We present a new approach based on the formulation of the integrodifferential quadrature method (hereafter called IDQ) to handle Volterra's and Fredholm's equations. This approach is constructed and tested with some realistic numerical examples using the basic computational aspects.
In the present work, we present a formulation of a new numerical approach which is based on the generalized integrodifferential quadrature method and applied to weakly singular Volterra and Fredholm integrodifferential equations in the linear case. This method studies the situation in which the unknown function is identified as the Lagrange polynomial and the interpolating points of the Tchebychev type are used. The accuracy and efficiency of solving integrodifferential equations are still an ongoing research in numerical analysis. Several different methods have also been modeled for the solutions of linear and nonlinear problem [
The central argument for the interest of the type of this problem comes naturally from its wide applications almost in any branches of science and engineering described by systems of ODEs and PDEs [
New calculations are performed for the construction of the solution by a suitable choice of the interpolating points using the unified integrodifferential quadrature method. Based on this fact, this technique is a very promising and powerful methodology in successfully locating best solutions of the problem. Our main purpose is to develop a general numerical schema for the integrodifferential equations that is universally applicable.
The contents of this paper are organized as follows. In Section
The description of generalized integrodifferential quadrature method is summarized as follows: the starting point of the weakly singular integrodifferential equation is written as
In order to avoid the singularity, (
At this stage, we introduce the quadrature aspect of solutions: the derivatives and integral in expression (
With little effort the original problems (
The relations (
Now the expressions (
The practical part of this study is examined in the following section.
We shall consider the solution of some crack situations in order to show the application and the effectiveness of the method described in Section
We write (
Evaluating (
The accuracy of solutions can be checked by using the error tolerance
In order to verify the efficiency of the method developed in the previous section, the following examples have been selected to provide a comparison with previously published work.
In this example we use Volterra’s integrodifferential equation [
Illustrated in Figure
Solution
We consider the numerical solution of the following Volterra’s integrodifferential equation [
The above equation has the exact solution
The result is displayed in Figure
Solution
Consider the Fredholm integrodifferential equation [
In Figure
Solution
We apply the mentioned technique to solve the fifth-order Fredholm integrodifferential equation [
Solution
This paper has introduced a new formulation that uses the unified integrodifferential quadrature method to handle some integrodifferential equations. The main advantage of Lagrange polynomial is that the weighting coefficients that have to be computed do not depend on the
The preliminary results, obtained through the use of this method, show that the resulting solutions are quite good for all examples which have been selected here as a testbed. We can note that similar patterns of convergence are seen for all four examples with a common tolerance
This method seems to be a powerful alternative and consequently gives very accurate solutions to the problems under consideration. It would be interesting to generalize this method to the nonlinear case. This matter deserves a further work.
The authors gratefully acknowledge helpful conversations with Professor W. Cramer. This work was sponsored in part by the M.E.R.S (Ministère de l’Enseignement et de la Recherche Scientifique): under contract no. D01420060012.