The Modified Degenerate Kernel Method for the Multi-Dimensional Fredholm Integral Equations of the Second Kind

Introduction

For the scientific study of a natural phenomenon it must be modeled. The resulting model is often expressed as a differential equation (DE), an integral equation (IE) or an integro-differential equation (IDE) or a system of these. Therefore, integral equations and their solutions play a major role in the fields of science and engineering. There are several numerical and analytical methods for solving above mentioned models. Analytical methods include homotopy and other methods that give the answer as a sequence or series. Two most important classes of numerical methods for integral equations are projection methods, including the Galerkin and collocation methods, and Nystrom methods. The degenerate kernel method (DKM) is a well-known classical method for solving Fredholm integral equations of the second kind, and it is one of the easiest numerical methods to define and analyze. This method for a given degenerate kernel is called direct computation method (DCM). In this paper, to investigate the multi-dimensional Fredholm integral equations of the second kind a modified degenerate kernel method (MDKM) is used. To construct the mentioned modification, the source function is approximated by the same method which employed to obtain a degenerate approximation of the kernel. Often dealing with nonlinear integral equations poses challenges, the most important of which is to find all the solutions or appropriate approximations of them. In this study, we demonstrate that for the linear and nonlinear equations that are "possible" to find the exact solution or solutions, the proposed method performs this well and without fail. The term "possibility" here is the solution to the equation under consideration, belonging to the subspace generated by the base. Also, the proposed method is able to provide an appropriate approximation of solutions that are not available for the "possibility" mentioned above.

Material and methods

In MDKM, the interpolation method is used to make a degenerate approximation of a non-degenerate kernel as well as source function. Lagrange polynomials are adopted for the interpolation. This method transforms an integral equation of the second kind, to a system of algebraic equations. The error and convergence of the algorithm are given strictly.

Results and discussion

The efficiency of the approach will be shown by applying the procedure on some prototype examples and then a comparison will be done with some other methods. The reported results demonstrate that the present method can obtain all the exact solutions for equations that have not previously been reported using other methods. Also, the results reported in the table of CPU time indicate that the computation cost of our method is very suitable.

Conclusion

The following conclusions were drawn from this research.

It is possible to obtain all the exact solutions of an integral equation with a non-degenerate kernel.
The presented method can give the closed form of the exact solution(s) of an integral equation.
The presented method shows that the nonlinearity of an integral equation cannot change the form of the exact solution(s).