HYBRIDIZATION OF HOMOTOPY PERTURBATION METHOD AND LAPLACE TRANSFORMATION FOR THE PARTIAL DIFFERENTIAL EQUATIONS 1
Zhen-jiang LIU1,M. Y. ADAMU2 and E. SULEIMAN2, Ji-Huan HE3* College of Mathematics and Statistics, Qujing Normal University, Qujing 655011, PR China
2.
Department of Mathematical Sciences, Abubakar Tafawa Balewa, University, Bauchi. Nigeria
3
National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, 215123, China Corresponding Author. Email:
[email protected] Homotopy perturbation method (HPM) is combined with Laplace transformation to obtain approximate analytical solutions of nonlinear differential equations. An example is given to elucidate the solution process and confirm reliability of the method. The result indicates superiority of the method over the conventional HPM due its flexibility in choosing its initial approximation. Key words: HPM, Laplace Transformation, He-Laplace method, Nonlinear Differential Equations.
Introduction The last two decades have experienced a rapid development of non-linear sciences arising in thermal science and other fields as well. Recently, several analytical methods were developed in order to find solutions of nonlinear partial differential equations (PDEs), for examples, the homotopy perturbation method [1,2], the variational iteration method [3-5], the exp-function method [6-8], the homotopy analysis method [9,10], and others [11,12]. The classic approach by the perturbation method is still widely used for weakly nonlinear equations and appeared in many textbooks. The drawback of the perturbation method is that their choice of a small parameter seems to be an art rather than a solution procedure, an inappropriate choice of such parameter leads to an inaccurate result or even a wrong one. The shortcoming can be easily solved by the homotopy perturbation method (HPM), which was proposed in later 1990s [1,2], and the method has been developed into a mature tool for almost all kinds of nonlinear equations. Laplace transform is accessible to all students, but it is suitable only for linear equations. Gondal and Khan[13] first coupled the homotopy perturbation method with the Laplace transform, and the coupled technology is widely used in nonlinear differential equations[14] and fractional calculus[11], and the method is generally called as He-Laplace method[15]. He-Laplace method[15] To illustrate the basic idea of this method, we consider the general form of one-dimensional non-homogeneous partial differential equations with a variable coefficient of the form 321
u 2u ( x) 2 ( x, t ) t x
(1)
2u 2u ( x) 2 ( x, t ) 2 t x
(2)
and
subject to the boundary conditions (3) u (0, t ) g 0 (t ), u (1, t ) g1 (t ). and the initial condition (4) u ( x , 0) f ( x ). The methodology consists of applying transform on both sides of equations (1) and (3) and in view of the initial condition[13-15]: d 2U su ( x, s) ( x, s) f ( x) (5) 0 dx 2 (x) ( x)
u (0, s) g0 ( s),
u (1, s) g1 ( s )
(6)
which is second-order boundary value problem. According to HPM[1,2], we construct a homotopy in the form d 2v d 2 u 0 d 2v sv ( x, s) f ( x) (7) H ( v , p ) (1 p ) 2 p 2 0 2 dx ( x) ( x) dx dx where
is the arbitrary function that satisfies boundary conditions equation (6), therefore
v( x, s)
8
p v ( x, s) v i
i
i0
0
( x , s ) p 1v1 ( x , s ) p 2 v 2 ( x , s ) ...,
(8)
Applying the inverse Laplace transform on both sides of equation (8), we get
v( x, t)
8
i0
Setting
p i v i ( x , s ) v 0 ( x , t ) p 1 v1 ( x , t ) p 2 v 2 ( x , t ) ...,
(9)
gives the approximate solution of equation (9)
u ( x, t ) u0 ( x, t ) u1 ( x, t ) u 2 ( x, t ) ...,
(10)
Implementation In order to ascertain the wider applicability of the method we consider a nonlinear differential equation of the form (11) as an example:
With the initial condition:
dU d 2U ( ae t e at ) dt dx 2
(11)
U ( x , 0) sin( x )
(12)
and boundary U (0, t ) 0, U x ( e t e at ) By applying the method subject to the initial condition, we have 1 d 2U a ( 2 1 a s )U 2 sin( x ) 0 2 dx s 1 s a 1 1 U (0, s ) 0, U x (0, s ) s 1 s a To solve equation (14) by means of HPM, a homotopy equation can easily be constructed as follows d 2V d 2 U 0 d 2V a 1 H (V , P ) (1 P ) 2 P ( 2 1 a s )V 2 sin( x ) 2 2 dx s 1 s a dx dx
We assume that the solution to equation (16) may be written as a power series in 322
(13) (14) (15)
(16)
P V ( x, s ) V
V ( x, s )
i
i0
i
0
(17)
( x , s ) P 1V1 ( x , s ) P 2V 2 ( x , s ) ....
