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.

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


I. S. Grant, W. R. Phillips, Electromagnetism. John Wiley & Sons, 2013.
V. S. Vladimirov, Equations of mathematical physics. Nauka, Moscow,
1881.
A. Dezhbord, Taher Lotfi, Katayoun Mahdiani, A new efficient method for
a case of the singular integral equation of the first kind. Journal of
Computational and Applied Mathematics. 296 (2016) 156-169.
B. L. Yung, S. Lee, U. J. Choi, A modified boundary integral method on
open arcs in the Plane. Computers Math. 31 (1996) 37-43.
E. S. Shoukralla, Approximate solution to weakly singular integral
equations, Journal of appl. Math Modelling. 20 (1996) 800-803.
E. Kendall, Atkinson, I. H. Sloan, The numerical solution of first-kind
logarithmic-kernel integral equations on smooth open arcs. Mathematics
of Computation. 56 (1991) 119-139.
G. Schmidt, B. N. Khoromoskij, Boundary integral equations for the
biharmonic Dirichlet problem on non-smooth domains, Journal of integral
equations and applications. 11 (1999).
K. Maleknejad , A. Ostadi, Using Sinc-collocation method for solving
weakly singular Fredholm integral equations of the first kind, Journal
Applicable Analysis, 96 (2017) 702-713.

"> S. Prössdore, J. Saranen, I. H. Sloan, A discrete method for the logarithmic
kernel integral equations on open arcs, J. Austral. Math. Soc. Ser. B 34
(1993) 401-418.
S. Christiansen, E. B. Hansen, Numerical Solution of boundary value
problem through integral equations, Apple. Math, and Mech., ZAMM. 58
(1978) 14-25.
V. Domı́nguez, High-order collocation and quadrature methods for some
logarithmic kernel integral equations on open arcs, Journal of
Computational & Applied Mathematics. 161 (2003) 145-159.
E. S. Shoukralla, S. A. El-Serafi, The Dirichlet Problem for Laplace
equation for an open boundary, Ain Shams University in Egypt,
Engineering Bulletin. 25 (1990) 544-551.
Y. Hayashi, The Dirichlet problem for the two-dimensional Helmholtz
equations for an open boundary, J. Math. Anal., and Appl. 44 (1973) 489-
530.
Y. V. Shestopalv, Y. G. Smirnov, E. R. Chernokozhin, Logarithmic
integral equations electromagnetics, VSP, 2000.
Y.V. Shestopalov, E.V. Chernokozhin, On the solution to integral
equations with a logarithmic singularity of the kernel on several intervals
of integration: elements of the spectral theory, Visnek, Kharkov National
university, Ukraine. 1058 (2013)