Comparison between the Homotopy Perturbation Method and Homotopy Perturbation Transform Method

In this paper, comparison of homotopy perturbation method (HPM) and homotopy perturbation transform method (HPTM) is made, revealing that homotopy perturbation transform method is very fast convergent to the solution of the partial differential equation. For illustration and more explanation of the idea, some examples are provided.


Introduction
Many problems in our life are modeled by linear and nonlinear partial differential equations.In recent years, various analytical methods proposed by researchers to solve these PDEs.However, it is still very difficult to obtain closed-form solutions.The homotopy perturbation method proposed by Ji-Huan He [1] [2] [3] [4] and employed by scientists and engineers [5] [6] [7] to solve many problems in engineering and science.And it has proved tremendously effective to solve these problems.In this letter, we use the coupling of homotopy perturbation method and Laplace transform named homotopy transform method (HPTM) [8]- [13] to compare the rate of convergent to the solution with standard homotopy perturbation method.
M. Elbadri DOI: 10.4236/am.2018.92009 131 Applied Mathematics ( ) ( ) ( ) 0 With the boundary conditions , 0, , u B u r n where L is linear, while Ν is nonlinear, ( ) f r is a known analytic function, Γ is the boundary of the domain Ω .Define a homotopy ( ) [ ] where The changing process of p from zero to unity is just that of ( ) Here the imbedding parameter p can be considered as "small parameter".Assume that the solution of Equation ( 3) can be written as a power series in p Setting 1 p = result in the approximate solution of Equation ( 6)

Homotopy Perturbation Transform Method (HPTM)
To illustrate the basic ideas of the (HPTM), we consider the following nonlinear differential equation with the initial conditions of the form where D is the second order linear differential operator Taking the Laplace transform (denoted by L) on both sides of Equation ( 8): 11)   Using the initial conditions: , G x t represents the term arising from the source term and the pre- scribed initial conditions.Now we apply the HPM And the nonlinear term can be decomposed as where ( ) n u Η are He's polynomials given by ( ) Substituting Equations ( 14) and ( 15) in Equation ( 13), we get Comparing the coefficient of like powers of p, the following approximations are obtained.

Application
Example 1.Consider the inhomogeneous Advection problem [14] ( ) ( ) ( ) Standard HPM: According to homotopy Equation (3) we have ( ) ( ) And the solution for first few steps reads: Taking inverse Laplace transform, we get Now, we apply the homotopy perturbation method; we have where n Η are He's polynomials that represent the nonlinear terms.
The first few components of He's polynomials, for example, are given by ( ) ( ) Standard HPM: According to homotopy Equation (3) we have And the solution for first few steps reads: Taking inverse Laplace transform, we get ( ) Now, we apply the homotopy perturbation method; we have The first few components of He's polynomials, for example, are given by ( ) ( ) The noise terms With the initial conditions ( ) ( ) Standard HPM: According to homotopy perturbation method we have: Let's ignore the first few steps and start from determining i v Taking inverse Laplace transform, we get ( ) Now, we apply the homotopy perturbation method; we have ( ) ( ) The first few components of He's polynomials, for example, are given by ( ) Therefore, the exact solution is given by ( )

Conclusion
In this work, we compared HPTM with standard HPM, it is clear that the rate of convergence of HPTM is faster than HPM.In most cases, the number of calculations in the HPTM is less than HPM.Furthermore, the exact solution can easily be obtained by using HPTM in comparison to HPM in some equations.The HPTM usually results in the exact solution for the inhomogeneous problem, even for the problem which HPM leads to an approximate solution.
of less order than D; N represents the general nonlinear differential operator and ( ) , g x t is the source term.
of like powers of p, we have the coefficient of like powers of p, we have