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Fixed Point Theory and Applications

, 2010:470149 | Cite as

Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation

Open Access
Research Article

Abstract

This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces Open image in new window and Open image in new window .

Keywords

Banach Space Fixed Point Theorem Distinct Point Dense Subset Lipschitz Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

The aim of this paper is to introduce a numerical method to approximate the solution of the nonlinear Volterra integro-differential equation, which generalizes that developed in [1]. Let us consider the nonlinear Volterra integro-differential equation
where Open image in new window and Open image in new window and Open image in new window are continuous functions satisfying a Lipschitz condition with respect to the last variables: there exist Open image in new window such that
for Open image in new window and for Open image in new window . In the sequel, these conditions will be assumed. It is a simple matter to check that a function Open image in new window is a solution of (1.1) if, and only if, it is a fixed point of the self-operator of the Banach space Open image in new window (usual supnorm) Open image in new window given by the formula

Section 2 shows that operator Open image in new window satisfies the hypothesis of the Banach fixed point theorem and thus the sequence Open image in new window converges to the solution Open image in new window of (1.1) for any Open image in new window However, such a sequence cannot be determined in an explicit way. The method we present consists of replacing the first element of the convergent sequence, Open image in new window by the new easy to calculate function Open image in new window and in such a way that the error Open image in new window is small enough. By repeating the same process for the function Open image in new window and so on, we obtain a sequence Open image in new window that approximates the solution Open image in new window of (1.1) in the uniform sense. To obtain such sequence, we will make use of some biorthogonal systems, the usual Schauder bases for the spaces Open image in new window and Open image in new window , as well as their properties. These questions are also reviewed in Section 2. In Section 3 we define the sequence Open image in new window described above and we study the error Open image in new window . Finally, in Section 4 we apply the method to two examples.

Volterra integro-differential equations are usually difficult to solve in an analytical way. Many authors have paid attention to their study and numerical treatment (see for instance [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] for the classical methods and recent results). Among the main advantages of our numerical method as opposed to the classical ones, such as collocation or quadrature, we can point out that it is not necessary to solve algebraic equation systems; furthermore, the integrals involved are immediate and therefore we do not have to require any quadrature method to calculate them. Let us point out that our method clearly applies to the case where the involved functions are defined in Open image in new window , although we have chosen the unit interval for the sake of simplicity. Schauder bases have been used in order to solve numerically some differential and integral problems (see [1, 16, 17, 18, 19, 20]).

2. Preliminaries

We first show that operator Open image in new window also satisfies a suitable Lipschitz condition. This result is proven by using an inductive argument. The proof is similar to that of the linear case (see [1, Lemma Open image in new window ]).

Lemma 2.1.

where Open image in new window

In view of the Banach fixed point theorem and Lemma 2.1, Open image in new window has a unique fixed point Open image in new window and
Now let us consider a special kind of biorthogonal system for a Banach space. Let us recall that a sequence Open image in new window in a Banach space Open image in new window is said to be a Schauder basis if for every Open image in new window there exists a unique sequence of scalars Open image in new window such that Open image in new window The associated sequence of (continuous and linear) projections Open image in new window is defined by the partial sums Open image in new window We now consider the usual Schauder basis for the space Open image in new window (supnorm), also known as the Faber-Schauder basis: for a dense sequence of distinct points Open image in new window with Open image in new window and Open image in new window we define Open image in new window and for all Open image in new window we use Open image in new window to stand for the piecewise linear function with nodes at the points Open image in new window with Open image in new window for all Open image in new window and Open image in new window It is straightforward to show (see [21]) that the sequence of projections Open image in new window satisfies the following interpolation property:
In order to define an analogous basis for the Banach space Open image in new window (supnorm), let us consider the mapping Open image in new window given by (for a real number Open image in new window , Open image in new window denotes its integer part)
If Open image in new window is a Schauder base for the space Open image in new window , then the sequence
with Open image in new window , is a Schauder basis for Open image in new window (see [21]). Therefore, from now on, if Open image in new window is a dense subset of distinct points in Open image in new window , with Open image in new window and Open image in new window , and Open image in new window is the associated usual Schauder basis, then we will write Open image in new window to denote the Schauder basis for Open image in new window obtained in this "natural" way. It is not difficult to check that this basis satisfies similar properties to the ones for the one-dimensional case: for instance, the sequence of projections Open image in new window satisfies, for all Open image in new window and for all Open image in new window with Open image in new window ,
Under certain weak conditions, we can estimate the rate of convergence of the sequence of projections. For this purpose, consider the dense subset Open image in new window of distinct points in Open image in new window and let Open image in new window be the set Open image in new window ordered in an increasing way for Open image in new window Clearly, Open image in new window is a partition of Open image in new window . Let Open image in new window denote the norm of the partition Open image in new window . The following remarks follow easily from the interpolating properties (2.3) and (2.6) and the mean-value theorems for one and two variables:

3. A Method for Approximating the Solution

We now turn to the main purpose of this paper, that is, to approximate the unique fixed point of the nonlinear operator Open image in new window given by (1.3), with the adequate conditions. We then define the approximating sequence described in the Introduction.

