Open Access

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

Fixed Point Theory and Applications20102010:470149

DOI: 10.1155/2010/470149

Received: 22 March 2010

Accepted: 14 June 2010

Published: 5 July 2010

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq2_HTML.gif .

1. Introduction

The aim of this paper is to introduce a numerical method to approximate the solution of the nonlinear Volterra integro-differential equation, which generalizes that developed in [1]. Let us consider the nonlinear Volterra integro-differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq5_HTML.gif are continuous functions satisfying a Lipschitz condition with respect to the last variables: there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq6_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ2_HTML.gif
(1.2)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq7_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq8_HTML.gif . In the sequel, these conditions will be assumed. It is a simple matter to check that a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq9_HTML.gif is a solution of (1.1) if, and only if, it is a fixed point of the self-operator of the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq10_HTML.gif (usual supnorm) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq11_HTML.gif given by the formula
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ3_HTML.gif
(1.3)

Section 2 shows that operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq12_HTML.gif satisfies the hypothesis of the Banach fixed point theorem and thus the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq13_HTML.gif converges to the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq14_HTML.gif of (1.1) for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq15_HTML.gif 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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq16_HTML.gif by the new easy to calculate function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq17_HTML.gif and in such a way that the error https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq18_HTML.gif is small enough. By repeating the same process for the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq19_HTML.gif and so on, we obtain a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq20_HTML.gif that approximates the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq21_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq23_HTML.gif , as well as their properties. These questions are also reviewed in Section 2. In Section 3 we define the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq24_HTML.gif described above and we study the error https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq25_HTML.gif . 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 [215] 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq26_HTML.gif , 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, 1620]).

2. Preliminaries

We first show that operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq27_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq28_HTML.gif ]).

Lemma 2.1.

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq30_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ4_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq31_HTML.gif

In view of the Banach fixed point theorem and Lemma 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq32_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq33_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ5_HTML.gif
(2.2)
Now let us consider a special kind of biorthogonal system for a Banach space. Let us recall that a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq34_HTML.gif in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq35_HTML.gif is said to be a Schauder basis if for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq36_HTML.gif there exists a unique sequence of scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq37_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq38_HTML.gif The associated sequence of (continuous and linear) projections https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq39_HTML.gif is defined by the partial sums https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq40_HTML.gif We now consider the usual Schauder basis for the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq41_HTML.gif (supnorm), also known as the Faber-Schauder basis: for a dense sequence of distinct points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq42_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq44_HTML.gif we define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq45_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq46_HTML.gif we use https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq47_HTML.gif to stand for the piecewise linear function with nodes at the points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq48_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq49_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq51_HTML.gif It is straightforward to show (see [21]) that the sequence of projections https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq52_HTML.gif satisfies the following interpolation property:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ6_HTML.gif
(2.3)
In order to define an analogous basis for the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq53_HTML.gif (supnorm), let us consider the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq54_HTML.gif given by (for a real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq56_HTML.gif denotes its integer part)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ7_HTML.gif
(2.4)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq57_HTML.gif is a Schauder base for the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq58_HTML.gif , then the sequence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ8_HTML.gif
(2.5)
with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq59_HTML.gif , is a Schauder basis for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq60_HTML.gif (see [21]). Therefore, from now on, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq61_HTML.gif is a dense subset of distinct points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq62_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq64_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq65_HTML.gif is the associated usual Schauder basis, then we will write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq66_HTML.gif to denote the Schauder basis for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq67_HTML.gif 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 instance, the sequence of projections https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq68_HTML.gif satisfies, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq69_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq70_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq71_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ9_HTML.gif
(2.6)
Under certain weak conditions, we can estimate the rate of convergence of the sequence of projections. For this purpose, consider the dense subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq72_HTML.gif of distinct points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq73_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq74_HTML.gif be the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq75_HTML.gif ordered in an increasing way for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq76_HTML.gif Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq77_HTML.gif is a partition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq78_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq79_HTML.gif denote the norm of the partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq80_HTML.gif . The following remarks follow easily from the interpolating properties (2.3) and (2.6) and the mean-value theorems for one and two variables:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ10_HTML.gif
(2.7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ11_HTML.gif
(2.8)

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq81_HTML.gif given by (1.3), with the adequate conditions. We then define the approximating sequence described in the Introduction.

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq83_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq84_HTML.gif be a set of positive numbers and, with the notation above, define inductively, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq86_HTML.gif the functions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ12_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ13_HTML.gif
(3.2)

where

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq87_HTML.gif is a natural number such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq88_HTML.gif

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq89_HTML.gif is a natural number such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq90_HTML.gif with

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ14_HTML.gif
(3.3)
Then, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq91_HTML.gif it is satisfied that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ15_HTML.gif
(3.4)

Proof.

In view of condition ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq92_HTML.gif ) we have, by applying (2.7), that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq93_HTML.gif , the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ16_HTML.gif
(3.5)
is valid. Analogously, it follows from condition ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq94_HTML.gif ) and (2.8) that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq95_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ17_HTML.gif
(3.6)
As a consequence, we derive that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq96_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ18_HTML.gif
(3.7)
and therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ19_HTML.gif
(3.8)

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.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq98_HTML.gif be any subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq99_HTML.gif Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ20_HTML.gif
(3.9)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq100_HTML.gif being the fixed point of the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq102_HTML.gif

Proof.

