Abstract
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.
1. Introduction
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 where is the solution to be calculated and , , and are known.
Observe that, for , the th equation of the system (1) adopts the form with .
The system (1) is linear when for all and we have that
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 [1] utilize the CAS wavelet operational matrix of integration for obtaining numerical solution of linear Fredholm integrodifferential equations. Jafarian and Measoomy Nia [2] offer an architecture of artificial neural networks (NNs) for finding approximate solution of linear Fredholm integral equations system of the second kind. In [3], Maleknejad et al. present a rationalized Haar functions method for solving linear Fredholm integrodifferential systems. In [4] Maleknejad and Tavassoli Kajani use the hybrid Legendre and block-pulse functions on interval to solve the systems of linear integrodifferential equations. In [5], a fully discrete version of a piecewise polynomial collocation method is constructed to solve initial or boundary value problems of linear Fredholm integrodifferential equations with weakly singular kernels. In [6], Pour-Mahmoud et al. extend the Tau method for the numerical solution of integrodifferential equations system (IDES). Yalçinbaş et al. [7] present a Legendre collocation matrix method to solve high-order linear Fredholm integrodifferential equations under the mixed conditions in terms of Legendre polynomials. Yusufoğlu in [8] introduce a numerical method for solving initial value problems for a system of integrodifferential equations (the main idea is based on the interpolations of unknown functions at distinct interpolation points). Yüzbaşı et al. [9] present a numerical matrix method based on collocation points for the approximate solution of the systems of high-order linear Fredholm integrodifferential equations with variable coefficients under mixed conditions in terms of the Bessel polynomials. Zarebnia and Ali Abadi [10] use the Sinc-collocation method to solve systems of nonlinear second-order integrodifferential equations. Berenguer et al. used in [11–14] von Neumann series, fixed-point techniques, and Faber-Schauder systems in Banach spaces to solve integro and integrodifferential equations.
In the present paper we approximate the solution of (1) and we extend the numerical approximation method given in [14]. This paper is organized as follows. In Section 2 we describe the proposed method and in Section 3 the convergence of the proposed method is investigated. In Section 4 some numerical examples are presented to show the efficiency of the proposed scheme. Finally, in Section 5, we end with some conclusions.
2. Description of the Proposed Method
We suppose that , satisfy a global Lipschitz condition in its last variable; that is, there exist such that for all and
If we reformulate the system (1) in terms of an adequate operator, we can derive its unique solvability under a suitable condition. To be more precise, if is the operator given for each as then solving (1) is equivalent to finding a fixed point of the operator .
A direct calculation over leads to for all , where , with and being the Lipschitz constants of and , respectively. Thus, according to the Banach fixed-point theorem (see [15]), (5) has one unique fixed point; equivalently, (1) has one and only one solution provided that . In addition, for each , and in particular .
Then, given , our next target is to obtain . We consider the functions and defined by
Observe that , where for all
Now we will make use of the usual Schauder basis in and the usual Schauder basis for the Banach space (see [16, 17]), although the numerical method given works equally well by replacing it with any complete biorthogonal system in this space. We denote by and the sequences of projections in and , respectively (see Section 3 in [11]).
Then, for all and , where and are the sequences of scalars satisfying and , where and , and for is and with .
In view of (10) we can calculate, at least in a theoretical way, . From a practical point of view, in general these calculations are not possible explicitly, since they are infinite sums. The idea of our numerical method is to truncate them by means of the projections of the Schauder bases , and approximate the solution in this way. Specifically, we consider the sequence defined as follows. Let , , and , be subsets of natural numbers and . Define inductively for and where Observe that for
3. Convergence of the Scheme
This section is devoted to provide a convergence analysis for the numerical scheme . To analyze the convergence we employ the following two results.
Theorem 1. Let and such that and , , , , and for each satisfy a global Lipschitz condition in the last variables. Then, maintaining the notation above, the sequences , , and , with , are bounded.
