The reduction method for approximative solution of systems of Singular Integro-Differential Equations in Lebesgue spaces(case γ 6= 0)

Abstract: In this article we have elaborated the numerical schemes of reduction methods for approximate solution of system of singular integro-differential equations when the kernel has a weak singularity. The equations are defined on the arbitrary smooth closed contour of complex plane. We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for the approximate solution of weakly singular integrodifferential equations defined on smooth closed contours in the complex plane. We use the cut-off technique kernel to reduce the weakly singular integrodifferential equation to the continuous one. Our approach is based on the Krykunov theory and Zolotarevski results. We have obtained the theoretical background for these methods in classical Lebesgue spaces.