Mathematical Problems in Engineering

Volume 2011 (2011), Article ID 296796, 7 pages

doi:10.1155/2011/296796

Letter to the Editor

Analyzing “Homotopy Perturbation Method for Solving Fourth-Order Boundary Value Problem"

Afgan Aslanov

Department of Mathematics and Computing, Beykent University, Istanbul, Turkey

Received 29 March 2011; Accepted 8 April 2011

Academic Editor: Jaromir Horacek

Copyright © 2011 Afgan Aslanov. Afgan Aslanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We analyze a previous paper by S. T. Mohyud-Din and M. A. Noor (2007) and show the mistakes in it. Then, we demonstrate a more efficient method for solving fourth-order boundary value problems.

1. Problem

Let us consider the fifth-order boundary problem of the type

with the boundary conditions

where and are continuous functions and , , , and are real constants.

The homotopy perturbation method (HPM) is employed in [1] for solving such problems. The purpose of this paper is to point out the mistakes in paper [1] and demonstrate more efficient method for solving the problems of type (1.1)-(1.2).

First of all, we show the mistakes.

(1)  In Example 3.2 the approximate solution

of the problem

(see formula (3.19) in [1]) has no relationship with the exact solution . And “error” is at least .

It is strange that the authors demonstrated the reliability of the errors in the table (see Table 3.2 in [1]).

(2)  In Example 3.3 the approximate solution

of the problem

(see formula (3.28) in [1]) has no relationship with the exact solution , since , .

2. Homotopy Perturbation Method

The basic ideas of the standard HPM were given by He [2, 3], and a new interpretation of HPM was given by He [4]. We introduce a new reliable procedure for choosing the initial approximation in HPM. To do so, we consider the following general nonlinear differential equation

with some initial boundary conditions, where and are, respectively, the linear and nonlinear operators.

According to HPM, we construct a homotopy which satisfies the following relations:

where is an embedding parameter and is an initial approximation. When we put and in (2.2), we obtain

respectively. In topology, this is called deformation and and are called homotopics.

The solution of (2.2) is expressed as

Hence, the approximate solution of (1.5) can be expressed as

Mohyud-Din and Noor [1] tried to rewrite the problem as a system of integral equations and then HPM applied for each equation. In the use of HPM, what we are mainly concerned about are the auxiliary operator and the initial guess . We take , and

where and are yet to be determined. Then using the boundary conditions , we determine and .

3. Applications

Here we apply the HPM to solve correctly the problems in [1].

Example 3.1 (see [1, Example 3.2]). We have

with boundary conditions

We construct a homotopy which satisfies the relation

where

Now substituting (3.4) into (3.3), we obtain

and, equating the coefficients of a like powers of , we get a system of equations:

Solving (3.6), we get

Using only three-term approximation, we have

Now it follows from conditions , that and and, therefore,

or, in power series form,

Higher accuracy level can be attained by evaluating some more terms of .

Example 3.2 (see [1, Example 3.3]). We have

with boundary conditions

(the exact solution of the problem is ).

We construct a homotopy which satisfies the relation

where

Substituting (3.14) into (3.13), we obtain

and, equating the coefficients of a like powers of , we get a system of equations:

Solving (3.16) we get

Using only three-term approximation we have

Now by using the conditions , , we have a system of equations of degree three. Solving this system numerically (applying some standard computer programs) we have that , and the series solution

4. Conclusion

In this paper we have used the homotopy perturbation method for finding the solution of fourth-order linear and nonlinear boundary value problems. We presented a simple way to choose and when we use the homotopy perturbation method. In most cases, our simple choice yields very good approximation of exact solution.

References

S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary value problems,” Mathematical Problems in Engineering, vol. 2007, Article ID 98602, 15 pages, 2007.

J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.

J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.

J. H. He, “New interpretation of homotopy perturbation method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561–2568, 2006.