Numerical Solution of Integro-Differential Equations for Fracture Mechanics Problems

Abstract

The paper is dedicated to the development of methods for solving one-dimensional singular and hypersingular integro-differential equations with generalized Cauchy kernels. Using the means of the theory of special functions, constructive methods of direct (without regularization) numerical solution of such equations are suggested. One-dimensional singular integro-differential equation (SIDE) with standardized integration interval in sufficiently general case is considered: $$ \begin{gathered} A\phi '(\eta ) + B\phi (\eta ) + C\int\limits_{ - 1}^1 {\frac{{\phi '(\xi )d\xi }} {{\xi - \eta }}} + D\int\limits_{ - 1}^1 {\frac{{\phi '(\xi )d\xi }} {{\xi - \eta }}} + \int\limits_{ - 1}^1 {K(\xi ,\eta )} \phi '(\xi )d\xi + \hfill \\ + \int\limits_{ - 1}^1 {L(\xi ,\eta )} \phi (\xi )d\xi = p(\eta ),{\text{ }} - 1 < \eta < 1,{\text{ }}\phi '{\text{(}}x{\text{) = }}\frac{{d\phi (x)}} {{dx}} \hfill \\ \end{gathered} $$ (1) Here φ (ξ) is unknown function, p (η) is bounded continuous function known on the interval [−1, 1], and kernels K (ξ, η) and L(ξ, η) can have “fixed” singularities on the end points of integration interval, i.e. becomes unbounded only if ξ and η approach simultaneously to one of end point of the interval[−1, 1] (ξ = η → ± 1 ). In the latter it is assumed, that K (ξ, η) and L (ξ, η) are generalized Cauchy kernels, i.e. their singularities have form ρ −1 (ρ → 0) (Erdogan et al. [1]). A, B, C, and D are constants, which may be functions of η under certain assumptions (see Muskhelishvili [2]). Constants and functions K (ξ, η), L (ξ, η) and p(η) and are real or complex