Abstract

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.

1. 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:

with boundary conditions , , , and continuity conditions of , , and at internal points and of the interval . Here and are real and finite constants, is a continuous function on and is a nonlinear function. Such type of problems arise in the study of obstacle, contact, unilateral, and equilibrium problems arising in economics, transportation, nonlinear optimization, oceanography, ocean wave engineering, fluid flow through porous media, and some other branches of pure and applied sciences; see [1, 2, 4–15] and references therein. Several techniques have been used to solve system of linear third-order boundary value problems associated with obstacle, contact, and unilateral problems. Noor et al. [7] applied finite difference method for unilateral problems; Al-Said et al. [14] used finite difference method for obstacle problems; Khalifa and Noor [15] applied quintic spline method for contact problems; Noor and Khalifa [8] applied quartic spline method for odd-order obstacle problems; Al-Said and Noor [13] used quartic spline method for obstacle problems. Gao and Chi [1] applied quartic B-spline method for third-order obstacle problems and Islam et al. [11] proposed nonpolynomial spline methods for solving system of third-order boundary value problems associated with obstacle, contact, and unilateral problems. Some of these methods are numerical and require huge computational work for finding the approximate solutions. Moreover, these numerical methods provide discrete point solution.

In this paper, we use the modified variation of parameters method to solve the systems of third-order nonlinear boundary value problems associated with obstacle problems. Noor et al. [4, 5, 10, 16, 17] have used variation of parameters method for solving a wide classes of higher-order initial and boundary value problems. Ma et al. [18–20] have applied variation of parameters method for solving some nonhomogenous partial differential equations. Ramos [21] has used this technique to find frequency of some nonlinear oscillators. Ramos [21] has also shown the equivalence of this technique with variational iteration method. It is further investigated by Noor et al. [4, 5, 10, 16, 17] that the proposed technique is distinctly different from variational iteration method in many aspects. The multiplier used in the variation of parameters method is obtained by Wronskian technique and is totally different from the Lagrange multiplier of variational iterative method. Moreover, the variation of parameters method (VPM) removes the higher-order derivative term from its iterative scheme which is clear advantage over the variational iteration method (VIM) as the term may cause of repeated computation and computations of unneeded terms, which consumes time and effort, in most of the cases, one of the cases is mentioned in [22]. Thus the variation of parameters method has reduced a lot of computational work involved due to this term as compared to some other existing techniques using this term. This shows that the variation of parameters method is better than the other techniques. The modified variation of parameters method is obtained by combining the variation of parameters method [3–5, 19, 20] and Adomian’s decomposition method [23]. It turned out that modified variation of parameters method is very flexible and is very efficient. The use of multiplier and Adomian’s polynomial together in the modified variation of parameters method increases the rate of convergence by reducing the number of iterations and successive application of integral operators. This technique makes the solution procedure simple while still maintaining the higher level of accuracy. Here, we implement this technique for solving systems of third-order nonlinear boundary value problems associated with obstacle, unilateral, and contact problems, which is the main motivation of this paper. Examples are given to illustrate the implementation and the efficiency of the proposed method. The comparison of the proposed method with other similar methods is the subject of another paper. It is well known that the obstacle problems can be studied in the general frame work of the variational inequalities. The applications 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.

2. 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 is a higher-order linear operator, is a linear operator of order less than , is a nonlinear operator, and is a source term. sing variation of parameters method [2–5, 10, 16, 17], we have following general solution of  (2.1) where is a order of given differential equation and are unknowns which can be further determined by initial/boundary conditions. Here 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. Noor et al. [4, 5, 16, 17] have obtained the following for finding the multiplier as

For different choices of , one can obtain the following values of

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, , for . However, we can modify the initial guess by dividing in two parts and using one of them as initial guess. It is more convenient way in case of more than two terms in . In a modified variation of parameters method, we define the solution by the following series:

and the nonlinear terms are decomposed by infinite number of polynomials as follows:

where is a function of and 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

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.

3. Numerical Results

Example 3.1. Consider following system of third-order nonlinear boundary value problems relevant to system (1.1): with boundary conditions .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): Since the governing equation is of 3rd order, using , we have

Case 1 (). In this case, we implement the modified variation of parameters method as follows: we take , for better approximation. we decompose initial guess as , and obtain further iterations as follows:

Case 2 (). In this case, we have following approximations:

Case 3 (). 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:

Hence, we have the following series solution after two iterations: By using boundary conditions and continuity conditions at and 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: 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)

Figure 1 a graphical representation of analytical solution of system of third-order nonlinear boundary value problem (3.1) by using modified variation of parameters method.

Figure 1 

Example 3.2. Consider following system of third-order nonlinear boundary value problem relevant to system (1.2): with boundary conditions .
Proceeding as before, we have a following iterative scheme to solve nonlinear system (3.13) by using the modified variation of parameters method:

Case 1 (). In this case, we implement MVPM as follows. We consider the initial value as and obtain further iterations as follows:

Case 2 (). In this case, we have following approximations:

Case 3 (). In this case, we proceed as follows:

Hence, we have the following series solution after two iterations

By using boundary conditions and continuity conditions at and 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:

By using values of unknowns from (3.20) into (3.19), we have following analytic solution of system of third-order nonlinear boundary value problem associated with obstacle problem  (3.13)