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.

1. Introduction

Most mathematical models used in many problems of physics, biology, chemistry, engineering, and in other areas are based on integral equations like the linear Volterra integral equation of the second kind:

where and are two known continuous functions and is the unknown function to be determined.

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., [14] and the references therein). Specifically, there are several numerical techniques to solve this equation such as the collocation method, finite element method, and spectral method.

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 [510]), although in each problem the analytical techniques are quite different, as fixed point theorems, duality mapping in a Banach space, and generalized least-squares methods.

Among the main advantages that our method presents over the classical ones, as collocation or quadrature (see [11]), we can point out that it is not necessary to solve linear equations systems. In addition, the involved integrals in our method are immediate and therefore, we do not have to use any quadrature method to calculate them.

The paper is organized as follows: some basic facts and properties on the Volterra equation (1.1) and on Schauder bases in and are reviewed in Section 2. In Section 3 we define the approximating functions and we study the error. Finally, in Section 4 two numerical examples taken from [12, 13] are given to illustrate the theoretical results.

2. Development of the Numerical Method: Classical Tools in Functional Analysis

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

Let be the Banach space of all continuous and real-valued functions on , endowed with its usual sup-norm:

Let us write

and let be the operator defined by

It can be shown by an induction argument and Fubini's theorem that

for all and (we adopt the convention ). Hence,

and therefore, where The convergence of the series of real numbers ensures that the series is convergent in .

This remark and the fact that (1.1) can be written equivalently as

lead us to consider the following result (see [14]).

Theorem 2.1 (geometric series theorem). Let be a Banach space and let be a continuous and linear operator such that the series converges. Then, is a continuous, linear, and bijective operator and

Then the unique solution of (2.7) is given by

Thus, the sequence defined by () converges to the solution of (2.7).

By making use of an appropriate Schauder basis in the space we will replace each () for a new function , easier to calculate, and in such a way the error is small enough.

Let us recall now that a sequence in a Banach space is said to be a Schauder basis provided that for all there exists a unique sequence of scalars in such a way that

The associated sequence of (continuous and linear) biorthogonal functionals in the topological dual of is given by

and the sequence of (continuous and linear) projections is defined by the partial sums:

We now consider the usual Schauder basis for the space (although the results in this paper work for equations stated over functions defined in , we shall just consider the Banach space for the sake of simplicity): for a dense sequence of distinct points in , with and we define

and for all we stand for the piecewise linear continuous function determined by the points satisfying

It is easy to obtain the sequence of biorthogonal functionals (see [15]): if then

and for all

In addition, the sequence of projections satisfies the following interpolation property: for all for all and for all , we have that

In what follows, for a real number , will denote its integer part and the bijective mapping defined by

If and are Schauder bases for the space , then the sequence

with is a Schauder basis for (the proof of this fact can be found in [15, 16]). Therefore, from now on, if is a dense subset of distinct points in , with and , and is the associated usual Schauder basis, then we will write to denote the Schauder basis for 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 all

and for

In particular, the sequence of biorthogonal functionals can be easily obtained: if , then

and for all if we have

As a consequence, the sequence of projections satisfies

whenever and

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 of distinct points in and let be the set ordered in an increasing way for Let denotes the maximum distance between two consecutive points of .

The following result is derived easily from (2.23) and the Mean-Value theorem.

Proposition 2.2. Let and write If then

3. Numerical Study of the Linear Volterra Integral Equation of the Second Kind: Convergence and Error

We are now in a position to define the functions announced in the preceding section. Let and, with the notation above, define inductively, for the functions

where for are natural numbers.

We obtain a first estimation of the error .

Proposition 3.1. Maintaining the notation,

Proof. The triangle inequality gives For the first summand we have For the second one, by an induction argument we can show that Indeed, for , the result is clearly true. Suppose that it holds for . Then,

In order to control the sum in the right-hand term of the inequality stated in Proposition 3.1, let us assume that and let us write

Then we derive from Proposition 2.2 that


and hence,

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

Proposition 3.2. The sequences and are bounded and, as a consequence, the sequence is also bounded.

Proof. First we show that for all , and, as a consequence, by using an inductive argument: since the Schauder basis is clearly monotone (norm-one projections), we have Suppose that the result holds for . Then On the other hand, and thus Finally, the bounding of follows from

Letting and , from (3.10) we have

Then Propositions 3.1 and 3.2 imply the following result, which gives an upper bound of the error.

Theorem 3.3. Suppose and let be the functions defined in (3.1). Then, there exists such that

4. Numerical Examples

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 is

To construct the functions introduced in Section 3, we have taken in the expression (3.1). For such a choice the value of in Section 3 is .

In both cases we exhibit, for and the absolute errors committed in eight representative points of when we approximate the exact solution by the function , where has been determined in the following way: for each we note

and choose satisfying

Example 4.1. The equation (see [12]), has exact solution ; see Table 1.
In Table 2 we compare the approximations obtained as described above with the collocation solution shown in [12] in terms of the norm introduced in the cited article. We point out that in both methods.

Example 4.2. Second example is taken from [13]. We consider the equation whose exact solution is ; see Table 3.
The computed results by the suggested method for () improve the obtained ones in [13] for .


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