In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra integral equation of the second kind.

Most mathematical models used in many problems of physics, biology, chemistry, engineering, and in other areas are based on integral equations like the

Many authors have paid attention to the study of linear Volterra integral equation of the second kind from the viewpoint of their theoretical properties, numerical treatment, as well as its applications (see e.g., [

In this paper a new technique for solving this linear Volterra integral equation is shown. The method is based on two classical analytical tools: the Geometric Series theorem and Schauder bases in a Banach space. Schauder bases in adequate Banach spaces have been used in other numerical methods for solving some integral, differential, or integrodifferential equations (see [

Among the main advantages that our method presents over the classical ones, as collocation or quadrature (see [

The paper is organized as follows: some basic facts and properties on the Volterra equation (

In this section we show some analytical techniques and some related results, useful for us in order to give our numerical method.

Let

This remark and the fact that (

Let

Then the unique solution

By making use of an appropriate Schauder basis in the space

Let us recall now that a sequence

In what follows, for a real number

Under some weak condition, we can estimate the rate of the convergence of the sequence of projections in the bidimensional case. To this purpose, consider the dense subset

The following result is derived easily from (

Let

We are now in a position to define the functions

We obtain a first estimation of the error

Maintaining the notation,

The triangle inequality gives

In order to control the sum in the right-hand term of the inequality stated in Proposition

To arrive at the announced estimation we finally have the following.

The sequences

First we show that for all

Letting

Then Propositions

Suppose

The behaviour of this method is illustrated by means of the following two examples. The computations associated with the numerical experiments were carried out using Mathematica 7.

The chosen dense subset of

To construct the functions

In both cases we exhibit, for

The equation (see [

In Table

Numerical results for Example

Compared results.

Second example is taken from [

The computed results by the suggested method for

Numerical results for Example

This Research is Partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533, and by Junta de Andalcía Grant FQM359.