On the solution of integral equations with strongly singular kernels

Abstract

In this paper some useful formulas are developed to evaluate integrals having a singularity of the form ( t − x ) − m , m ≥ 1 {\left ( {t - x} \right )^{ - m}},m \ge 1 . Interpreting the integrals with strong singularities in the Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term ( t − x ) − m {\left ( {t - x} \right )^{ - m}} , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.