Advances in Numerical Analysis

Volume 2013 (2013), Article ID 470258, 5 pages

http://dx.doi.org/10.1155/2013/470258

Research Article

Iterative Method for Solving a Beam Equation with Nonlinear Boundary Conditions

Quang A. Dang¹ and Nguyen Thanh Huong²

¹Institute of Information Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi 10000, Vietnam

²College of Science, Thai Nguyen University, Thai Nguyen City 23000, Vietnam

Received 3 June 2013; Accepted 16 July 2013

Academic Editor: Michele Benzi

Copyright © 2013 Quang A. Dang and Nguyen Thanh Huong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we propose an iterative method for solving a beam problem which is described by a nonlinear fourth-order equation with nonlinear boundary conditions. The method reduces this nonlinear fourth-order problem to a sequence of linear second-order problems with linear boundary conditions. The convergence of the method is proved, and some numerical examples demonstrate the efficiency of the method.

1. Introduction

In the paper, we consider the following boundary value problem (BVP):

which models bending equilibrium of elastic beams on nonlinear supports [1]. Here, represents the deflection of an elastic beam of length , subjected to a force exerted by a foundation, and and express the influence of fixed torsional springs at the ends of the beam.

In [2], the problem (1) by means of a Green function was reduced to the problem of finding fixed point of a nonlinear integral equation by successive approximations. Therefore, at each iteration, it is needed to compute a single and a double integrals and a derivative. Differently from this approach in this paper following the methodology of [3, 4], we lead the solution of problem (1) to a sequence of simple linear boundary value problems (BVPs) for second-order equation, which are easily solved numerically. The convergence of the iterative method is established by the contraction principle. Some performed numerical examples demonstrate the efficiency of the method.

It should be noticed that the idea of reduction of linear BVPs for fourth partial differential equations, namely, for the biharmonic and biharmonic-type equations to operator equations for investigating iterative methods for their solution was successfully used by ourselves in many works, for example, in [5, 6].

2. Iterative Method

First, we reduce problem (1) to an operator equation, and then apply the successive approximation method to the latter one. For this purpose, we set

Then, due to the boundary conditions for the function we have

and the function has the property

By the setting (2), problem (1) is decomposed to the problems

Obviously, the solution from these problems depends on the function . Consequently, its derivative also depends on . Therefore, we can represent this dependence by an operator

Combining with the first relation in (2), we get the operator equation for :

That is, is a fixed point of .

Now, we consider properties of the operator . For this purpose, we introduce the space

Next, we make the following assumptions on the given functions in Problem (1): there exist constants , , and such that

for any ,  and  .

Proposition 1. The operator maps the space to and is a contraction operator in if the number

Proof. Since by definition , then . Due to the boundary conditions for , we have ; that is, .

Now, we prove the contraction property of . Let ,   be two functions from , ,   and , and let   be the corresponding solutions of the problems (5), (6). Then, using Appendix A, it is easy to get

Consequently, due to the second inequality in , we have

Here and in sequel, . For the solution of (5) by Appendix A, we have the representation

Therefore,

Using the assumptions (10) and Lemma B.1 in Appendix B, we have the estimate

Combining this estimate with (13), we obtain

So, under the assumptions (10), the operator is contractive with the contraction coefficient .

Now, we consider the following iterative method for finding the fixed point of .

(i)Given an initial approximation , for example, , in .(ii)Knowing (), solve consecutively two BVPs

(iii)Update the new approximation

From the fixed point theorem and , we have the following.

Theorem 2. Under the assumptions (10) and (11) the iterative method (5), (6) converges with rate of geometric progression with the quotient , and there hold the estimates

where is the exact solution of the original problem (1).

3. Numerical Examples

In order to realize the iterative process, we use the finite difference method [7] for solving BVPs (5) and (6), where we use formulas of second-order approximation for the second derivatives and the trapezium formula for computing definite integral on a uniform grid with the stepsize , where is the number of grid points. For computing the first derivative in (19), we also use formulas of second-order approximation.

For testing the convergence of the method, we perform some experiments for the case of the known exact solutions and also for the case of the unknown exact solutions.

Example 1. We take as in [2]

The exact solution in is

The convergence of the method is presented in Table 1, where .

Table 1: The convergence in Example 1 for the starting approximation .

Number of iterations

80
16

100
16

150
18

200
20

250
22

Example 2. We take as in [2]

The exact solution in is

The convergence of the method is presented in Table 2.

Table 2: The convergence in Example 2 for the starting approximation .

Number of iterations

80
4
0.00026809
0.0003125

100
4
0.000012153
0.0002

140
4

160
4

200
4

250
4

In the next examples, the exact solution of Problem (1) is unknown, and we carry out iterations until .

Example 3. We consider the influence of the right-hand side on the solution. For this purpose, we take two functions

with the same

The convergence of the iterative method in these cases is given in Table 3, where and are the number of iterations for the cases of and , respectively. The graph of the approximate solutions for these cases is depicted in Figure 1.

Table 3: The convergence in Example 3 for the starting approximation .

80
6
6

100
6
6

150
6
6

200
7
7

250
7
7

Figure 1: Graph of approximate solutions in Example 3 for the cases and .

Example 4. We consider the influence of the functions and in boundary conditions on the solution. For this purpose we take

and two combinations of and

The convergence of the iterative method in these cases is given in Table 4, where and are the number of iterations for the cases of and , respectively. The graph of the approximate solutions for these cases is depicted in Figure 2.

Table 4: The convergence in Example 4 for the starting approximation .

80
6
6

100
6
6

150
7
6

200
7
7

250
7
7

Figure 2: Graph of approximate solutions in Example 4 for different pairs of .

4. Concluding Remarks

In this paper, we propose an iterative method for solving a nonlinear beam equation with nonlinear boundary conditions. The idea of the method is the reduction of the problem for the fourth-order equation to a sequence of linear second-order problems, which are easily solved numerically. The method can be successfully applied to other nonlinear beam problems, for example, the problems considered in [8–10]. For the problems in [8, 9], the nonlinear BVPs are reduced to sequences of Cauchy problems for linear second-order ODE. The application of the proposed method to these problems will be investigated in the future.

Appendices

A. Green Function and Its Estimates (See [2])

The Green function associated to the second-order problem

is

So, the solution of (A.1) is represented in the form

For the function Green, there hold the estimates

B. Norm in

In the space

there holds the estimate

where (see [2]).

Lemma B.1. For any function from the space defined by (9), there holds the estimate

Indeed, it is valid because for this function if setting

then we have

and (B.3) follows immediately from (B.2).

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