Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He’s Polynomial

Dinkar Sharma; Prince Singh; Shubha Chauhan

doi:10.1515/nleng-2016-0011

Article Open Access

Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He’s Polynomial

Dinkar Sharma , Prince Singh and Shubha Chauhan

Published/Copyright: January 27, 2017

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From the journal Nonlinear Engineering Volume 6 Issue 2

Abstract

In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is applied to solve nonlinear fifth order Korteweg de Vries (KdV) equations. The method is known as homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He’s polynomials. Two test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM).

Keywords: Homotopy perturbation method; Laplace transform method; Non-linear partial differential equations; He’s polynomials

1 Introduction

Nonlinear phenomena play a crucial role in applied mathematics, physics and issues related to engineering. It is very difficult to solve nonlinear problems and it is often more difficult to find an analytic solution. Various methods were proposed to find approximate solutions of nonlinear equations δ-expansion method [1], Hirota bilinear techniques [2], Lyapunov’s small parameter method [3], perturbation techniques [4–6], He’s Semi-inverse method [7], variational iteration method (VIM) [8–15], Adomain decomposition method (ADM) [16, 17], Laplace decomposition method [18], sinh-cosh method, pseudospectral method, spectral collacation method, differential quadrature method [19–21] wavelet method [22, 23] etc . He developed the homotopy perturbation method (HPM) by combining the homotopy in topology and classical perturbation techniques, which has been applied to solve many linear and nonlinear differential equations [24–41]. In the recent years, the homotopy perturbation method is combined with Laplace transformation method and the variational iteration method to produce a highly effective technique for handling nonlinear terms is known as homotopy perturbation transform method (HPTM). This method provides the solution in rapid convergent series which leads the solution in a closed form. The use of He’s polynomials in the nonlinear terms was first introduced by Ghorbani [42, 43] . Later on many researcher use homotopy perturbation transform method for different type of linear and nonlinear differential equations [44–51].

In this paper, HPTM is applied to find the solution of fifth order KdV equation. This equation was first introduced by Korteweg and de Vries (1895). This equation has many applications to macroscopic phenomena and used to describe a large number of physical phenomena, for examples: shallow water waves, ion acoustic plasma waves, bubble-liquid mixtures and wave phenomena in enharmonic crystals. Many researcher use different techniques to solve the KdV equation [52–58]. Present study shows that HPTM is highly efficient for solving nonlinear equations.

2 Homotopy perturbation transform method

To illustrate the basic idea of this method, we consider a general form of the fifth-order KdV equation:

ut+Au2ux+Buxuxx+Cuuxxx+Duxxxxx=0(1)

subjected to the initial condition

u(x,0)=h(x)(2)

The above equation is known as Lax’s fifth order KdV equation for A = 30, B = 30, C = 10, D = 1 and is known as Sawada-Kotera equation with A = 45, B = 15, C = 15, D = 1 [58]. Taking the Laplace transform on both sides of Eq. (1):

L[ut]=−L[Nu(x,t)]+L[Ru(x,t)](3)

Where N represents the general non-linear differential operator, R is the linear differential operator. Using the differentiation property of the Laplace transform, we have

L[u(x,t)]=h(x)s−1sL[Nu(x,t)]−1sL[Ru(x,t)](4)

Taking Laplace inverse on both sides

u(x,t)=G(x,t)−L−11sL[Nu(x,t)]+1sL[Ru(x,t)](5)

where G(x, t) represents the term arising from the prescribed initial condition. Now, we apply the homotopy perturbation method

u(x,t)=∑n=0∞pnun(x,t)(6)

and the nonlinear term can be decomposed as

Nu(x,t)=∑n=0∞pnHn(u)(7)

for some He’s polynomials H_n that are given by

Hn(u0,...,un)=1n!∂n∂pnN∑i=0∞(piui)p=0,(8)

n=0,1,2,3...(9)

Substituting Eqs. (6) and (7)in Eq. (5), we get

∑n=0∞pnun(x,t)=G(x,t)−pL−11sLR∑n=0∞pnun(x,t)+∑n=0∞pnHn(u)(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

p0:u0(x,t)=G(x,t),p1:u1(x,t)=−L−11sL[Ru0(x,t)+H0(u)],p2:u2(x,t)=−L−11sL[Ru1(x,t)+H1(u)],p3:u3(x,t)=−L−11sL[Ru2(x,t)+H2(u)],⋮(11)

3 Application

In order to educidate the solution procedure of the homotopy perturbation transform method, we consider the following two examples:

Example 3.1

Consider the Sawada Kotera Equation, given by

ut+45u2ux+15uxuxx+15uuxxx+uxxxxx=0(12)

Subject to the initial condition

u(x,0)=2k2sech2(kx)(13)

By applying the aforesaid method subject to the initial condition, we get

u(x,s)=1s(2k2sech2(kx))−1sLuxxxxx+15uuxxx+15uxuxx+45u2ux(14)

The inverse Laplace transform implies that

u(x,t)=2k2sech2(kx)−L−11sLuxxxxx+15uuxxx+15uxuxx+45u2ux(15)

