International Journal of Differential Equations
Volume 2018 (2018), Article ID 8725014, 11 pages
https://doi.org/10.1155/2018/8725014

Research Article

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

Muhammad Asim Khan,¹ Shafiq Ullah,² and Norhashidah Hj. Mohd Ali¹

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

Correspondence should be addressed to Muhammad Asim Khan; asim.afg@gmail.com

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 form

Clearly 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 form

Substituting 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 . And

where 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 becomes

Generally, the order solution of the problem can be obtained in the form

Putting 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 minimized

where and are two constants depending on the given problem. The values of can be optimally found by the condition

After 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 order

The exact solution of problem (14) is . Now according to OHAM , the nonlinear part and .

The zeroth-order problem is

The solution of (15) is

The first-order problem is

The solution of (17) is

The second-order problem is

The 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.

Table 1: Comparison of the third-order OHAM solution with the exact solution and HPM.

OHAM Solution ()

HPM [1]

Figure 1: Comparison between exact solution (dashed line) and approximate solution (dotted line) for example 1.

3.2. Example 2

Consider the nonlinear two-point boundary value problem [1] of the type

According to OHAM and , while . The exact solution of (23) is . Now proceeding with the same lines as above we have the following zeroth-order problem:

The solution of (24) is

Now the first-order problem is

The solution of (26) is

The second-order problem is

The solution of (29) is

The third-order problem is

The solution of the third-order problem results a large output, therefore not included here.

Now the third-order approximate solution is

has the following values and then substituting in the above solution we will get the approximate solution. is given in Appendix (A.1).

The solution at the points given in Table 2 and the graph of the solution is shown in Figure 2. Here it is third-order OHAM solution while the HPM [1] gives the accuracy up to 9 decimal places in 7th order.

Table 2: Comparison of second-order OHAM solution with the exact solution for example 2.

OHAM Solution ()

Figure 2: Comparison between exact solution (dashed line) and approximate solution (dotted line) for example 2.

3.3. Example 3

Now we consider higher order TPBVP of order four. The problem is

with the boundary conditions , , , and .

Where , , and , the exact solution of problem (34) is . After solving this by the method described in Section 2, we have the following zeroth-order problem:

The solution to (35) is

The first-order problem is

The second-order problem is

The solutions of problem (38) and (40) are very large; therefore we did not write it here. The constants and have the values and , respectively. Table 3 and Figure 3 show a good agreement with the exact values. The approximate solution is given in Appendix (A.2).

Table 3: Comparison of second-order OHAM solution with the exact solution for example 3.

OHAM Solution ()

Figure 3: Comparison between exact solution (dashed line) and approximate solution (dotted line) for example 3.

3.4. Example 4

At last, consider the second-order nonlinear TPBVP[1]

The exact solution of (42) is . Solving (42) by the method depicted in Section 2, we have the following zeroth order problem:

The solution of (44) is given by

The first-, second-, and third-order problems are given in (46), (47), and (48) respectively.

The solutions of problem (46), (47), and (48) are very large and therefore cannot be written here but the table of values and the graph are shown in Table 4 and Figure 4, respectively. The approximate solution is written in Appendix (A.3). The values of the constants can be found by (13) which are given as follows:

Table 4: Comparison of third-order OHAM solution with the exact solution.

OHAM Solution ()

Figure 4: Comparison between exact solution (dashed line) and approximate solution (dotted line) for example 4.

4. Conclusion

This paper reveals that OHAM is a very strong method for solving TPBVP and gives us a more accurate solution as compared to other methods. In these examples only second- and third-order solution gives us the accuracy up to 8 or 10 decimal places; therefore it is concluded that this method converges very fast to the exact solution and in some problems like example 1 it gives us the exact solution. The plots and tables show well agreement with the exact solution.

Appendix

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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