Using fixed-point techniques and Faber-Schauder systems in adequate Banach spaces, we approximate the solution of a system of nonlinear Fredholm integrodifferential equations of the second kind.
An important area of research interest is the study of systems of nonlinear Fredholm integrodifferential equations. A system of nonlinear Fredholm integrodifferential equations can be written in vectorial form as
Observe that, for
The system (
Many problems of physics and engineering lead to the solution of integro or integrodifferential equations or systems of such equations. In most cases, these cannot be solved by direct methods, and this, together with the powerful computer tools available, has led to the development of numerical methods that allow obtaining approximate solutions of these equations or systems of equations. In literature it is easy to find many of them.
Danfu and Xufeng [
In the present paper we approximate the solution of (
We suppose that
If we reformulate the system (
A direct calculation over
Then, given
Observe that
Now we will make use of the usual Schauder basis
Then, for all
In view of (
This section is devoted to provide a convergence analysis for the numerical scheme
Let
Let us fix
For all
Now we will show that the sequence
From the monotonicity of the Schauder bases
Since the sequence
For
Meanwhile,
Therefore,
Next, we will show that the sequence
Given
In view of the monotonicity of the Schauder bases
Therefore, the sequence
We will prove that the sequences
For
By repeating the previous argument we obtain
Therefore, the sequences
Meanwhile,
In view of the identities (
For a dense subset
With the previous notation and the same hypothesis as in Theorem
The announced estimation follows from the inequalities obtained in Propositions 4 and 5 in [
In the result below we show that the sequence defined in (
With the same hypothesis as in Theorem
For
First we deal with proving (
And, in turn, applying (
Finally, using the triangle inequality,
Observe that under the hypotheses of Theorem
If we consider an interval
We now turn our attention to the application of the method presented in this paper for the numerical solution of six test problems. In order to construct the Schauder basis, we consider the subset
Consider the Fredholm integrodifferential equation appearing in [
Absolute errors for Example
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Consider the Fredholm integrodifferential equation:
Absolute errors for Example
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Consider the Fredholm integrodifferential equation:
Absolute errors for Example
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Consider now the following system of Fredholm integrodifferential equations with the exact solutions
Absolute errors for Example
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Consider now the following system of Fredholm integrodifferential equations with the exact solutions
Absolute errors for Example
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Consider now the following system of Fredholm integrodifferential equations with the exact solutions
Absolute errors for Example
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In this paper we have successfully approximated the solution of systems of nonlinear Fredholm integrodifferential equations. To this end, we have used the Banach fixed-point theorem and the Schauder basis. Moreover, the convergence of the proposed scheme is analyzed and some illustrative examples were included to demonstrate the validity and applicability of the method. The approximating functions
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is partially supported by Junta de Andalucía Grant FQM359 and the ETSIE of the University of Granada, Spain.