An Algorithm : Optimal Homotopy Asymptotic Method for Solutions of Systems of Second-Order Boundary Value Problems

Optimal homotopy asymptotic method (OHAM) is proposed to solve linear and nonlinear systems of second-order boundary value problems. OHAM yields exact solutions in just single iteration depending upon the choice of selecting some part of or complete forcing function. Otherwise, it delivers numerical solutions in excellent agreement with exact solutions. Moreover, this procedure does not entail any discretization, linearization, or small perturbations and therefore reduces the computations a lot. Some examples are presented to establish the strength and applicability of this method. The results reveal that the method is very effective, straightforward, and simple to handle systems of boundary value problems.


Introduction
In this presentation, we study the nonlinear system of secondorder differential equations [1][2][3] of the subsequent type:   +  1 ()   +  2 ()  +  3 () V  +  4 () V  +  5 () V +  1 (, , V) =  1 () , with boundary conditions: where 0 ≤  ≤ 1 and  1 and  2 are nonlinear functions of  and V, respectively.Also,  1 and  2 are known forcing functions of the system and   (),   () for  = 1, 2, . . ., 5 are given continuous functions.In [1], the analytical solution of the above problem is illustrated in the form of series under the assumption that the solution is unique.In [4,5], Sinccollocation and the Chebyshev finite difference methods, respectively, are used to solve the same systems.A numerical method based on the cubic B-spline scaling functions is proposed in [6] to find the solutions of (1)- (2).Also, in [2], the variational iteration method is presented to elucidate problem (1)-( 2).He's homotopy perturbation method (HPM) is proposed to solve the same nonlinear systems of secondorder boundary value problems [3].
In the present work, optimal homotopy asymptotic method (OHAM) is extended to demonstrate the solutions of systems of boundary value problems (BVP).This method finds the exact solutions of system (1) having boundary conditions (2) for the choice of selecting some part of or complete forcing function; otherwise, it produces numerical solutions in excellent agreement with exact solutions.In recent years, the application of OHAM in linear and nonlinear problems has been developed by scientists and engineers, because this method deforms the difficult problems under study into simple problems, which are easy to solve.The OHAM was proposed first by the Romanian researcher Marinca et al. [7] in 2008.The advantage of OHAM is integrated convergence criteria similar to HAM but flexible to a greater extent in implementation.Marinca et al. [8][9][10] and Iqbal et al. [11][12][13][14][15] in a series of papers have established validity, usefulness, simplification, and consistency of the method and acquired reliable solutions of currently significant applications in science and technology.
The organization of this presentation is as follows: In Section 2, extended algorithm of OHAM is illustrated for our subsequent progress.As a result, systems of simple differential equations are formed and the exact solutions of the considered problems are introduced.In Section 3, some examples of linear and nonlinear systems are answered by the extended algorithm, results to clarify the method with existing exact results.Section 4 ends this paper with a brief conclusion.
Optimal values of auxiliary constants for solutions of system can be determined by minimizing functional (11) as follows: Note.To determine the solutions of systems of boundary value problems, the choice of  1 () from   () of the systems is very important.It has been observed in this article that exact solutions may be obtained by the proper choice of  1 ().Example 3 has been discussed for different  1 ().

Explanatory Examples
This section is devoted to explanatory examples.The extended method presented in this paper is applied to the three systems of boundary value problems.These problems are chosen such that they have exact solutions.
Example 1.A linear system of second-order boundary value problems is as follows [2,3]: subject to the boundary conditions The exact solutions are () =  2 −  and V() =  −  2 .
A series of problems are generated by OHAM formulation presented in the previous section.The expressions for zerothorder, first-order, and second-order problems of the system and their solutions are given below: By using (10), (e) part of the OHAM algorithm, solutions of the system are Equations ( 16) show the exact solutions of the system given in Example 1.In this system, (f) and (g) parts of the algorithms are not used.
Example 3. Consider the following nonlinear system [2,3]: subject to the following boundary conditions: where  1 () =  3 − 2 .The expressions for zeroth-order, first-order, and second-order problems of the system and their solutions are given below: (24) By using (10), (e) part of the OHAM algorithm, solutions of the system are (25) Equations ( 25) show the exact solutions of the system given in Example 3.

Conclusions
In this paper, it has been revealed that the optimal homotopy asymptotic method can be applied effectively for solving the systems of second-order boundary value problems.This method is straightforward and easy to practice for solving the problems without any requirement of discretization of the variables.Therefore, it is not exaggerated by computational round of errors.As an advantage of the generalized optimal homotopy asymptotic method over the other procedures, it delivers the exact solutions of the problems depending upon the selection of  1 ().Moreover, the proposed method is free from rounding off errors and does not require excessive computer power or memory for its implementation.

Table 1 :
Absolute errors of 2nd-order solutions of () for different choices of  1 () and also comparison with the 9th-order HPM solutions.

Table 2 :
Absolute errors of 2nd-order solutions of V() for different choices of  1 () and also comparison with the 9th-order HPM solutions.