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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:
Homotopy on OHAM can be constructed as
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 zeroth-order problem is
Table 1
Comparison of the third-order OHAM solution with the exact solution and HPM.
| OHAM Solution ( | | HPM [1] | |
---|---|---|---|---|
| | | | |
| ||||
| | | | |
| ||||
| | | | |
| ||||
| | | | |
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| | | | |
3.2. Example 2
Consider the nonlinear two-point boundary value problem [1] of the type
Now the third-order approximate solution is
Table 2
Comparison of second-order OHAM solution with the exact solution for example 2.
| OHAM Solution ( | | |
---|---|---|---|
| | | |
| |||
| | | |
| |||
| | | |
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| | | |
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| | | |
3.3. Example 3
Now we consider higher order TPBVP of order four. The problem is
Where
Table 3
Comparison of second-order OHAM solution with the exact solution for example 3.
| OHAM Solution ( | | |
---|---|---|---|
| | | |
| | | |
| | | |
| | | |
3.4. Example 4
At last, consider the second-order nonlinear TPBVP[1]
Table 4
Comparison of third-order OHAM solution with the exact solution.
| OHAM Solution ( | | |
---|---|---|---|
| | | |
| |||
| | | |
| |||
| | | |
| |||
| | | |
| |||
| | | |
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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Appendix
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[9] H. Yaghoobi, M. Torabi, "The application of differential transformation method to nonlinear equations arising in heat transfer," International Communications in Heat and Mass Transfer, vol. 38 no. 6, pp. 815-820, DOI: 10.1016/j.icheatmasstransfer.2011.03.025, 2011.
[10] D. D. Ganji, G. A. Afrouzi, R. A. Talarposhti, "Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations," Physics Letters Section A: General, Atomic and Solid State Physics, vol. 368 no. 6, pp. 450-457, DOI: 10.1016/j.physleta.2006.12.086, 2007.
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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).
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