The 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, were analyzed. The problems were formulated in terms of partial differential equations which involved singular integral operators. The self-similarity of these equations led to singular integro-differential equations which were solved in closed form by reduction to an exceptional case of the Riemann-Hilbert boundary-value problem in the theory of analytical functions on an open contour. Series representation, and a full asymptotic expansion of the solution, were also given for the case of large arguments.

Exact Solution of Integro-Differential Equations of Diffusion along a Grain Boundary. Y.A.Antipov, H.Gao: Quarterly Journal of Mechanics and Applied Mathematics, 2000, 53[4], 645-74