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.

For this work, the main idea is to make an adapted modification to the Newton-Kantorovich method destined to solve a nonlinear integral equations, so that by this technical method we obtain a simple application to this solution. Moreover, we compare the numerical results obtained by this method against ones obtained by another authors. This comparison showed the efficiency of this method.

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].

In this work, we present a modified linear approximation for solving the first and the second kind Abel–Volterra integral equations. This approximation was used by the author to approximate a weakly singular integral on the curve. Noting that this new technique gives a good approximation of these solutions compared with several methods in several numerical examples.

In this work we present a numerical solution for singular integral equations of the first kind on the oriented smooth contour with Cauchy type kernel. For this one we use an adapted quadratic approximation constructed by the author for this goal, based on the Simpson rule. Many examples are treated in order to prove the efficiency of this approximation.