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Mathematical Problems in Engineering
Volume 2011, Article ID 296796, 7 pages
http://dx.doi.org/10.1155/2011/296796
Letter to the Editor

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

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

Received 29 March 2011; Accepted 8 April 2011

Copyright © 2011 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

  1. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. H. He, “New interpretation of homotopy perturbation method,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561–2568, 2006. View at Publisher · View at Google Scholar · View at MathSciNet