An Efficient Method for Solving System of Third-Order Nonlinear Boundary Value Problems

we use the modified variation of parameters method for finding the analytical solution of a system of third-order nonlinear boundary value problems associated with obstacle, unilateral, and contact problems. The results are calculated in terms of convergent series with easily computable components. The suggested technique is applied without any discretization, perturbation, transformation, and restrictive assumptions. Moreover, it is free from round off errors. Some examples are given to illustrate the implementation and efficiency of the modified variation of parameters method.


Introduction
In recent years, much attention has been given to solve system of third-order boundary value problems, see 1-11 .In this paper, we consider the following systems of third-order nonlinear boundary value problems: of the modified variation of parameters methods for solving the variational inequalities is an open and interesting problem for future research.For the formulation, applications, and numerical techniques for solving the variational inequalities and related optimization problems, see 2-4, 7, 24, 25 and the references therein.

Modified Variation of Parameters Method
To illustrate the basic concept of the variation of parameter method for differential equations, we consider the general differential equation in operator form where L is a higher-order linear operator, R is a linear operator of order less than L, N is a nonlinear operator, and g is a source term.sing variation of parameters method 2-5, 10, 16, 17 , we have following general solution of 2.1 where n is a order of given differential equation and Bi s are unknowns which can be further determined by initial/boundary conditions.Here λ x, s is multiplier which can be obtained with the help of Wronskian technique.This multiplier removes the successive application of integrals in iterative scheme, and it depends upon the order of equation.
For different choices of n, one can obtain the following values of λ n 1, λ x, s 1,

2.4
Hence, we have the following iterative scheme from 2.2 : It is observed that the fix value of initial guess in each iteration provides the better approximation, that is, u k x u 0 x , for k 1, 2, . ... However, we can modify the initial guess by dividing u 0 x in two parts and using one of them as initial guess.It is more convenient way in case of more than two terms in u 0 x .In a modified variation of parameters method, we define the solution u x by the following series: and the nonlinear terms are decomposed by infinite number of polynomials as follows: where u is a function of x andA k are the so-called Adomian's polynomials.These polynomials can be generated for various classes of nonlinearities by specific algorithm developed in 23 as follows: Hence, we have the following iterative scheme for finding the approximate solution of 2.1 as 2.9 We would like to mention that the modified variation of parameters method for solving the system of third-order nonlinear boundary value problems may be viewed as an important and significant improvement as compared with other similar method.

Numerical Results
Example 3.1.Consider following system of third-order nonlinear boundary value problems relevant to system 1.1 : We will use modified variation of parameters method for solving system of third-order nonlinear boundary value problems 3.1 .By using the modified variation of parameters method, we have following iterative scheme to solve nonlinear system 3.1 :

3.3
Case 1 −1 ≤ x < −1/2 .In this case, we implement the modified variation of parameters method as follows: we take u 0 c 1 x 2 /2! c 2 x c 3 , for better approximation.we decompose initial guess as u 0 c 2 x, and obtain further iterations as follows: . . .

3.7
Case 3 1/2 ≤ x ≤ 1 .In this case, we proceed as follows: By using MVPM, we have following formula for getting series solution in the whole domain from the above cases: 3.9 Hence, we have the following series solution after two iterations:

Mathematical Problems in Engineering
By using boundary conditions and continuity conditions at x −1/2 and x 1/2 and we have a system of nonlinear equations.By using Newton's method for system of nonlinear equations, we have the following values of unknown constants:

3.11
By using values of unknowns from 3.11 into 3.10 , we have following analytic solution of system of forth-order nonlinear boundary value problem associated with obstacle problem 3.1 .003347168075x 6 .0004439485602x 7  Example 3.2.Consider following system of third-order nonlinear boundary value problem relevant to system 1.2 :

3.13
with boundary conditions u −1 u 1 0, u −1 1. Proceeding as before, we have a following iterative scheme to solve nonlinear system 3.13 by using the modified variation of parameters method:

3.14
Case 1 −1 ≤ x < −1/2 .In this case, we implement MVPM as follows.We consider the initial value as and obtain further iterations as follows: . . .

3.18
Mathematical Problems in Engineering 11 Hence, we have the following series solution after two iterations x 3

3.19
By using boundary conditions and continuity conditions at x −1/2 and x 1/2 and we have a system of nonlinear equations.By using Newton's method for system of nonlinear equations, we have the following values of unknown constants:

3.21
Figure 2 is a graphical representation of analytical solution of system of thirdorder nonlinear boundary value problem 3.13 by using modified variation of parameters method.

Conclusion
In this paper, we have used the modified variation of parameters method, which is a combination of variation of parameters method and Adomian's decomposition method for solving system of third-order nonlinear boundary value problem.It is worth mentioning that we have solved nonlinear systems of boundary value problem by our proposed technique while most of the methods in the literature are proposed to solve linear systems of boundary value problems associated with obstacle problems.We took two examples for both the systems which are highly nonlinear in their nature.After applying our proposed technique we obtained series solutions as well as their graphical representation over the whole domain.We analyze that our proposed method is well suited for such physical problems as it provides best solution in less number of iterations.It is worth mentioning that the method is capable of reducing the volume of the computational work as compared to the existing classical methods.The use of multiplier gives this technique a clear edge over the decomposition method by removing successive application of integrals.Therefore, it may be concluded that modified variation of parameters method is very powerful and efficient technique for finding the analytical solutions for a wide class of systems of nonlinear boundary value problems.We would also like to mention that Ma et al. 26, 27 have used the multiple expo function method and linear superposition principle for solving the Hirota bilinear equations for constructing a specific subclass of N-soliton solutions.It is an interesting and open problems to compare the modified variation of parameters method with the technique of Ma et al. 26, 27 for solving the system of third-order nonlinear boundary value problems associated with variational inequalities.Applications of the multi-expo function method for solving the variational inequalities and related optimization problems is an interesting problem for future research.Results proved in this paper may inspire the research for novel and innovative applications of these techniques.

Figure 1 Figure 1 a
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