On spectral polar fractional Laplacian

Abstract

In this paper, we introduce the fractional Sturm–Liouville operators and present the polar fractional Laplacian as a particular case of these operators. We also consider the distributed order space–time fractional diffusion equation involving the polar fractional Laplacian and propose three approaches for studying this problem on the circle and ring domains. First, we apply the integral transforms as the analytical methods to get the solution in terms of the Laplace-type integrals using the Titchmarsh theorem. Second, we use the finite difference method for discretization of the space variable and employ the matrix transfer technique to obtain a system of the distributed order fractional equations. For this system, we modify the Putzer’s algorithm and get the solution in terms of the eigenvalues of discretization matrix. Third, we employ the backward Euler numerical method for discretization of the time variable and find the corresponding error bound. Moreover, we compare and verify the approximate solutions with the exact solutions in analytical form.