Now, we apply the homotopy perturbation method

∑n=0∞pnun(x,t)=2k2sech2(kx)−pL−11sL(∑n=0∞pnun(x,t))xxxxx+1sL∑n=0∞pnHn(u)(16)

where H_n(u) are He’s polynomials that represent the nonlinear terms. The first few components of He’s polynomials, for example, are given by

H0(u)=15u0u0xxx+15u0xu0xx+45u02u0xH1(u)=15(u1u0xxx+u1xxxu0)+15(u0xu1xx+u1xu0xx)+45(2u0u1u0x+u02u1x)

H2(u)=15(u0u2xxx+u1u1xxx+u2u0xxx)+15(u0xu2xx+u1xu1xx+u2xu0xx)(17)

+45(2u1u2+2u0u3)u0x+(u12+2u0u2)u1x+2u0u1u2x+u02u3x(18)

Comparing the coefficient of like powers of p, we have

p0:u0(x,t)=2k2sech2(kx)

p1:u1(x,t)=64k7tanh⁡(kx)sech2(kx)t

p2:u2(x,t)=−512k12sech4(kx)3−2cosh2⁡(kx)t2⋮

Therefore, solution of above problem when p → 1 is:

u(x,t)=2k2sech2(kx)+64k7tanh⁡(kx)sech2(kx)t−512k12sech4(kx)3−2cosh2⁡(kx)t2+…(19)

Using Taylor series, the closed form solution is as follows:

u(x,t)=2k2sech2kx−16k5t(20)

which is the same as that obtained by ADM and HPM [54].

Example 3.2

We now consider the Lax’s fifth order Equation

ut+30u2ux+30uxuxx+10uuxxx+uxxxxx=0(21)

Subject to the initial condition

u(x,0)=2k23sech2(kx)−1(22)

By applying the aforesaid method subject to the initial condition, we get

u(x,s)=1s[u(x,0)]−1sLN(u(x,t))−1sLuxxxxx(23)

where N(u(x,t))=30u2ux+30uxuxx+10uuxxx

The inverse Laplace transform implies that

u(x,t)=u(x,0)−L−11sL[N(u(x,t))]+1sL[uxxxxx](24)

Now, we apply the homotopy perturbation method

∑n=0∞pnun(x,t)=u(x,0)−pL−11sL∑n=0∞pnHn(u)+1sL∑n=0∞pnun(x,t)xxxxx(25)

where H_n(u) are He’s polynomials that represent the nonlinear terms. The first few components of He’s polynomials are given by

H0(u)=30u02u0x+30u0xu0xx+10u0u0xxx

H1(u)=30(2u0u1u0x+u02u1x)+30(u1xu0xx+u0xu1xx)+10(u1u0xxx+u0u1xxx)

H2(u)=30[(2u0u2+u12)u0x+2u0u1u1x+u02u2x]+30[u2xu0xx+u1xu1xx+u2xu0xx]+10[u2u0xxx+u1u1xxx+u0u2xxx]⋮(26)

Comparing the coefficients of like powers of p, we have

p0:u0(x,t)=2k23sech2(kx)−1

p1:u1(x,t)=6k7tsech7(kx)7sinh⁡(5kx)+141sinh⁡(3kx)−586sinh⁡(kx)

p2:u2(x,t)=k12t22sech12(kx)[1602472−19327698cosh⁡(2kx)+3754368cosh⁡(4kx)−330327cosh⁡(6kx)+8568cosh⁡(8kx)−63cosh⁡(10kx)]⋮(27)

Therefore, solution of above problem when p → 1 is:

u(x,t)=2k23sech2(kx)−1+6k7tsech7(kx)7sinh⁡(5kx)+141sinh⁡(3kx)−586sinh⁡(kx)+k12t22sech12(kx)[1602472−19327698cosh⁡(2kx)+3754368cosh⁡(4kx)−330327cosh⁡(6kx)+8568cosh⁡(8kx)−63cosh⁡(10kx)]+…(28)

Using Taylor series, the closed form solution is as follows:

u(x,t)=2k23sech2(kx−56k5t)−1(29)

which is the same as that obtained by HPM [58].

4 Conclusion

In this work, homotopy perturbation transform method (HPTM) has been successfully applied to approximate the solution of fifth order nonlinear KdV equation. It is extremely simple, easy to use and very accurate for solving nonlinear equations. Maple 10 package is used to calculate series obtained from iteration. Moreover, a comparison with HPM shows that although the results of both methods are same, HPTM is simple and easily handle the nonlinear terms, which arises difficulties in the calculation of HPM. As the method needs much less computational work as compare to other traditional methods. It shows that HPTM is very fast convergent, precise and cost efficient tool for solving nonlinear problems.

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Received: 2016-2-25

Accepted: 2016-12-17

Published Online: 2017-1-27

Published in Print: 2017-6-27

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Articles in the same Issue

https://doi.org/10.1515/nleng-2016-0011

Keywords for this article

Homotopy perturbation method; Laplace transform method; Non-linear partial differential equations; He’s polynomials

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