Numerical solution of singular Fredholm integral equations of the first kind using Newton interpolation

Shoukralla, Email; Elganaini, W.; Markos, M.

doi:10.21608/mjeer.2019.62756

Numerical solution of singular Fredholm integral equations of the first kind using Newton interpolation

Document Type : Original Article

Authors

¹ Dept. of Physics and Engineering Mathematics, Faculty of Engineering and Technology, Future University.

² Dept. of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University.

10.21608/mjeer.2019.62756

Abstract

In this paper a computational technique is presented for the
numerical solution of a certain potential-type singular Fredholm
integral equation of the first kind with singular unknown density
function, and a weakly singular logarithmic kernel. This equation is
equivalent to the solution of the Dirichlet boundary value problem
for Laplace equation for an open contour in the plane. The
parameterization of the open contour facilitates the treatment of the
density function’s singularity in the neighborhood of the end-points
of the contour, and the kernel’s singularity. The unknown density
function is replaced by a product of two functions; the first explicitly
expresses the bad behavior of the density function, while the
second is a regular unknown function, which will be interpolated
using Newton interpolation in a matrix form. The singularity of the
parameterized kernel is treated by expanding the two argument
parametric functions into Taylor polynomial of the first degree about
the singular parameter. Moreover, two asymptote formulas are
used for the approximation of the kernel. In addition, an adaptive
Gauss–Legendre formula, is applied for the computations of the
obtained convergent integrals. Thus the required numerical solution
is found to be equivalent to the solution of a system of algebraic
equations. The numerical solution of the illustrated example is
closer to the exact solution; which ensures the high accuracy of the
presented computational technique.

Keywords

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