Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation
- MI Berenguer1,
- AI Garralda-Guillem1 and
- M Ruiz Galán1Email author
https://doi.org/10.1155/2010/470149
© M. I. Berenguer et al. 2010
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
and
.
Keywords
1. Introduction









Section 2 shows that operator
satisfies the hypothesis of the Banach fixed point theorem and thus the sequence
converges to the solution
of (1.1) for any
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,
by the new easy to calculate function
and in such a way that the error
is small enough. By repeating the same process for the function
and so on, we obtain a sequence
that approximates the solution
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
and
, as well as their properties. These questions are also reviewed in Section 2. In Section 3 we define the sequence
described above and we study the error
. 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 [2–15] 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
, 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, 16–20]).
2. Preliminaries
We first show that operator
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
]).
Lemma 2.1.













































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
given by (1.3), with the adequate conditions. We then define the approximating sequence described in the Introduction.
Theorem 3.1.





where
(1)
is a natural number such that
(2)
is a natural number such that
with
Proof.
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.
with
being the fixed point of the operator
and
Proof.



where
In particular, it follows from this inequality that given
there exists
such that
















Finally, since the sequence
is bounded,
also is. Similarly, one proves that
is bounded (sequences
and
are bounded and
and
are Lipschitz at their second variables) and
is bounded (sequences
and
are bounded and
,
and
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
and
.
4. Numerical Examples
The behaviour of the numerical method introduced above will be illustrated with the following two examples.
Example 4.1.
Example 4.2.

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
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