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Abstract and Applied Analysis
Volume 2010, Article ID 135216, 11 pages
http://dx.doi.org/10.1155/2010/135216
Research Article

Nonlinear Volterra Integral Equation of the Second Kind and Biorthogonal Systems

E.U. Arquitectura Técnica, Departamento de Matemática Aplicada, Universidad de Granada, c/ Severo Ochoa, s/n, 18071 Granada, Spain

Received 9 April 2010; Accepted 17 June 2010

Academic Editor: D. Anderson

Copyright © 2010 M. I. Berenguer et al. 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

We obtain an approximation of the solution of the nonlinear Volterra integral equation of the second kind, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.

1. Introduction

This paper puts forth a new method in order to numerically solve the nonlinear Volterra integral equation of the second kind where  :  and the kernel  :  are assumed to be known continuous functions, and the unknown function to be determined is  :  .

Modeling many problems of science, engineering, physics, and other disciplines leads to linear and nonlinear Volterra integral equations of the second kind. These are usually difficult to solve analytically and in many cases the solution must be approximated. Therefore, in recent years several numerical approaches have been proposed (see, e.g., [48]). The numerical methods usually transform the integral equation into a linear or nonlinear system that can be solved by direct or iterative methods. In a recent work [9], the authors use a new technique for solving the linear Volterra integral equation. The method is based on two classical analytical tools: the Geometric Series theorem and Schauder bases in a Banach space. The purpose of this paper is to develop, and generalize to the nonlinear case, an effective method for approximating the solution using biorthogonal systems and another classical tool in analysis: the Banach fixed point theorem.

The work is structured in three parts: in Section 2, we will recall one well-known result and some useful definitions needed later. In Section 3, we define the approximating functions and we study the error. Finally, the numerical results given in Section 4 show the high accuracy of the method.

2. Preliminaries

Let be the Banach space of all continuous and real-valued functions on , endowed with its usual supnorm. Let us start by observing that (1.1) is equivalent to the problem of finding fixed points of the operator defined by To establish the existence of fixed points of (2.2), we will use the version of the Banach fixed-point theorem (see [10]) which we enunciate below: let be a Banach space, let  :  and let be a sequence of nonnegative real numbers such that the series is convergent and for all and for all , . Then has a unique fixed point . Moreover, if is an element in , then we have that for all , In particular, .

On the other hand, we recall briefly some definitions on the theory of Schauder bases and biorthogonal systems in general (see [11]), which are central areas of research, and also some important tools in Functional Analysis. The use of Schauder bases in the numerical study of integral and differential equations has been previously considered in [1214].

Let us start by recalling the notion of biorthogonal system of a Banach space. Let be a Banach space and its topological dual space. A system , where , and ( is Kronecker's delta), is called a biorthogonal system in . We say that the system is a fundamental biorthogonal system if .

We will work with a particular type of fundamental biorthogonal systems. Let us recall that a sequence of elements of a Banach space is called a Schauder basis of if for every there is a unique sequence of scalars such that . A Schauder basis gives rise to the canonical sequence of (continuous and linear) finite dimensional projections , and the associated sequence of (continuous and linear) coordinate functionals in is given by Note that a Schauder basis is always a fundamental biorthogonal system, under the interpretation of the coordinate functionals as biorthogonal functionals.

3. Main Results

We begin this section making use of a Schauder basis in the Banach space endowed with its usual supnorm. To construct such a basis, we recall that a usual Schauder basis in can be obtained from a dense sequence of distinct points from such that and . We set for , and for , we let be a piecewise linear continuous function on with nodes at , uniquely determined by the relations and for . For this basis, the sequence of biorthogonal functionals satisfies (see [1]) for all In addition, the sequence of associated projections satisfies for all for all and for all such that

From the Schauder basis in , we can build another Schauder basis of (see [1, 2]). It is sufficient to consider for all with , where for a real number , will denote its integer part and is the bijective mapping defined by

Remark 3.1. The Schauder basis of has similar properties to the ones for the one-dimensional case.
For all , and for ,
If , then , and for all if
The sequence of associated projections satisfies whenever and
This Schauder basis is monotone, that is, .

Remark 3.2. We have chosen the Schauder basis above for simplicity in the exposition, although the method to be presented also works considering any fundamental biorthogonal system in .
With the previous notation, our first result enables us to obtain the image under operator defined in (2.2) of any continuous function in terms of certain sequences of scalars, sequences which are obtained just by evaluating some functions at adequate points.