Theorem 3.1.

Let Open image in new window and Open image in new window Let Open image in new window be a set of positive numbers and, with the notation above, define inductively, for Open image in new window and Open image in new window the functions

where

(1) Open image in new window is a natural number such that Open image in new window

(2) Open image in new window is a natural number such that Open image in new window with

Then, for all Open image in new window it is satisfied that

Proof.

In view of condition ( Open image in new window ) we have, by applying (2.7), that for all Open image in new window , the inequality
is valid. Analogously, it follows from condition ( Open image in new window ) and (2.8) that for all Open image in new window
As a consequence, we derive that for all Open image in new window we have
and therefore,

as announced.

The next result is used in order to establish the fact that the sequence defined in Theorem 3.1 approximates the solution of the nonlinear Volterra integro-differential equation, as well as giving an upper bond of the error committed.

Proposition 3.2.

with Open image in new window being the fixed point of the operator Open image in new window and Open image in new window

Proof.

We know from Lemma 2.1 that
The proof is complete by applying (2.2) to Open image in new window and taking into account that
As a consequence of Theorem 3.1 and Proposition 3.2, if Open image in new window is the exact solution of the nonlinear Volterra integro-differential (1.1), then for the sequence of approximating functions Open image in new window the error Open image in new window is given by

where Open image in new window In particular, it follows from this inequality that given Open image in new window there exists Open image in new window such that Open image in new window

In order to choose Open image in new window and Open image in new window (projections Open image in new window and Open image in new window in Theorem 3.1), we can observe the fact, which is not difficult to check, that the sequences Open image in new window and Open image in new window are bounded (and hence conditions (1.1) and (1.3)) in Theorem 3.1 are easy to verify), provided that the scalar sequence Open image in new window is bounded, Open image in new window and Open image in new window are Open image in new window functions, and Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window satisfy a Lipschitz condition at their last variables. Indeed in view of inequality (3.13),
and in particular Open image in new window is bounded. Therefore, taking into account that the Schauder bases considered are monotone (norm-one projections, see [21]), we arrive at
Take Open image in new window and Open image in new window to derive from the triangle inequality and the last inequality that

Finally, since the sequence Open image in new window is bounded, Open image in new window also is. Similarly, one proves that Open image in new window is bounded (sequences Open image in new window and Open image in new window are bounded and Open image in new window and Open image in new window are Lipschitz at their second variables) and Open image in new window is bounded (sequences Open image in new window and Open image in new window are bounded and Open image in new window , Open image in new window and Open image in new window are Lipschitz at the third variables).

We have chosen the Schauder bases above for simplicity in the exposition, although our numerical method also works by considering fundamental biorthogonal systems in Open image in new window and Open image in new window .

4. Numerical Examples

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

Example 4.1.

([22, Problem Open image in new window ]). The equation

has exact solution Open image in new window

Example 4.2.

Consider the equation

whose exact solution is Open image in new window

The computations associated with the examples were performed using Mathematica 7. In both cases, we choose the dense subset of Open image in new window
to construct the Schauder bases in Open image in new window and Open image in new window . To define the sequence Open image in new window introduced in Theorem 3.1, we take Open image in new window and Open image in new window (for all Open image in new window ) in the expression (3.2), that is
In Tables 1 and 2 we exhibit, for Open image in new window and Open image in new window , the absolute errors committed in eight points ( Open image in new window ) of Open image in new window when we approximate the exact solution Open image in new window by the iteration Open image in new window . The results in Table 1 improve those in [22].

Notes

Acknowledgment

This research is partially supported by M.E.C. (Spain) and FEDER, project MTM2006-12533, and by Junta de Andaluca Grant FQM359.

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Copyright information

© M. I. Berenguer et al. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • MI Berenguer
    • 1
  • AI Garralda-Guillem
    • 1
  • M Ruiz Galán
    • 1
  1. 1.Departamento de Matemática Aplicada, Escuela Universitaria de Arquitectura TécnicaUniversidad de GranadaGranadaSpain

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