Exact solution of integro-differential equations of diffusion along a grain boundary

We analyse model problems of stress-induced atomic diffusion from a point source or from the surface of a material into an infinite or semi-infinite grain boundary, respectively. The problems are formulated in terms of partial differential equations which involve singular integral operators. The self-similarity of these equations leads to singular integro-differential equations which are solved in closed form by reduction to an exceptional case of the Riemann–Hilbert boundary-value problem of the theory of analytic functions on an open contour. We also give a series representation and a full asymptotic expansion of the solution in the case of large arguments. Numerical results are reported.