Boundary Value Problems and Approximate Solutions

In this paper, we discuss about some basic things of boundary value problems. Secondly, we study boundary conditions involving derivatives and obtain finite difference approximations of partial derivatives of boundary value problems. The last section is devoted to determine an approximate solution for boundary value problems using Variational Iteration Method (VIM) and discuss the basic idea of He’s Variational Iteration Method and its applications.


Introduction
Solutions of Boundary Value Problems can sufficiently closely be approximated by simple and efficient numerical methods.Among these numerical methods are finite difference method, standard 5-point formula, iteration method, relaxation method and standard analytic method.But here the finite-difference method and Variational Iteration Method will be considered.There exist several methods to solve second order boundary value problem.One of these is the finite difference method, which is most popular.Amann (1986) contributed to Quasilinear Parabolic systems under nonlinear boundary conditions.Deng and Levine (2000) studied about the role of critical exponents in blow-up theorems.Friedman (1967) made an introduction to partial differential equations of parabolic type.Friedman and McLeod (1985) developed the blow-up of positive solutions of semi-linear heat equations.Keller (1969) studied elliptic boundary value problems suggested by nonlinear diffusion process.Lady`zenskaja et al. (1988) developed the concept of linear and quasilinear equations of parabolic type.Le Roux (1994) made a semi-discretization in time of nonlinear parabolic equations with blowup of the solutions.
Le Roux (2000) derived a numerical solution of nonlinear reaction diffusion processes.Levine (1990) identified the role of critical exponents in blow-up theorems.Mochizuki and Suzuki (1997) find a critical exponents and critical blow up for quasi-linear parabolic equations.Qi (1991) studied the asymptotics of blow-up solutions of a degenerate parabolic equations.Samarskii et al. (1995) observed a blow-up in quasilinear parabolic equations.Sperp (1980) studied the maximum principles and their applications.Zhang (1997) achieved on blow-up of solutions for a class of nonlinear reaction diffusion equations.

METHODOLOGY
The finite difference method for the solution of a two point boundary value problem consists in replacing the derivatives present in the differential equation and the boundary conditions with the help of finite difference approximations and then solving the resulting linear system of equations by a standard method.
It is assumed that y 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 (Usmani, 1977a;Usmani, 1977b;Momani, 1991).Usmani (1977aUsmani ( , 1977b) ) discussed sixth order methods for the linear differential equation subject to the boundary conditions .The method described by Usmani (1977a) leads to five diagonal linear systems and involves p', p'', q', q'' at a and b, while the method described in Usmani (1977b) leads to nine diagonal linear systems.that there is no bending moment at x= 1, the beam is resting on the bearing g.
Solving (1.3) by means of iterative procedures, Ma and Silva (2004) 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).In this paper, it is aimed to apply the variational iteration method proposed by He (1999) to different forms of (1.1) subject to boundary conditions of physical significance.

Definition:
A boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions.

Definition:
A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Note:
Using the Taylor's series, we have (2.9) Now equation (2.8) under the conditions (2.9) gives rise to a tridiagonal system which can be easily solved by the model method.

OBTAINING FINITE DIFFERENCE APPROXIMATIONS OF PARTIAL DERIVATIVES
Let the x-y plane be divided into a network of rectangles of side and by drawing the set of lines and The points of intersection of these lines are called mesh points (lattice points or grid points).
Then we have the finite difference approximations for the partial derivatives in x-direction, as Further writing as simply , the above approximations are reduced to: where −∞ < a ≤ x ≤ b < ∞, , are finite constants.

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: (5.4) where λ is a general Lagrange multiplier which can be identified optimally via the variational theory and the subscript n indicates the nth 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 (5.4),upon making it stationary leads to the following: (5.5) .
Solving the system of (5.5) yields (5.6) and the variational iteration formula is obtained in the form (5.7)

The Applications of Variational Iteration Method
In this section, the Variational Iteration Method is applied to different forms of the fourth-order boundary value problem introduced in through (5.1).
Example 1: Consider the following linear boundary value problem: The exact solution for this problem is (5.10)According to (5.7), the following iteration formulation is achieved: . (5.11) Now it is assumed that an initial approximation has the form (5.12) where a, b, c, and d are unknown constants to be further determined.
By the iteration formula (5.11), the following first-order approximation may be written: = (5.13) Incorporating the boundary conditions (5.9), into , the following coefficients can be obtained: (5.14) 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: (5.16) Therefore, the second-order approximate solution may be written as

CONCLUSION
This study showed that the finite difference method for the solution of a two point boundary value problem consists in replacing the derivatives present in the differential equation and the boundary conditions with the help of finite difference approximations and then solving the resulting linear system of equations by a standard method.The Variational Iteration Method is remarkably effective for solving boundary value problems.
A fourth-order differential equation with particular engineering applications was solved using the VIM in order to prove its effectiveness.Different forms of the equation having boundary conditions of physical significance were considered.
Comparison between the approximate and exact solutions showed that one-iteration is enough to reach the exact solution.Therefore the Variational Iteration Method is able to solve partial differential equations using a minimum calculation process.This method is a very promoting method, which promises to find wide applications in engineering problems.
Boundary value problems arise in several branches of physics as any physical differential equation will have them.Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.A large class of important boundary value problems are; the Sturm-Liouville problems.The analysis of these problems involves the Eigen functions of a differential operator.Consider the second linear boundary problem: Ma and Silva (2004) 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 E is the Young modulus of elasticity and I is the inertial moment.Considering the deformation caused by a load they deduced, from a free-body diagram, that and where v denotes the shear force.For u representing an elastic beam of Gebreslassie, T., Venketeswara Rao, J., Ataklti, A and Daniel, T (MEJS) Volume 4 (1):102-114, 2012 © CNCS, Mekelle University ISSN: 2220-184X 104 length L= 1, which is clamped at its left side x= 0, and resting on an elastic bearing at its right side x =1, and adding a load f along its length to cause deformations, Ma and Silva [2004] arrived at the following boundary value problem assuming an EI = 1(R) are real functions.The physical interpretation of the boundary conditions is that is the shear force at x = 1, 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 ., Venketeswara Rao, J., Ataklti, A and Daniel, T (MEJS) Volume 4 (1):102-114, 2012 © CNCS, Mekelle University ISSN: 2220-184X 105 (2.3)This is the forward difference approximation for .Also we have (2.4)This gives the back-ward difference approximation for .Then the central difference approximation for is obtained by subtracting (2.2) from (2.1);(2.5)Thisgives us a better approximation to as compared to (2.3) or (2.4).Further adding (2.1) and (2.2), we get;(2.6)Similarlywe can obtain finite-difference approximations of higher derivatives.To solve the boundary-value problem given by with the boundary conditions: and , we divide the range in to n-equal sub intervals of width so that .The corresponding value of are then substituting the values of and in (2.7), we get ;Gebreslassie, T., Venketeswara Rao, J., Ataklti, A and Daniel, T Figure 1.Mesh points of when The general form of the equation for a fixed positive integer n, n ≥ 2 is a differential equation of order 2n

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

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