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
.
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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ1_HTML.gif)
where and
and
are continuous functions satisfying a Lipschitz condition with respect to the last variables: there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ2_HTML.gif)
for and for
. In the sequel, these conditions will be assumed. It is a simple matter to check that a function
is a solution of (1.1) if, and only if, it is a fixed point of the self-operator of the Banach space
(usual supnorm)
given by the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ3_HTML.gif)
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.
For any and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ4_HTML.gif)
where
In view of the Banach fixed point theorem and Lemma 2.1, has a unique fixed point
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ5_HTML.gif)
Now let us consider a special kind of biorthogonal system for a Banach space. Let us recall that a sequence in a Banach space
is said to be a Schauder basis if for every
there exists a unique sequence of scalars
such that
The associated sequence of (continuous and linear) projections
is defined by the partial sums
We now consider the usual Schauder basis for the space
(supnorm), also known as the Faber-Schauder basis: for a dense sequence of distinct points
with
and
we define
and for all
we use
to stand for the piecewise linear function with nodes at the points
with
for all
and
It is straightforward to show (see [21]) that the sequence of projections
satisfies the following interpolation property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ6_HTML.gif)
In order to define an analogous basis for the Banach space (supnorm), let us consider the mapping
given by (for a real number
,
denotes its integer part)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ7_HTML.gif)
If is a Schauder base for the space
, then the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ8_HTML.gif)
with , is a Schauder basis for
(see [21]). 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 instance, the sequence of projections
satisfies, for all
and for all
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ9_HTML.gif)
Under certain weak conditions, we can estimate the rate of convergence of the sequence of projections. For this purpose, consider the dense subset of distinct points in
and let
be the set
ordered in an increasing way for
Clearly,
is a partition of
. Let
denote the norm of the partition
. The following remarks follow easily from the interpolating properties (2.3) and (2.6) and the mean-value theorems for one and two variables:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ11_HTML.gif)
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.
Let and
Let
be a set of positive numbers and, with the notation above, define inductively, for
and
the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ13_HTML.gif)
where
(1) is a natural number such that
(2) is a natural number such that
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ14_HTML.gif)
Then, for all it is satisfied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ15_HTML.gif)
Proof.
In view of condition () we have, by applying (2.7), that for all
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ16_HTML.gif)
is valid. Analogously, it follows from condition () and (2.8) that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ17_HTML.gif)
As a consequence, we derive that for all we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ18_HTML.gif)
and therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ19_HTML.gif)
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 and
be any subset of
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ20_HTML.gif)
with being the fixed point of the operator
and
Proof.
We know from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ21_HTML.gif)
for , which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ22_HTML.gif)
The proof is complete by applying (2.2) to and taking into account that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ23_HTML.gif)
As a consequence of Theorem 3.1 and Proposition 3.2, if is the exact solution of the nonlinear Volterra integro-differential (1.1), then for the sequence of approximating functions
the error
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ24_HTML.gif)
where In particular, it follows from this inequality that given
there exists
such that
In order to choose and
(projections
and
in Theorem 3.1), we can observe the fact, which is not difficult to check, that the sequences
and
are bounded (and hence conditions (1.1) and (1.3)) in Theorem 3.1 are easy to verify), provided that the scalar sequence
is bounded,
and
are
functions, and
,
,
,
and
satisfy a Lipschitz condition at their last variables. Indeed in view of inequality (3.13),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ25_HTML.gif)
and in particular is bounded. Therefore, taking into account that the Schauder bases considered are monotone (norm-one projections, see [21]), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ26_HTML.gif)
Take and
to derive from the triangle inequality and the last inequality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ27_HTML.gif)
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.
([22, Problem ]). The equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ28_HTML.gif)
has exact solution
Example 4.2.
Consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ29_HTML.gif)
whose exact solution is
The computations associated with the examples were performed using Mathematica 7. In both cases, we choose the dense subset of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ30_HTML.gif)
to construct the Schauder bases in and
. To define the sequence
introduced in Theorem 3.1, we take
and
(for all
) in the expression (3.2), that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470149/MediaObjects/13663_2010_Article_1289_Equ31_HTML.gif)
In Tables 1 and 2 we exhibit, for and
, the absolute errors committed in eight points (
) of
when we approximate the exact solution
by the iteration
. The results in Table 1 improve those in [22].
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Acknowledgment
This research is partially supported by M.E.C. (Spain) and FEDER, project MTM2006-12533, and by Junta de Andaluca Grant FQM359.
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Berenguer, M., Garralda-Guillem, A. & Ruiz Galán, M. Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation. Fixed Point Theory Appl 2010, 470149 (2010). https://doi.org/10.1155/2010/470149
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DOI: https://doi.org/10.1155/2010/470149