Solution of the space-fractional diffusion equation on bounded domains of superdiffusive phenomena.

Abstract

Space-fractional diffusion equations find widespread application in nature. They govern the anomalous dynamics of many stochastic processes, generalizing the standard diffusion equation to superdiffusive behavior. Strikingly, the solution of space-fractional diffusion equations on bounded domains is still an open problem. This is in part due to the difficulty of handling nonlocal boundary conditions ascribed to the space-fractional derivative, leading to the failure of standard methods. Here we revisit the space-fractional diffusion equation in one spatial dimension with bounded domains and present a solution in terms of weighted Jacobi polynomials. Calculated eigenvalues and eigenfunctions in the superdiffusive regime show remarkable agreement with results from numerical discretization of the space-fractional derivative operator and Monte Carlo simulations. To exemplify, we apply the proposed solution to obtain the exact mean residence time or mean first-passage time, first-passage-time distribution, and survival probability, in agreement with known results for the superdiffusive regime. The system of equations converges rather fast for the first eigensolutions, as is desirable for practical application purposes in superdiffusive phenomena.