Substituting equation (17) into (16) and equating the terms with the identical powers of
we have
0 d 2V d 2 U 0 1 1 0, V0 (0, s ) 0, V0 x (0, s ) p : 2 2 dx dx s 1 s a 1 d 2V1 a 1 2 2 V1 (0, s ) 0,V1x (0, s ) 0 p : 2 V0 ( 1 a s) 2sin x(1 ) dx s 1 s a 0 ' 2 2 d V2 ( 2 1 a s )V2 0, V2 (0, s ) 0,V2 x (0, s ) 0 p : dx 2 2 p 3 : d V3 ( 2 1 a s )V 0, V (0, s ) 0, V (0, s ) 0 2 3 3x dx 2
(18)
The initial approximation V0 ( x, s ) or U 0 ( x, s ) can freely be chosen, and we can set
V0 ( x, s ) U 0 ( x, s )
sin x sin x s 1 sa
(19)
which satisfies the boundary conditions (15) Substituting equation (19) into equation (18), the first component of the homotopy perturbation solution for equation (14) are derived as follows: sin x 1 1 sin x ax 2 x4 2 2 ( )( 1 ) (1 ) v1 ( x, s ) ( 2 1 a s ) a s 4 2 (20) 24( s 1) 24( s a ) s 1 s a 1 1 1 1 x ( 2 1 a s ) 2 ( )( 2 1 a s ) 2 (1 2 ) s 1 s a
By taking the inverse Laplace of equation (20) yields: v0 ( x, t ) e t sin x e at sin x v1 ( x , y )
sin x
(1 2 )
2 which is the exact solution.
ax 2 t x 2 at sin x sin x e e ( 2 2 a ) e t ( 2 1) e at 2 2 2 2
Conclusion In this work we employed He-Laplace method[15] to find the exact solutions of some nonlinear partial differential equations. The method is very easy to implement due to its flexibility in freely chosen its initial approximation. References [1]He, J. H., Homotopy Perturbation Technique, Compt. Appl. Mech. Engrg. 178(1999), 3-4, 257-262. [2] He, J. H., A Coupling Method of Homotopy Technique and A Perturbation Technique For Non-Linear Problems, International Journal Of Non Linear Mechanics 35 (2000), 35, 37-43 [3] He, J.H., Variational iteration method - a kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics, 34 (1999),4, 699-708 [4] Lu, J.F., Ma, L. Analytical approach to a generalized Hirota-Satsuma coupled Korteweg-de Vries equation by modified variational iteration method, Thermal Science, 20(2016),3,885-888 [5]Lu, J.F., Modified variational iteration method for variant Boussinesq equation, Thermal Science, 19(2015),4,1195-1199 [6] He, J. H. and Wu, X. H., Exp-function method for nonlinear wave equations. Chaos Solitons and Fractals 30(2006), 3, 700-708 [7] Güner, Ö., Bekir, A. Exp-function method for nonlinear fractional differential equations, Nonlinear 323
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Paper submitted: July 15, 2016 Paper revised: August 26, 2016 Paper accepted: Sept. 4, 2016
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