Proposition 3.3. Let  :  be the continuous integral operator defined in (2.2). Let and let us consider the function defined by . Let be the sequences of scalars satisfying Then for all we have that where and for

Proof. The result follows directly from the expression in the integral appearing in the definition of .

On the other hand, in order to discuss the application of Banach's fixed point theorem to find a fixed point for the operator defined in (2.2), we establish the following result.

Proposition 3.4. Assume that in (1.1) the kernel satisfies a Lipschitz condition in its third variable: for some constant . Then the integral equation (1.1) has a unique solution . In addition, for each the sequence in converges uniformly to the unique solution and for all ,

Proof. For all and for all we obtain by a mathematical induction (see [3, Theorem ]) that for all . In particular, Since converges for any and , by Banach's fixed point theorem, we will derive the existence and uniqueness of a solution of the integral equation (1.1). From (3.10) and (2.3), we deduce (3.9).

In view of Propositions 3.3 and 3.4, (3.4) gives the unique solution of (1.1). The problem is that generally this expression cannot be calculated explicitly. The idea of the proposed method is to truncate to calculate approximately a sequence of iterations and projections that converge to the solution. More specifically, let  :  be a continuous function, and Consider the continuous functions and for , we define

In order to obtain the convergence of the sequence to the unique solution of (1.1), we need, under some weak condition, to uniformly estimate the rate of the convergence of the sequence of projections in the bidimensional case. To this end, we introduce the following notation that will be used in the next results: if is the dense subset of distinct points in we considered to define the Schauder basis, let be the set ordered in an increasing way for Let denote the maximum distance between two consecutive points of .

Proposition 3.5. Let such that , , satisfy a global Lipschitz condition in the third variable. Then and are uniformly bounded.

Proof. From (3.12), we have that for all , Let and we have for all and with as the Lipschitz constant of .
As a consequence of the monotonicity of the Schauder basis, we have If one applies (3.15) and by repeating the previous argument, Applying recursively this process and the Fubini theorem, we get Thus for all and Hence the sequence is uniformly bounded.
Meanwhile , and in view of the monotonicity of the Schauder basis and the fact that for all and we have that , the boundedness of follows from that of (3.15) and of , which is done below.
Then with and being the Lipschitz constant of .
Similarly, with and as the Lipschitz constant of .
Finally with and as the Lipschitz constant of .
Hence, and are uniformly bounded.

Theorem 3.6. With the previous notation, let ,  , and with , , , satisfying the Lipschitz global condition of the third variable. Then, there is such that for all and

Proof. We use Proposition 3.5. and the inequality resulting from Remark 3.1.(c) and the Mean Value Theorem to get for and

The main result that establishes that the sequence defined in (3.11) and (3.13) approximates the solution of (1.1) as well as giving an upper bond of the error committed is given below.

Theorem 3.7. Let such that satisfies a global Lipschitz condition in the third variable and let . Let and assume that certain positive numbers satisfy and let be the exact solution of the integral equation (1.1). Then where is the Lipschitz constant of .

Proof. On one hand, Proposition 3.4 gives On the other hand, in view of (3.10) for all we have that Hence Then we use the triangular inequality and the proof is complete in view of (3.27) and (3.29).

Remark 3.8. Under the hypothesis of Theorem 3.6, can be estimated as follows: there is such that for all and , Hence, given certain , we can find positive integers such that and by Theorem 3.7, we can state the convergence of and an estimation of the error.

4. Some Examples

The behaviour of the method introduced above will be illustrated with the following three examples.

Example 4.1. The equation has the exact solution

Example 4.2. Consider the equation whose exact solution is

Example 4.3. Consider the equation whose exact solution is

To construct the Schauder basis in , we considered the particular choice , and for , if where are integers. To define the sequence , we take and (for all ). In Tables 1, 2, and 3, we exhibit, for and , the absolute errors committed in eight representative points ( ) of when we approximate the exact solution by the iteration . The computations associated with the examples were performed using Mathematica 7.

tab1
Table 1: Absolute errors for Example 4.1.
tab2
Table 2: Absolute errors for Example 4.2.
tab3
Table 3: Absolute errors for Example 4.3.

5. Conclusions

In this paper, we introduce a new numerical method which approximates the solution of the nonlinear Volterra integral equation of the second kind (1.1). Unlike what happens in the classical methods, as in the collocation one, we do not need to solve high-order nonlinear systems of algebraical equations: for our method we just calculate linear combinations of scalar obtained by evaluating adequate functions. This is done due to the properties of the Schauder basis considered in its development.

Acknowledgments

The research is partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533, and by Junta de Andaluca Grant FQM359.

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