Adapted quadratic approximation for singular integrals

Abstract

Abstract. The goal of this work is to present an adapted modification to the parabolic approximation of the density function for singular integrals of Cauchy type. This approximation serves to eliminate the singularity of the integral and gives the help to obtain the numerical solution of singular integral equations with Cauchy type kernel on an oriented smooth contour.Mathematics subject classification (2010): Primary 45D05, 45E05, 45L05; Secondary 65R20.

A new technical for solving a weakly singular integro-differential equations

“…In this section we describe some of the numerical experiments performed in solving the weakly singular integro-differential equations (1), using collocation methods with the approximation technical in [5,7]. In all cases, the curve is taking the unit circle and we chose the right hand side f(t) in such way that we know the exact solution.…”

“…In each table, ϕ represents the given exact solution of the weakly singular integro-differential equations and ϕ corresponds to the approximate solution of the equation produced by the approximation method for singular integral with logarithmic kernel in [5,7].…”

“…It is known that, the most effective methods for the approximate solution of weakly singular integrodifferential equations consists in their reduction to a system of linear algebraic equations by the replacement of the integral with a proper quadrature sum [5][6][7].…”

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

“…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…”

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).…”