Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He’s Polynomial
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Dinkar Sharma
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).
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:
subjected to the initial condition
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):
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
Taking Laplace inverse on both sides
where G(x, t) represents the term arising from the prescribed initial condition. Now, we apply the homotopy perturbation method
and the nonlinear term can be decomposed as
for some He’s polynomials Hn that are given by
Substituting Eqs. (6) and (7)in Eq. (5), we get
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
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
Subject to the initial condition
By applying the aforesaid method subject to the initial condition, we get
The inverse Laplace transform implies that
Now, we apply the homotopy perturbation method
where Hn(u) are He’s polynomials that represent the nonlinear terms. The first few components of He’s polynomials, for example, are given by
Comparing the coefficient of like powers of p, we have
Therefore, solution of above problem when p → 1 is:
Using Taylor series, the closed form solution is as follows:
which is the same as that obtained by ADM and HPM [54].
Example 3.2
We now consider the Lax’s fifth order Equation
Subject to the initial condition
By applying the aforesaid method subject to the initial condition, we get
where
The inverse Laplace transform implies that
Now, we apply the homotopy perturbation method
where Hn(u) are He’s polynomials that represent the nonlinear terms. The first few components of He’s polynomials are given by
Comparing the coefficients of like powers of p, we have
Therefore, solution of above problem when p → 1 is:
Using Taylor series, the closed form solution is as follows:
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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Artikel in diesem Heft
- Frontmatter
- Chaos Suppression in Fractional order Permanent Magnet Synchronous Generator in Wind Turbine Systems
- Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He’s Polynomial
- A study on validating KinectV2 in comparison of Vicon system as a motion capture system for using in Health Engineering in industry
- MHD Flow with Hall current and Joule Heating Effects over an Exponentially Stretching Sheet
- Solitons and other solutions to the coupled nonlinear Schrödinger type equations
- Nonlinear Dynamic Characteristics of the Railway Vehicle
- Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate in a Non-Darcy Porous Medium with Convective Boundary Condition
- Viscous Dissipation and Thermal Radiation effects in MHD flow of Jeffrey Nanofluid through Impermeable Surface with Heat Generation/Absorption
Artikel in diesem Heft
- Frontmatter
- Chaos Suppression in Fractional order Permanent Magnet Synchronous Generator in Wind Turbine Systems
- Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He’s Polynomial
- A study on validating KinectV2 in comparison of Vicon system as a motion capture system for using in Health Engineering in industry
- MHD Flow with Hall current and Joule Heating Effects over an Exponentially Stretching Sheet
- Solitons and other solutions to the coupled nonlinear Schrödinger type equations
- Nonlinear Dynamic Characteristics of the Railway Vehicle
- Quadratic Convective Flow of a Micropolar Fluid along an Inclined Plate in a Non-Darcy Porous Medium with Convective Boundary Condition
- Viscous Dissipation and Thermal Radiation effects in MHD flow of Jeffrey Nanofluid through Impermeable Surface with Heat Generation/Absorption
