On the Numerical Solution of Singular Integral Equations Using Sandikidze's Approximaton

Abstract

Abstract:The aim of this work is to solve singular integral equations (S.I.E), of Cauchy type on a smooth curve by pieces. This method is based on the approximation of the singular integral of the dominant part [6], where the (S.I.E) is reduced to a linear system of equations and to realize this approach numerically by the means of a program [3,5].

Numerical Solution of the Singular Integral Equations of the First Kind on the Curve

“…This exact solution is used only to show that the numerical solution obtained with our approximation is correct. We apply the algorithms described in [3] and [6] to solve S.I.E. of the first kind and we present results concerning the accuracy of the calculations; in this numerical experiments it is easily to see that the matrix of the system of algebraic equation given by our approximation is invertible, confirmed in [3] and [7].…”

“…Assuming that, for the indices σ, ν = 0, 1, 2, ...., N − 1, the points t and t 0 belong respectively to the arcs t σ t σ+1 and t ν t ν+1 where t α t α+1 designates the smallest arc with ends t α and t α+1 [3], [5], [6] and [7]. Following [6], we define the approximation ψ σν (ϕ; t, t 0 ) for the density ϕ(t) by the following expression…”

“…We apply the algorithms described in [3] and [6] to solve S.I.E. of the first kind and we present results concerning the accuracy of the calculations; in this numerical experiments it is easily to see that the matrix of the system of algebraic equation given by our approximation is invertible, confirmed in [3] and [7]. In each table, ϕ represents the exact solution given in the sense of the principal value of Cauchy and ϕ corresponds to the approximate solution produced by the approximation at points values interpolation [3] and [6].…”

A Study on Singular Integro-Differential Equation of Abel's Type by Iterative Methods

“…The mathematical formulation of solid state physics,plasma physics, fluid mechanics, chemical kinetics and mathematical biology often involve singular integral and integro-differential equations. In the three decades , many powerful and simple methods have been proposed and applied successfully to approximate various type of singular integral and integro-differential equations with a wide range of applications [1][2][3][4][5][6][7][8][9][10].In this work,we discuss the three different methods such as Adomian decomposition method (ADM) that introduced in 1986 [11],variation iteration method (VIM) and homotopy analysis method (HAM) that proposed by Chinese mathematician Ji-Huan He [12][13][14][15] and apply these to solve singular integro-differential equation whit Abel's kernel as follows:…”

Quadratic Approximation for Singular Integrals

“…where, x(s) and y(s) are continuous functions on the finite interval of definition [a, b] and have a continuous first derivatives x'(s) and y'(s) never simultaneously null. Let N be an arbitrary natural number, generally we take it large enough and divide the interval [a, b] Assuming that for the indices σ,v = 0,1,2,…N-1 the points t and t 0 belong respectively to the arcs designates the smallest arc with ends t a and t a+1 (Nadir, 1985;1998;Nadir and Antidze, 2004;Nadir and Lakehali, 2007;Sanikidze, 1970;Antidze, 1975).…”