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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 470258, 5 pages
Iterative Method for Solving a Beam Equation with Nonlinear Boundary Conditions
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.
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.
In the paper, we consider the following boundary value problem (BVP): which models bending equilibrium of elastic beams on nonlinear supports . 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 , 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  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.
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.
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.
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.
A. Green Function and Its Estimates (See )
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 ).
Lemma B.1. For any function from the space defined by (9), there holds the estimate
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