Local Interpolation Splines and Solution of Integro-Differential Equations of Mechanic’s Problems

Abstract

Integro-differential equations are encountered when solving various problems of mechanics. Although Integro-Differential equations are encountered frequently in mathematical analysis of mechanical problems, very few of these equations will ever give us analytic solutions in a closed form. So that construction of numerical methods is the only way to find the approximate solution. This paper discusses the calculation schemes for solving integro-differential equations using local polynomial spline approximations of the Lagrangian type of the fourth and fifth orders of approximation. The features of solving integro-differential equations with the first derivative and the Fredholm and Volterra integrals of the second kind are discussed. Using the proposed spline approximations, formulas for numerical differentiation are obtained. These formulas are used to approximate the first derivative of a function. The numerical experiments are presented.