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...

Abstract We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove Sobolev-type inequality, weak Harnack local Hölder estimate.

The main purpose of this article is to present an approximation method of for singular integrodifferential equations with Cauchy kernel in the most general form under the mixed conditions in terms of the second kind Chebyshev polynomials. This method transforms mixed singular integro-differential equations with Cauchy kernel and the given conditions into matrix equation and using the zeroes of ...

The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel (t − s)−μ with 0 < μ < 1. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both th...

We study a generalized extrapolated Crank–Nicolson scheme for the time discretization of a semilinear integro-differential equation with a weakly singular kernel, in combination with a space discretization by linear finite elements. The scheme uses variable grids in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assumin...

We predict theoretically and numerically the existence of incoherent dispersive shock waves. They manifest themselves as an unstable singular behavior of the spectrum of incoherent waves that evolve in a noninstantaneous nonlinear environment. This phenomenon of "spectral wave breaking" develops in the weakly nonlinear regime of the random wave. We elaborate a general theoretical formulation of...

In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....