International Journal of Differential Equations

Volume 2018, Article ID 8725014, 11 pages

https://doi.org/10.1155/2018/8725014

## Application of Optimal Homotopy Asymptotic Method to Some Well-Known Linear and Nonlinear Two-Point Boundary Value Problems

^{1}School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia^{2}Department of Mathematics, University of Peshawar, Pakistan

Correspondence should be addressed to Muhammad Asim Khan; moc.liamg@gfa.misa

Received 20 May 2018; Accepted 21 October 2018; Published 3 December 2018

Guest Editor: Dongfang Li

Copyright © 2018 Muhammad Asim Khan et al. 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

The objective of this paper is to obtain an approximate solution for some well-known linear and nonlinear two-point boundary value problems. For this purpose, a semianalytical method known as optimal homotopy asymptotic method (OHAM) is used. Moreover, optimal homotopy asymptotic method does not involve any discretization, linearization, or small perturbations and that is why it reduces the computations a lot. OHAM results show the effectiveness and reliability of OHAM for application to two-point boundary value problems. The obtained results are compared to the exact solutions and homotopy perturbation method (HPM).

#### 1. Introduction

Two-point boundary value problems (TPBVP) have many applications in the field of science and engineering [1, 2]. These problems arise in many physical situations like modeling of chemical reactions, heat transfer, viscous fluids, diffusions, deflection of beams, the solution of optimal control problems, etc. Due to the wide applications and importance of boundary value problems (BVP) in science and engineering we need solutions to these problems.

There are many techniques available for the solution of-of BVP like Adomian Decomposition Method (ADM) [3–7], Extended Adomian Decomposition Method (EADM)[8], Differential Transformation Method (DTM) [9], Variational Iteration Method (VIM) [10], Perturbation methods(PMs) [1, 11–13], and so on. Perturbation methods are easy to solve but they require small parameters which are sometimes not an easy task. Recently V. Marinca et al. presented optimal homotopy asymptotic method (OHAM) [14] for the solution of BVP, which did not require small parameters. The method can also be applied to solve the stationary solution of some partial differential equations, e.g., gKdv equation, nonlinear parabolic problems, and so on [15–20]. In OHAM, the concept of homotopy is used together with the perturbation techniques. Here, OHAM is applied to TPBVP to check the applicability of OHAM for TPBVP.

#### 2. Basics of OHAM

Let us take the BVP whose general form is the following:where is a linear operator, is independent variable, is the nonlinear operator, is a known function, and is a boundary operator.

Homotopy on OHAM can be constructed as where is an embedding parameter, is an unknown function, is a nonzero auxiliary function for , and is of the formClearly when then . And obviously, when then When then . So as increases from to , the solution varies from to the exact solution , where is obtained from (2) for The proposed solution of (1) will be of the formSubstituting this value of into (1), after some calculations, we can obtain the governing equations of by using (4) and , that is,where is the coefficient of in the series expansion of with respect to the embedding parameter . Andwhere is given by (5). The convergence of series (5) depends on the convergence of the constants , if these constants are convergent at , then the solution becomesGenerally, the order solution of the problem can be obtained in the formPutting this solution in (1) we get the following residual:If , then the solution is going to be exact, but generally, such a situation does not arise in nonlinear problems but the functional defined below can be minimizedwhere and are two constants depending on the given problem. The values of can be optimally found by the conditionAfter knowing these constants, the solution (10) is well determined.

#### 3. Examples

To check the applicability of OHAM for TPBVP, in this section four examples of TPBVP are presented in which one example is linear and the remaining are nonlinear.

##### 3.1. Example 1

Let us consider the linear problem [1] of second orderThe exact solution of problem (14) is . Now according to OHAM , the nonlinear part and .

The zeroth-order problem isThe solution of (15) isThe first-order problem isThe solution of (17) isThe second-order problem isThe solution of (19) is And the third-order approximate solution of the bvp (14) is as follows:Table 1 shows the comparison between the exact solution and the approximate solution obtained by OHAM. Figure 1 of the solution also shows well agreement with the exact solution.