Development of direct numerical integration methods for one-dimensional integro-differential equations in mechanics

Abstract

We propose constructive methods for direct numerical integration of one-dimensional singular and hypersingular integro-differential equations for the case in which their solution has an asymptotics of power-law type at the endpoints of the integration interval. The approaches are qualitatively divided into two types, typical of complex and real asymptotics. In the first case, the solution is constructed as an expansion with respect to a finite system of orthogonal polynomials (with the endpoint asymptotics explicitly taken into account), the singular and hypersingular integrals are calculated, the regular (generalized) kernel is replaced by a degenerate kernel of special form, and then the integral containing this kernel is calculated analytically (or by direct numerical computation). The application of the collocation method to the functional equation thus constructed permits obtaining a system of linear algebraic equations for the coefficients of the solution expansion. For the real asymptotics, we develop a direct approach based on the approximation of the unknown function by the Lagrange polynomial (with the endpoint asymptotics taken into account), the use of quadrature formulas of interpolation type, and the construction of a linear algebraic system for the values of the unknown function on a discrete set of points by using the collocation method. We present the results of numerical computations and compare them with the analytic solutions.