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

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).


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.

Basics of OHAM
Let us take the BVP whose general form is the following: where L is a linear operator,  is independent variable, N is the nonlinear operator, ϝ() is a known function, and  is a boundary operator.Homotopy on OHAM can be constructed as Clearly when þ = 0 then (0) = 0.And obviously, when þ = 0 then (, 0) =  0 ().When þ = 1 then (, 1) = ().So as þ increases from 0 to 1, the solution (, þ) varies from  0 () to the exact solution (), where  0 () is obtained from (2) for þ = 0 The proposed solution of (1) will be of the form Substituting this value of (, þ, C  ) into (1), after some calculations, we can obtain the governing equations of  0 () by using (4) and   (), that is, + N −1 ( 0 () ,  1 () ,  2 () , . . .,  −1 ()) ,  = 2, 3, 4, . . ., where N  ( 0 (),  1 (),  2 (), . . .,   ()) is the coefficient of þ  in the series expansion of N((, þ, C  )) with respect to the embedding parameter þ.And where (, þ, C  ) is given by (5).The convergence of series (5) depends on the convergence of the constants C   , if these constants are convergent at þ = 1, then the solution becomes Generally, the ℎ order solution of the problem can be obtained in the form Putting this solution in (1) we get the following residual: If R(, C  ) = 0, 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  0 and  1 are two constants depending on the given problem.The values of C    can be optimally found by the condition After knowing these constants, the solution ( 10) is well determined.

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.
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.
. .Example .Consider the nonlinear two-point boundary value problem [1] of the type According to OHAM L(()) =   () and N(()) = u()u  () −  3 (), while ϝ() = 0.The exact solution of ( 23) is 1/( + 1).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 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 C    has the following values and then substituting in the above solution we will get the approximate solution. (3) () 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.
. .Example .At last, consider the second-order nonlinear TPBVP [1]   () =  2 () + 2 2 cos (2) − sin 2 (2) , 0 ≤  ≤ 1 (42) International Journal of Differential Equations The exact solution of ( 42) is sin 2 ().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  (3) () is written in Appendix (A.3).The values of the constants C    can be found by (13) which are given as follows:

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.

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

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

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

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

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

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

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