The present paper presents the application of the polynomial least squares method to nonlinear integral equations of the mixed Volterra-Fredholm type. For this type of equations, accurate approximate polynomial solutions are obtained in a straightforward manner and numerical examples are given to illustrate the validity and the applicability of the method. A comparison with previous results is also presented and it emphasizes the accuracy of the method.
In this paper, we consider nonlinear integral Volterra-Fredholm equations of the general form:
Equations of this type are frequently used to model applications from various fields of science such as elasticity, electricity, and magnetism, fluid dynamics, the dynamic of populations, and mathematical economics.
In general, the exact solution of these nonlinear integral equations cannot be found and thus it is often necessary to find approximate solutions for such equations. In this regard, many approximation techniques were employed over the years. Some of the approximation methods employed in recent years include the following (see the examples in Section Rationalized Haar functions method ([ Chebyshev polynomials method ([ Triangular functions (TF) method ([ Sinc approximation method ([ Collocation methods ([ Optimal control method ([ Radial basis functions method ([ Bernoulli matrix method ([ Homotopy analysis method ([
In the next section, we will present the polynomial least squares method (PLSM), which allows us to determine analytical approximate polynomial solutions for nonlinear integral equations. In the third section, we will compare approximate solutions obtained using PLSM with approximate solutions computed recently for several test problems. If the exact solution of the test problem is polynomial, PLSM is able to find the exact solution. If not, PLSM allows us to obtain approximations with an error relative to the exact solution smaller than the errors obtained using other methods. In most cases, the approximate solutions obtained not only are more precise but also present a simpler expression in comparison to previous ones.
We consider the following operator, corresponding to (
We also consider the so-called remainder associated to (
Before we present the actual steps of the method, we introduce the following types of solutions.
One calls an
One calls a
One also considers the following type of convergence.
One considers the sequence of polynomials
The aim of PLSM is to find a weak
The values of the constants
By substituting the approximate solution (
Next, we attach to (
We compute
Using the constants
The following convergence theorem holds.
The necessary condition for (
Based on the way the coefficients of polynomial
Taking into account the fact that any
In this section, we compute approximate polynomial solutions for several test problems previously solved using other methods and compare the results.
Our first application is a simple nonlinear Fredholm integral equation ([
The exact solution of (
Since the solution is a polynomial, we expected that, by using PLSM, we would be able to find, if not the exact solution, at least a very accurate approximation.
In the following, in order to obtain our approximation, we will perform the steps described in the previous section. The computations were performed using the SAGE open source software (v5.5, available at
We choose the polynomial (
In Step
The corresponding functional (
In Step
For relatively simple problems such as (
The critical points corresponding to a functional
For the problem (
Using the “solve” command in SAGE and excluding the complex solutions, we find the critical points:
In order to find the minimum, we use the second partial derivative test, which is easy enough to be implemented in SAGE, and find that both
Moreover, we find that
We remark that the exact solutions can be found this way even if the initial polynomial
Generally speaking, if the degree of
In this situation, it is still possible to find good approximations of the solutions of the problem solving the system (
In this subsection, we will find approximate solutions for the given problem (
As the following results will show, the Newton method is able to find approximate solutions of (
In order for the sequence of approximations given by Newtons’ formula to converge to the solution(s) of the system (
In the case of the problem (
More precisely, we found the following approximate solutions:
The absolute errors for these approximations, computed as the differences in absolute value between the approximate solutions and the corresponding exact solutions, are of the order of
We remark that, for polynomials
A third approach in finding the minimum of the functional
In the case of the problem (
In the conclusion of this first application, we remark that, in the following applications, depending on the problem and also depending on the precision sought for the approximate solution, we presented one of the three approaches presented above. If the known solution of the problem is polynomial one, we search for the exact solution. If the solution is not polynomial, from the other two approaches, we presented the one which gave the most accurate approximation, as in the case of application 4.
Our second application is a nonlinear Volterra integral equation ([
The exact solution of (
In the following, we will compute the exact solution of the problem (
The critical points of the corresponding functional (
Using the “solve” command in SAGE and excluding the complex solutions, we obtain the following critical points:
Using the second partial derivative test, we deduce that only the first two critical points are minimum points. Computing the values of
The third application is a nonlinear mixed Volterra-Fredholm integral equation ([
The exact solution of (
We will compute the exact solution of the problem (
The critical points of the corresponding functional (
Using the “solve” command in SAGE and again excluding the complex solutions, we obtain the following critical points:
Using the second partial derivative test, it follows that only the first two critical points are minimum points and, by computing the values of
The next application is the nonlinear Volterra-Fredholm integral equation ([
The exact solution of (
Using the PLSM, we computed a seventh order polynomial approximate solution of (
Table
Comparison of the absolute errors of the approximate solutions for Problem (
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The paper presents the computation of approximate polynomial solutions for nonlinear integral equations of mixed Volterra-Fredholm type by using the polynomial least squares method, which is presented as a straightforward and efficient method.
The test problems solved clearly illustrate the accuracy of the method, since, in all of the cases, we were able to compute better approximations than the ones computed in previous papers, and, in most cases, the exact solutions were found. Moreover, the expressions of the approximations computed by PLSM are also simpler than the expressions of the approximations computed by using other methods.
The authors declare that there is no conflict of interests regarding the publication of this paper.