He-Laplace Method for Linear and Nonlinear Partial Differential Equations

A new treatment for homotopy perturbation method is introduced. The new treatment is called He-Laplacemethodwhich is the coupling of the Laplace transform and the homotopy perturbation method using He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The method is implemented on linear and nonlinear partial differential equations. It is found that the proposed scheme provides the solution without any discretization or restrictive assumptions and avoids the round-off errors.


Introduction
Many important phenomena occurring in various field of engineering and science are frequently modeled through linear and nonlinear differential equations.However, it is still very difficult to obtain closed-form solutions for most models of real-life problems.A broad class of analytical methods and numerical methods were used to handle such problems.In recent years, various methods have been proposed such as finite difference method 1, 2 adomian decomposition method 3-8 , variational iteration method 9-12 , integral transform 13 , weighted finite difference techniques 14 , Laplace decomposition method 15-17 , but all these methods have some limitations.The homotopy perturbation method was first introduced by Chinese mathematician  The essential idea of this method is to introduce a homotopy parameter p, say which takes the values from 0 to 1.When p 0, the system of equations usually reduces to a simplified form which normally admits a rather simple solution.As p gradually increases to 1, the system goes through a sequence of deformation and the solution of each of which is close to that at the previous stage of deformation.Eventually at p 1, the system takes the

Basic Idea of Homotopy Perturbation Method
Consider the following nonlinear differential equation with the boundary conditions of where A, B, f r , and Γ are a general differential operator, a boundary operator, a known analytic function and the boundary of the domain Ω, respectively.The operator A can generally be divided into a linear part L and a nonlinear part N. Equation 2.1 may therefore be written as.
By the homotopy technique, we construct a homotopy v r, p : where p ∈ 0, 1 is an embedding parameter, while y 0 is an initial approximation of 2.1 , which satisfies the boundary conditions.Obviously, from 2.4 and 2.5 we will have

2.6
The changing process of p from zero to unity is just that of v r, p from y 0 to y r .In topology, this is called deformation, while L v − L y 0 and A v − f r are called homotopy.
If the embedding parameter p is considered as a small parameter, applying the classical perturbation technique, we can assume that the solution of 2.4 and 2.5 can be written as a power series in p: Setting p 1 in 2.7 , we have The combination of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while keeping all its advantages.The series 2.8 is convergent for most cases.However, the convergent rate depends on the nonlinear operator A v .Moreover, He 21 made the following suggestions.
1 The second derivative of N v with respect to v must be small because the parameter may be relatively large; that is, p → 1.
2 The norm of L −1 ∂N/∂v must be smaller than one so that the series converges.

He-Laplace Method
To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous partial differential equation with initial conditions of the form

3.2
Applying the initial conditions given in 3.1 , we have Operating the inverse Laplace transform on both sides of 3.3 , we have where F x, t represents the term arising from the source term and the prescribed initial conditions.Now, we apply the homotopy perturbation method: and the nonlinear term can be decomposed as p n H n y .

3.6
For some He's polynomials H n see 31, 32 with the coupling of the Laplace transform and the homotopy perturbation method are given by Comparing the coefficients of like powers of p, we have the following approximations:

Application
To demonstrate the applicability of the above-presented method, we have applied it to two linear and three nonlinear partial differential equations.These examples have been chosen because they have been widely discussed in literature.with the following conditions:

4.2
By applying the aforesaid method subject to the initial condition, we have The inverse of the Laplace transform implies that Now, we apply the homotopy perturbation method; we have Comparing the coefficient of like powers of p, we have

4.6
Proceeding in a similar manner, we have By applying the aforesaid method subject to the initial condition, we have The inverse of the Laplace transform implies that Now, we apply the homotopy perturbation method; we have Comparing the coefficient of like powers of p, we have

4.14
Proceeding in a similar manner, we have  . . .

4.22
Comparing the coefficient of like powers of p, we have

4.23
Proceeding in a similar manner, we have . . .

4.24
so that the solution y x, t is given by

4.28
The inverse of the Laplace transform implies that

4.29
Now, we apply the homotopy perturbation method; we have where H n y are He's polynomials.The first few components of He's polynomials are given by . . .

4.31
Comparing the coefficient of like powers of p, we have

4.32
Proceeding in a similar manner, we have

4.33
so the solution y x, t is given by which is the exact solution of the problem.

Comparison of Rate of Convergence of HPM and He-Laplace Method
Example 5.1.Consider the following nonhomogeneous nonlinear PDE: with the following conditions:

5.2
According to the homotopy perturbation method, we have

5.3
The initial approximation is chosen y 0 at.By equating the coefficients of p to zero, we obtain Coefficient of p 0 :

5.5
Therefore, we obtain

5.6
Note.Now we solve the same problem using the He-Laplace method.

5.14
so that the solution y x, t is given by

5.15
which is the exact solution of the problem.
Remark 5.3.From comparison, it is clear that the rate of convergence of He-Laplace method is faster than homotopy perturbation method HPM .Also it can be seen the following demerits in the HPM.
1 Choice of initial approximation is compulsory.
2 According to the steps of homotopy, perturbation procedure operator L should be "easy to handle."We mean that it must be chosen in such a way that one has no difficulty in subsequently solving systems of resulting equations.It should be noted that this condition does not restrict L to be linear.In some cases, as was done by He to solve the Lighthill equation, a nonlinear choice of L may be more suitable, but its strongly recommended for beginners to take a linear operator as L.

Conclusions and Discussions
In this paper, the He-Laplace method is employed for solving linear and nonlinear partial differential equations, that is, heat and wave equations.In previous papers 6, 15, 34-38 many authors have already used Adomian polynomials to decompose the nonlinear terms in equations.The solution procedure is simple, but the calculation of Adomian polynomials is complex.To overcome this shortcoming, we proposed a He-Laplace method using He's polynomials 31, 32, 39 .It is worth mentioning that the method is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the numerical results.

Example 4 . 1 .
Consider the following homogeneous linear PDE 33 : /∂x 2 and R 2 ∂/∂x are the linear differential operators, N represents the general nonlinear differential operator and f x, t is the source term.Taking the Laplace transform denoted by L on both sides of 3.1