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.

This paper puts forth a new method in order to numerically solve the

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., [

The work is structured in three parts: in Section

Let

On the other hand, we recall briefly some definitions on the theory of Schauder bases and biorthogonal systems in general (see [

Let us start by recalling the notion of biorthogonal system of a Banach space. Let

We will work with a particular type of fundamental biorthogonal systems. Let us recall that a sequence

We begin this section making use of a Schauder basis in the Banach space

From the Schauder basis

The Schauder basis

For all

If

The sequence of associated projections

This Schauder basis is monotone, that is,

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

Let

The result follows directly from the expression

On the other hand, in order to discuss the application of Banach's fixed point theorem to find a fixed point for the operator

Assume that in (

For all

In view of Propositions

In order to obtain the convergence of the sequence

Let

From (

As a consequence of the monotonicity of the Schauder basis, we have

Meanwhile

Then

Similarly,

Finally

Hence,

With the previous notation, let

We use Proposition

The main result that establishes that the sequence defined in (

Let

On one hand, Proposition

Under the hypothesis of Theorem

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

The equation

Consider the equation

Consider the equation

To construct the Schauder basis in

Absolute errors for Example

Absolute errors for Example

Absolute errors for Example

In this paper, we introduce a new numerical method which approximates the solution of the nonlinear Volterra integral equation of the second kind (

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