We know from Lemma 2.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ21_HTML.gif
(3.10)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq103_HTML.gif , which implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ22_HTML.gif
(3.11)
The proof is complete by applying (2.2) to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq104_HTML.gif and taking into account that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ23_HTML.gif
(3.12)
As a consequence of Theorem 3.1 and Proposition 3.2, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq105_HTML.gif is the exact solution of the nonlinear Volterra integro-differential (1.1), then for the sequence of approximating functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq106_HTML.gif the error https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq107_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ24_HTML.gif
(3.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq108_HTML.gif In particular, it follows from this inequality that given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq109_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq110_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq111_HTML.gif

In order to choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq113_HTML.gif (projections https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq115_HTML.gif in Theorem 3.1), we can observe the fact, which is not difficult to check, that the sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq117_HTML.gif are bounded (and hence conditions (1.1) and (1.3)) in Theorem 3.1 are easy to verify), provided that the scalar sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq118_HTML.gif is bounded, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq119_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq120_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq121_HTML.gif functions, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq123_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq126_HTML.gif satisfy a Lipschitz condition at their last variables. Indeed in view of inequality (3.13),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ25_HTML.gif
(3.14)
and in particular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq127_HTML.gif is bounded. Therefore, taking into account that the Schauder bases considered are monotone (norm-one projections, see [21]), we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ26_HTML.gif
(3.15)
Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq129_HTML.gif to derive from the triangle inequality and the last inequality that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ27_HTML.gif
(3.16)

Finally, since the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq130_HTML.gif is bounded, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq131_HTML.gif also is. Similarly, one proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq132_HTML.gif is bounded (sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq134_HTML.gif are bounded and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq136_HTML.gif are Lipschitz at their second variables) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq137_HTML.gif is bounded (sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq139_HTML.gif are bounded and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq142_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq144_HTML.gif .

4. Numerical Examples

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

Example 4.1.

([22, Problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq145_HTML.gif ]). The equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ28_HTML.gif
(4.1)

has exact solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq146_HTML.gif

Example 4.2.

Consider the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ29_HTML.gif
(4.2)

whose exact solution is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq147_HTML.gif

The computations associated with the examples were performed using Mathematica 7. In both cases, we choose the dense subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq148_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ30_HTML.gif
(4.3)
to construct the Schauder bases in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq150_HTML.gif . To define the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq151_HTML.gif introduced in Theorem 3.1, we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq153_HTML.gif (for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq154_HTML.gif ) in the expression (3.2), that is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ31_HTML.gif
(4.4)
In Tables 1 and 2 we exhibit, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq156_HTML.gif , the absolute errors committed in eight points ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq157_HTML.gif ) of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq158_HTML.gif when we approximate the exact solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq159_HTML.gif by the iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq160_HTML.gif . The results in Table 1 improve those in [22].
Table 1

Absolute errors for Example 4.1.

  

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq161_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq162_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq163_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq164_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq165_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq166_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq167_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq168_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq169_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq170_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq171_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq172_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq173_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq174_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq175_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq176_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq177_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq178_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq179_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq180_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq181_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq182_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq183_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq184_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq185_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq186_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq187_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq188_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq189_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq190_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq191_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq192_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq193_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq194_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq195_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq196_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq197_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq198_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq199_HTML.gif

Table 2

Absolute errors for Example 4.2.

  

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq200_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq201_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq202_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq203_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq204_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq205_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq206_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq207_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq208_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq209_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq210_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq211_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq212_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq213_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq214_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq215_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq216_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq217_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq218_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq219_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq220_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq221_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq222_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq223_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq224_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq225_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq226_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq227_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq228_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq229_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq230_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq231_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq232_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq233_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_IEq234_HTML.gif

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Declarations

Acknowledgment

This research is partially supported by M.E.C. (Spain) and FEDER, project MTM2006-12533, and by Junta de Andaluca Grant FQM359.

Authors’ Affiliations

(1)
Departamento de Matemática Aplicada, Escuela Universitaria de Arquitectura Técnica, Universidad de Granada

