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

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## 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## 1. Introduction

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

*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:

## 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.

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

Proof.

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.

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

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.

has exact solution Open image in new window

Example 4.2.

whose exact solution is Open image in new window

Absolute errors for Example 4.1.

Absolute errors for Example 4.2.

## 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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