Proof. Let us fix and write , . Making use of definitions (11), it follows that, for all , , and ,
where “” stands for the usual inner product in .
For all and , we have
with and being the Lipschitz constant of and analogous
with and being the Lipschitz constant of .
Now we will show that the sequence is bounded.
From the monotonicity of the Schauder bases , and the recursive application of this inequality and the following one,
we have
with , and . Applying it inductively, we arrive at
for all , and therefore the sequence is bounded.
Since the sequence in (15) and (16) is bounded it follows that and are uniformly bounded.
For , we have
with and as the Lipschitz constant of .
Meanwhile,
with and as the maximum of the Lipschitz constants for each , .
Therefore, and are bounded.
Next, we will show that the sequence is bounded.
Given , taking into account the definition of , we have for all that
In view of the monotonicity of the Schauder bases and and (15), (16), and (19), we obtain
Therefore, the sequence is also bounded.
We will prove that the sequences , , and are bounded.
For , we have
with and as the Lipschitz constant of .
By repeating the previous argument we obtain
with and as the Lipschitz constant of .
Therefore, the sequences and are bounded.
Meanwhile,
with and as the maximum of the Lipschitz constants for each , . Therefore, is bounded.
In view of the identities (14), we have that the sequences, , , and , with , are bounded.
For a dense subset of distinct points in , let be the set ordered in an increasing way for . Let denote the maximum distance between two consecutive points of .
Theorem 2. With the previous notation and the same hypothesis as in Theorem 1, for all , there are and such that
Proof. The announced estimation follows from the inequalities obtained in Propositions 4 and 5 in [11], respectively, and applying Theorem 1.
In the result below we show that the sequence defined in (11) approximates the solution of (1).
Theorem 3. With the same hypothesis as in Theorem 1, suppose that is the integral operator (5), , and that is the sequence defined by (11). Let us also assume that , , and is a set of positive numbers such that for all we have Then Moreover, if is the exact solution of the integral equation (1), then the error is given by
Proof. For , from (7), we have
First we deal with proving (29). For all and , Theorem 2 gives
And, in turn, applying (29) and recursively (6), we obtain
Finally, using the triangle inequality,
the proof is complete in view of (31) and (33).
Observe that under the hypotheses of Theorem 3, by inequality (30), we have Therefore, given , there exists such that for sufficiently small , since the points of the partition can be chosen in such a way that and become so close to zero as we desire and the first sum on the right hand side approach zero when increases.
Remark 4. If we consider an interval , then and the bound obtained in Theorem 3 for is given by when
4. Numerical Examples
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 defined by , and for , , if , where are integers. To define the sequence , we take and (for all ). We include, for different values of , the absolute errors committed in some representative points of when we approximate the exact solution by the iteration , where is shown in each table. The algorithms associated with the numerical methods were performed using Mathematica 7. In Examples 1, 2, and 3, and . In the other examples, and .
Example 1. Consider the Fredholm integrodifferential equation appearing in [1]: whose exact solution is . Numerical results obtained for this problem when we apply the method described in this paper and the results obtained in [1] are given in Table 1.
Example 2. Consider the Fredholm integrodifferential equation: where is chosen so that the exact solution is given by . The numerical results are given in Table 2.
Example 3. Consider the Fredholm integrodifferential equation: whose exact solution is . The numerical results are given in Table 3.
Example 4. Consider now the following system of Fredholm integrodifferential equations with the exact solutions and : The numerical results are given in Table 4.
Example 5. Consider now the following system of Fredholm integrodifferential equations with the exact solutions and : The numerical results are given in Table 5.
Example 6. Consider now the following system of Fredholm integrodifferential equations with the exact solutions and : The numerical results are given in Table 6.
5. Conclusion
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 and are the sum of integrals of piecewise univariate and bivariate polynomials of degree 2 and the calculation of the coefficients of such polynomials just requires linear combinations of several evaluations of the basic functions at sufficient number of points.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is partially supported by Junta de Andalucía Grant FQM359 and the ETSIE of the University of Granada, Spain.