14 ,
Differential transform method 15,16,17 etc. 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) 11,12,13,14 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.
2. HOMOTOPY PERTURBATION METHOD (HPTM)
TRANSFORM
2u 2u b1u b2 g (u ) f ( x, t ) t 2 x 2
To illustrate the basic idea of the method, we consider a general non-homogeneous partial differential equation with initial conditions of the form
(1) where u is the function of x and t,g(u) is the nonlinear function and f(x,t) is the known analytic function. The Klein-Gordan equations model many problems in classical and quantum mechanics, solitons and condensed matter physics. In recent years HPTM has been successfully employed to solve many types of nonlinear homogeneous or non-homogeneous partial differential equations. The HPTM has certain advantages over routine numerical methods. Numerical methods use discretization
Du ( x, t ) Ru ( x, t ) Nu ( x, t ) g ( x, t )
Volume 1, Issue 4 November - December 2012
(2)
u ( x,0) h( x), ut ( x,0) f ( x) where D is the second order linear differential operator
D
2 ,R is the linear differential operator of less t 2
order than D, N represents the general non-linear differential operator and g(x,t) is the source term. Taking Page 150
International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email:
[email protected],
[email protected] Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 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)) (3) Using the differentiation property of Laplace transform, we have
ETD EE …… 10) so on.
L(u ( x, t )) h( x) f ( x) 1 2 2 L( Ru ( x, t )) s s s 1 1 2 L( g ( x, t )) 2 L( Nu ( x, t )) s s
3. APPLICATIONS transform method. Example:3.1 Consider the Klein-Gordon equation
(4) Operating with Laplace inverse on both sides of Eq (3) gives
1 u ( x, t ) G ( x, t ) L ( 2 L( Ru ( x, t Nu ( x, t ))) s 1
(5) where G(x,t) represent the term arising from the source term and the prescribed initial conditions. Now we apply Homotopy perturbation method
u ( x, t ) = p n u n ( x , t )
2 u 2u u t 2 x 2 (11) Subject to the initial conditions u(x,0)=0 ut(x,0)=1+cosx We consider the initial condition u0=t+cosxt By applying aforesaid method we have n
p u ( x, t ) n
0
1 1 t t cos x pL 2 s
n0
(6) And the nonlinear term can be decomposed as
n n L (( p un ( x, t )) xx p un ( x, t )) 0
u1(x,t)=tcosx+t+t3/3!
Nu ( x, t ) p n H n (u ),
(12) u2(x,t)=tcosx+t+t3/3!+t5/5!
n 0
(7) For some He’s polynomials-that are given by
1 n H n (u0 .......un ) n! p n
1 LRu1 ( x, t ) H 1 (u ) s2 1 p 3 : u3 ( x, t ) 2 LRu2 ( x, t ) H 2 (u ) s p 2 : u 2 ( x, t )
i N ( ( p ui )) i0
p 0
n=0,1,2,3….. (8) Substituting Eqs.(6) and (5) in Eq.(4) we get
pn un ( x, t ) n 0
1 G ( x, t ) p ( L1 2 L R p n un ( x, t ) p n H n (u ) ) n 0 s n 0
(9) 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 p1 : u1 ( x, t ) 2 LRu0 ( x, t ) H 0 (u ) , s
........... The final solution is
n
p u ( x, t ) n
0
x
= e t 1 pL1 2 s
n n n L (( p un ( x, t )) xx p H n (u ) p An (u )) 0
u(x,t)= tcosx+t+t3/3!+t5/5!+t7/7!+........ u(x,t)=cosxt+sinht (13) Example:3.2 Now we consider homogeneous Klein-Gordan equation
another
linear
2u 2u u x 2u t 2 x 2 (14)
Volume 1, Issue 4 November - December 2012
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International Journal of Emerging Trends & Technology in Computer Science (IJETTCS) Web Site: www.ijettcs.org Email:
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[email protected] Volume 1, Issue 4, November – December 2012 ISSN 2278-6856 u(x,t)=ex(t+t3/3!+t5/5!+........................) Subject to initial conditions =exsinht x
u(x,0)=e , ut(x,0)=0
(17)
Now we consider the u0=ex
Example:3.4 Now we consider inhomogeneous Klein-Gordan equation
linear
By applying aforesaid method
2 u 2u u 2 sin x t 2 x 2
n
p u ( x, t ) n
(18)
0
1 e x pL1 2 L (( pn un ( x, t )) xx ( p nun ( x, t )) x p n 2u n ( x, t )) s 0
Subject to the initial conditions u(x,0)=sinx, ut(x,0)=1
x
x
2
(x,t)=e +esuccessive (2t) /2! approximations heu1following u0(x,t)=ex u2(x,t)= ex+ex(2t)2/2!+ex(2t)4/4!
Now we can select u0(x,0)=sinx+t By applying aforesaid method we have
............. The final solution is
n
p u ( x, t ) n
x
2
0
4
u(x,t)=e (1+(2t) /2!+(2t) /4!+........................)
1 sin x t pL1 2 L(( pnun (x,t))xx pnun (x,t) 2t sin x) s 0
=excosh2t (15) Example:3.3 Now we consider nonlinear homogeneous Klein-Gordan equation
2 u 2u 2 2 ux u2 2 t x
u0(x,t)=sinx+tu1(x,t)=sinx+t-t3/3! of ‘p’ we get tollowing successive approximations u2(x,t)= sinx+t-t3/3!+t5/5! ..................... The final solution is
(16)
u(x,t)=sinx+t-t3/3!+t5/5!t /7!+............................ 7
Subject to the initial conditions u(x,0)=0, ut(x,0)=ex Now we can select u0=ext By applying aforesaid method n
p u ( x, t ) n
=sinx+sint (19) Example:3.5 Now we consider another linear inhomogeneous Klein-Gordan equation
2u 2u sin x , 0
