The Galerkin–Fourier method for the study of nonlocal parabolic equations

Abstract

The aim of this paper is the study of a type of nonlocal parabolic equation. The formulation includes a convolution kernel k in the diffusion term and a design function h that plays the role of the diffusion coefficient. The main goal is twofold: On the one hand, the existence and uniqueness of nonlocal solution are deduced. Also, a comprehensive and rigorous procedure, which is based on the classical Galerkin–Fourier Method, is performed. As in the classical setting, the appropriate choice of the Gelfand triplet will guarantee the differentiation and therefore the operational technique for the study of the parabolic equation. On the other hand, the convergence of the nonlocal solution as the kernel k converges to a Dirac Delta is studied. The series expansion of the nonlocal solution allows us, in an easy way, to show its convergence to the solution of the corresponding local parabolic equation.