Abstract
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.
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: 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., 1-4 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.
Introduction
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 5-10 , 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.
Introduction
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.
Introduction
The paper is organized as follows: some basic facts and properties on the Volterra equation 1.1 and on Schauder bases in C 0, 1 and C 0, 1 2 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.
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 C a, b be the Banach space of all continuous and real-valued functions on a, b , endowed with its usual sup-norm:
Development of the Numerical Method: Classical Tools in Functional Analysis
Let us write
Development of the Numerical Method: Classical Tools in Functional Analysis
T is linear and continuous}, 2.2 and let T ∈ L C a, b be the operator defined by
Development of the Numerical Method: Classical Tools in Functional Analysis
It can be shown by an induction argument and Fubini's theorem that
Development of the Numerical Method: Classical Tools in Functional Analysis
Abstract and Applied Analysis 3 for all n ≥ 1, x ∈ C a, b and t ∈ a, b we adopt the convention t 0 t . Hence,
Development of the Numerical Method: Classical Tools in Functional Analysis
and therefore,
Development of the Numerical Method: Classical Tools in Functional Analysis
where Then the unique solution x of 2.7 is given by
Development of the Numerical Method: Classical Tools in Functional Analysis
Thus, the sequence {y n } n∈N defined by y n y 0 Ty n−1 n ≥ 1 converges to the solution x of 2.7 . By making use of an appropriate Schauder basis in the space C a, b 2 we will replace each Ty j−1 j ≥ 1 for a new function w j ∈ C a, b , easier to calculate, and in such a way the error x − y 0 w j is small enough. Let us recall now that a sequence {b n } n≥1 in a Banach space E is said to be a Schauder basis provided that for all x ∈ E there exists a unique sequence of scalars {β n } n≥1 in such a way that
Development of the Numerical Method: Classical Tools in Functional Analysis
The associated sequence of continuous and linear biorthogonal functionals {b * n } n≥1 in the topological dual E * of E is given by
Development of the Numerical Method: Classical Tools in Functional Analysis
Abstract and Applied Analysis and the sequence of continuous and linear projections {P n } n≥1 is defined by the partial sums:
Development of the Numerical Method: Classical Tools in Functional Analysis
We now consider the usual Schauder basis for the space C 0, 1 although the results in this paper work for equations stated over functions defined in a, b , we shall just consider the Banach space C 0, 1 for the sake of simplicity : for a dense sequence of distinct points {t i } i≥1 in 0, 1 , with t 1 0 and t 2 1, we define
Development of the Numerical Method: Classical Tools in Functional Analysis
and for all n ≥ 2 we stand b n for the piecewise linear continuous function determined by the points {t 1 , . . . , t n } satisfying
2.13
It is easy to obtain the sequence of biorthogonal functionals see 15 : if x ∈ C 0, 1 , then
2.13
14 and for all n ≥ 2,
2.13
In addition, the sequence of projections {P n } n≥1 satisfies the following interpolation property: for all x ∈ C 0, 1 , for all n ≥ 1, and for all i ≤ n, we have that
2.13
In what follows, for a real number x, x will denote its integer part and σ σ 1 , σ 2 : N → N × N the bijective mapping defined by σ n :
2.17
If whenever n, i, j ∈ N and σ −1 i, j ≤ n. 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 {t i } i≥1 of distinct points in 0, 1 and let T n be the set {t 1 , . . . , t n } ordered in an increasing way for n ≥ 2. Let ΔT n denotes the maximum distance between two consecutive points of T n .
2.17
The following result is derived easily from 2.23 and the Mean-Value theorem. 
2.24
If n ≥ 2, then
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 w j ∈ C 0, 1 announced in the preceding section. Let m ∈ N, w 0 0 and, with the notation above, define inductively, for j ∈ {1, . . . , m}, the functions
Numerical Study of the Linear Volterra Integral Equation of the Second Kind: Convergence and Error
where n j for j ∈ {1, . . . , m} are natural numbers. We obtain a first estimation of the error x − y 0 w m . 
3.6
In order to control the sum in the right-hand term of the inequality stated in Proposition 3.1, let us assume that k, y 0 ∈ C 1 0, 1 2 and hence,
3.6
To arrive at the announced estimation we finally have the following.
Proposition 3.2.
The sequences { w j } j∈N and { w j } j∈N are bounded and, as a consequence, the sequence {M j } j∈N is also bounded.
Abstract and Applied Analysis
Proof. First we show that for all t ∈ 0, 1 ,
Abstract and Applied Analysis
and, as a consequence,
Abstract and Applied Analysis
by using an inductive argument: since the Schauder basis {B n } n≥1 is clearly monotone normone projections , we have
Abstract and Applied Analysis
Suppose that the result holds for p ≤ j − 1. Then
3.14
On the other hand, 
3.17
Letting β sup{M j } j∈N and δ 4β j≥0 T j , from 3.10 we have
3.17
Then Propositions 3.1 and 3.2 imply the following result, which gives an upper bound of the error. 
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.
Numerical Examples
The chosen dense subset of 0, 1 is
Numerical Examples
To construct the functions w j introduced in Section 3, we have taken n j n in the expression 3.1 . For such a choice the value of h in Section 3 is 1/ n − 1 .
Numerical Examples
In both cases we exhibit, for n 9, 17, 33, and 65 the absolute errors committed in eight representative points of 0, 1 when we approximate the exact solution x by the function y 0 w m , where m has been determined in the following way: for each j ∈ N we note 1.82 × 10 2.14 × 10 The computed results by the suggested method for n 9 h 1/8 improve the obtained ones in 13 for h 1/10.

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