# Analysis of the new homotopy perturbation method for linear and nonlinear problems

- Ali Demir
^{1}Email author, - Sertaç Erman
^{1}, - Berrak Özgür
^{1}and - Esra Korkmaz
^{2}

**2013**:61

**DOI: **10.1186/1687-2770-2013-61

© Demir et al.; licensee Springer 2013

**Received: **14 December 2012

**Accepted: **6 March 2013

**Published: **26 March 2013

The Erratum to this article has been published in *Boundary Value Problems* 2015 **2015**:72

## Abstract

In this article, a new homotopy technique is presented for the mathematical analysis of finding the solution of a first-order inhomogeneous partial differential equation (PDE) ${u}_{x}(x,y)+a(x,y){u}_{y}(x,y)+b(x,y)g(u)=f(x,y)$. The homotopy perturbation method (HPM) and the decomposition of a source function are used together to develop this new technique. The homotopy constructed in this technique is based on the decomposition of a source function. Various decompositions of source functions lead to various homotopies. Using the fact that the decomposition of a source function affects the convergence of a solution leads us to development of a new method for the decomposition of a source function to accelerate the convergence of a solution. The purpose of this study is to show that constructing the proper homotopy by decomposing $f(x,y)$ in a correct way determines the solution with less computational work than using the existing approach while supplying quantitatively reliable results. Moreover, this method can be generalized to all inhomogeneous PDE problems.

## 1 Introduction

Many problems in the fields of physics, engineering and biology are modeled by linear and nonlinear PDEs. In recent years, the homotopy perturbation method (HPM) has been employed to solve these PDEs [1–6]. HPM has been used extensively to solve nonlinear boundary and initial value problems. Therefore HPM is of great interest to many researchers and scientists. HPM, first presented by Ji Huan He [7–9], is a powerful mathematical tool to investigate a wide variety of problems arising in different fields. It is obtained by successfully coupling homotopy theory in topology with perturbation theory. In HPM, a complicated problem under study is continuously deformed into a simple problem which is easy to solve to obtain an analytic or approximate solution [10].

In this work, a new homotopy perturbation technique is proposed to find the solution $u(x,y)$ based on the decomposition of a right-hand side function $f(x,y)$ which leads to the construction of new homotopies. It is shown that decomposing $f(x,y)$ in a correct way, requiring less computational work than the existing approach, helps us to determine the solution. The results reveal that the proposed method is very effective and simple.

The paper is organized as follows. In Section 2, an analysis of the new homotopy perturbation method is given. The construction of a new homotopy based on the decomposition of a source function for a first-order inhomogeneous PDE is covered in Section 3. Some examples are given to illustrate the construction of a new homotopy in Section 4. Finally, some concluding remarks are given in Section 5.

## 2 New homotopy perturbation method

where *A* is a general differential operator, *B* is a boundary operator, $f(r)$ is a known analytical function, and Γ is the boundary of domain Ω.

*A*in (1) can be rewritten as a sum of

*L*and

*N*, where

*L*and

*N*are linear and nonlinear parts of

*A*, respectively, as follows:

*p* is an embedding parameter, ${u}_{0}$ is an initial approximation of (1), which satisfies the boundary conditions. As *p* changes from zero to unity, $v(r,p)$ changes from ${u}_{0}$ to $u(r)$. In this technique, the convergence of a solution depends on the choice of ${u}_{0}$, that is, we can have different approximate solutions for different ${u}_{0}$.

As the embedding parameter *p* changes from zero to unity, $v(r,p)$ changes from ${L}^{-1}({f}_{1}(r))$ to $u(r)$.

*p*as a small parameter and assume that the solution of (6) can be written as a power series in

*p*as follows:

In the next section, it is shown that the decomposition of the right-hand side function has a great effect on the amount of calculation and the speed of convergence of the solution.

## 3 Construction of a new homotopy based on the decomposition of a source function

*a*,

*b*,

*g*and

*f*are continuous functions in some region of the plane and $g(0)=0$. By solving this boundary value problem by the homotopy perturbation method, we obtain an approximate or exact solution $u(x,y)$. Before proceeding further, let us introduce the integral operator

*S*defined in the following form:

*S*with respect to

*y*is defined as

*p*, we have

Therefore equation (15) is a necessary condition to accelerate the convergence. If equation (15) has a solution, then we obtain the approximate or exact solution of problem (8)-(9) in two steps.

In this case, we get the approximate or exact solution of the problem in more than two steps. Moreover, we can get the solution in the form of series.

## 4 Analysis of the new homotopy perturbation method for linear problems

where *a* and *b* are constants.

### 4.1 Case 1: the source function $f(x,y)$ is a polynomial

*n*th-order polynomial in problem (17)-(18). Hence $f(x,y)$ can be written in the following form:

where ${A}_{\alpha \beta}$ are constants and *α*, *β* are natural numbers.

*n*th-order polynomial given as

leads us to reaching the solution in two steps.

### 4.2 Case 2: the source function $f(x,y)$ is a sum of two functions $r(x)$ and $t(y)$

we reach the solution in two steps.

## 5 Numerical examples

### 5.1 Example 1

which is the exact solution of the problem with minimum amount of calculation.

### 5.2 Example 2

with short-length calculation.

### 5.3 Example 3

which is obtained by minimum amount of calculations.

### 5.4 Example 4

with short-length calculation.

## 6 Conclusion

In this paper, we employ a new homotopy perturbation method to obtain the solution of a first-order inhomogeneous PDE. In this method, each decomposition of the source function $f(x,y)$ leads to a new homotopy. However, we develop a method to obtain the proper decomposition of $f(x,y)$ which lets us obtain the solution with minimum computation and accelerate the convergence of the solution. This study shows that the decomposition of the source function has a great effect on the amount of computations and the acceleration of the convergence of the solution. Comparing to the standard one, decomposing the source function $f(x,y)$ properly is a simple and very effective tool for calculating the exact or approximate solutions with less computational work.

## Notes

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The research was supported by parts by the Scientific and Technical Research Council of Turkey (TUBITAK).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.