Mathematical Problems in Engineering

1. Introduction

This paper discusses the analytical approximate solution for fourth-order equations with nonlinear boundary conditions involving third-order derivatives. The general form of the equation for a fixed positive integer , is a differential equation of order : subject to the boundary conditions where are finite constants.

It is assumed that is sufficiently differentiable and that a unique solution of (1.1) exists. Problems of this kind are commonly encountered in plate-deflection theory and in fluid mechanics for modeling viscoelastic and inelastic flows [1–3]. Usmani [1, 2] discussed sixth order methods for the linear differential equation subject to the boundary conditions , , . The method described in [1] leads to five diagonal linear systems and involves at and , while the method described in [2] leads to nine diagonal linear systems.

Ma and Silva [4] adopted iterative solutions for (1.1) representing beams on elastic foundations. Referring to the classical beam theory, they stated that if denotes the configuration of the deformed beam, then the bending moment satisfies the relation where is the Young modulus of elasticity and is the inertial moment. Considering the deformation caused by a load they deduced, from a free-body diagram, that and where denotes the shear force. For u representing an elastic beam of length which is clamped at its left side and resting on an elastic bearing at its right side =1, and adding a load along its length to cause deformations (Figure 1), Ma and Silva [4] arrived at the following boundary value problem assuming an : the boundary conditions were taken as where and are real functions. The physical interpretation of the boundary conditions is that is the shear force at and the second condition in (1.5) means that the vertical force is equal to which denotes a relation, possibly nonlinear, between the vertical force and the displacement Furthermore, since indicates that there is no bending moment at the beam is resting on the bearing .

Figure 1: Beam on elastic bearing.

Solving (1.3) by means of iterative procedures, Ma and Silva [4] obtained solutions and argued that the accuracy of results depends highly upon the integration method used in the iterative process.

With the rapid development of nonlinear science, many different methods were proposed to solve differential equations, including boundary value problems (BVPS). These two methods are the homotopy perturbation method (HPM) [5–7] and the variational iteration method (VIM) [8–17]. In this paper, it is aimed to apply the variational iteration method proposed by He [14] to different forms of (1.1) subject to boundary conditions of physical significance.

2. Basic Idea of He’s Variational Iteration Method

To clarify the basic ideas of He's VIM, the following differential equation is considered: where L is a linear operator, N is a nonlinear operator, and is an inhomogeneous term. According to VIM, a correction functional could be written as follows:where is a general Lagrange multiplier which can be identified optimally via the variational theory. The subscript indicates the th approximation and is considered as a restricted variation, that is, .

For fourth-order boundary value problem with suitable boundary conditions, Lagrangian multiplier can be identified by substituting the problem into (2.2), upon making it stationary leads to the following: Solving the system of (2.3) yields and the variational iteration formula is obtained in the form

3. The Applications of VIM Method

In this section, the VIM is applied to different forms of the fourth-order boundary value problem introduced in through (1.1).

Example 3.1. Consider the following linear boundary value problem:subject to the boundary conditions The exact solution for this problem is According to (2.5), the following iteration formulation is achieved: Now it is assumed that an initial approximation has the form where , and are unknown constants to be further determined.
By the iteration formula (3.4), the following first-order approximation may be written: Incorporating the boundary conditions (3.2), into , the following coefficients can be obtained: Therefore, the following first-order approximate solution is derived: Comparison of the first-order approximate solution with exact solution is tabulated in Table 1, showing a remarkable agreement.
Similarly, the following second-order approximation is obtained: Therefore, the second-order approximate solution may be written as Again, the obtained solution is of distinguishing accuracy, as indicated in Table 2 and Figure 2.

Table 1: Comparison of the first-order approximate solution with exact solution.

Table 2: Comparison of the second-order approximate solution with exact solution.

Figure 2: Comparison between different solutions.

Example 3.2. Consider the following linear boundary value problem: subject to the boundary conditions The exact solution for this problem is According to (2.5), the iteration formulation may be written as Now it is assumed that an initial approximation has the form Where , and are unknown constants to be further determined.
By the iteration formula (3.14), the following first-order approximation is developed: Incorporating the boundary conditions (3.12), into , it can be written as Therefore, the following first-order approximate solution is obtained: Comparison of the first-order approximate solution with exact solution is tabulated in Table 3, again showing a clear agreement. Even higher accurate solutions could be obtained without any difficulty.
Similarly, the following second-order approximation can be written as Incorporating the boundary conditions, (3.12), into , yields The following second-order approximate solution is then achieved in the following form: The obtained solution is of evident accuracy, as shown in Table 4 and Figure 3.