References

  1. Berenguer MI, Fortes MA, Garralda-Guillem AI, Ruiz Galán M: Linear Volterra integro-differential equation and Schauder bases. Applied Mathematics and Computation 2004,159(2):495–507. 10.1016/j.amc.2003.08.132MathSciNetView ArticleMATHGoogle Scholar
  2. Bertola M, Gekhtman M, Szmigielski J: Cubic string boundary value problems and Cauchy biorthogonal polynomials. Journal of Physics A 2009,42(45):-13.
  3. Brunner H: The numerical treatment of Volterra integro-differential equations with unbounded delay. Journal of Computational and Applied Mathematics 1989, 28: 5–23. 10.1016/0377-0427(89)90318-XMathSciNetView ArticleMATHGoogle Scholar
  4. Brunner H: High-order methods for the numerical solution of Volterra integro-differential equations. Journal of Computational and Applied Mathematics 1986,15(3):301–309. 10.1016/0377-0427(86)90221-9MathSciNetView ArticleMATHGoogle Scholar
  5. Brunner H: A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations. Journal of Computational and Applied Mathematics 1982,8(3):213–229. 10.1016/0771-050X(82)90044-4MathSciNetView ArticleMATHGoogle Scholar
  6. Brunner H, Pedas A, Vainikko G: A spline collocation method for linear Volterra integro-differential equations with weakly singular kernels. BIT 2001,41(5):891–900. 10.1023/A:1021920724315MathSciNetView ArticleMATHGoogle Scholar
  7. Brunner H, van der Houwen PJ: The Numerical Solution of Volterra Equations, CWI Monographs. Volume 3. North-Holland, Amsterdam, The Netherlands; 1986:xvi+588.Google Scholar
  8. Crisci MR, Russo E, Vecchio A: Time point relaxation methods for Volterra integro-differential equations. Computers & Mathematics with Applications 1998,36(9):59–70. 10.1016/S0898-1221(98)00192-8MathSciNetView ArticleMATHGoogle Scholar
  9. Darani MRA, Adibi H, Lakestani M: Numerical solution of integro-differential equations using flatlet oblique multiwavelets. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2010,17(1):55–74.MathSciNetMATHGoogle Scholar
  10. Lin T, Lin Y, Rao M, Zhang S: Petrov-Galerkin methods for linear Volterra integro-differential equations. SIAM Journal on Numerical Analysis 2000,38(3):937–963. 10.1137/S0036142999336145MathSciNetView ArticleMATHGoogle Scholar
  11. Luo Z, Nieto JJ: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009,70(6):2248–2260. 10.1016/j.na.2008.03.004MathSciNetView ArticleMATHGoogle Scholar
  12. Nieto JJ, Rodríguez-López R: New comparison results for impulsive integro-differential equations and applications. Journal of Mathematical Analysis and Applications 2007,328(2):1343–1368. 10.1016/j.jmaa.2006.06.029MathSciNetView ArticleMATHGoogle Scholar
  13. Pour-Mahmoud J, Rahimi-Ardabili MY, Shahmorad S: Numerical solution of Volterra integro-differential equations by the tau method with the Chebyshev and Legendre bases. Applied Mathematics and Computation 2005,170(1):314–338. 10.1016/j.amc.2004.11.039MathSciNetView ArticleMATHGoogle Scholar
  14. Ramos JI: Iterative and non-iterative methods for non-linear Volterra integro-differential equations. Applied Mathematics and Computation 2009,214(1):287–296. 10.1016/j.amc.2009.03.067MathSciNetView ArticleMATHGoogle Scholar
  15. Song Y, Baker CTH: Qualitative behaviour of numerical approximations to Volterra integro-differential equations. Journal of Computational and Applied Mathematics 2004,172(1):101–115. 10.1016/j.cam.2003.12.049MathSciNetView ArticleMATHGoogle Scholar
  16. Berenguer MI, Gámez D, Garralda-Guillem AI, Ruiz Galán M, Serrano Pérez MC: Analytical techniques for a numerical solution of the linear Volterra integral equation of the second kind. Abstract and Applied Analysis 2009, Article ID 149367 2009:-12 Pages.Google Scholar
  17. Berenguer MI, Fernández Muñoz MV, Garralda-Guillem AI, Ruiz Galán M: Numerical treatment of fixed point applied to the nonlinear Fredholm integral equation. Fixed Point Theory and Applications 2009, Article ID 735638 2009:-8 Pages.Google Scholar
  18. Gámez D, Garralda-Guillem AI, Ruiz Galán M: High-order nonlinear initial-value problems countably determined. Journal of Computational and Applied Mathematics 2009,228(1):77–82. 10.1016/j.cam.2008.08.039MathSciNetView ArticleMATHGoogle Scholar
  19. Gámez D, Garralda-Guillem AI, Ruiz Galán M: Nonlinear initial-value problems and Schauder bases. Nonlinear Analysis: Theory, Methods & Applications 2005,63(1):97–105. 10.1016/j.na.2005.05.005MathSciNetView ArticleMATHGoogle Scholar
  20. Palomares A, Ruiz Galán M: Isomorphisms, Schauder bases in Banach spaces, and numerical solution of integral and differential equations. Numerical Functional Analysis and Optimization 2005,26(1):129–137. 10.1081/NFA-200051625MathSciNetView ArticleMATHGoogle Scholar
  21. Semadeni Z: Schauder Bases in Banach Spaces of Continuous Functions, Lecture Notes in Mathematics. Volume 918. Springer, Berlin, Germany; 1982:v+136.Google Scholar
  22. Feldstein A, Sopka JR: Numerical methods for nonlinear Volterra integro-differential equations. SIAM Journal on Numerical Analysis 1974, 11: 826–846. 10.1137/0711067MathSciNetView ArticleMATHGoogle Scholar

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© M. I. Berenguer et al. 2010

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