A unified spectral method for FPDEs with two-sided derivatives; part I: A fast solver

We develop a unified Petrov–Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form Dt2τ0u+∑i=1d[cliaiDxi2μiu+crixiDbi2μiu]+γu=∑j=1d[κljajDxj2νju+κrjxjDbj2νju]+f, where 2τ∈(0,2), 2τ≠1, 2μi∈(0,1) and 2νj∈(1,2), in a (1+d)-dimensional space–time hypercube, d=1,2,3,⋯, subject to homogeneous Dirichlet initial/boundary conditions. We employ the eigenfunctions of the fractional Sturm–Liouville eigen-problems of the first kind in [1], called Jacobi poly-fractonomials, as temporal bases, and the eigen-functions of the boundary-value problem of the second kind as temporal test functions. Next, we construct our spatial basis/test functions using Legendre polynomials, yielding mass matrices being independent of the spatial fractional orders (μi,νj,i,j=1,2,⋯,d). Furthermore, we formulate a novel unified fast linear solver for the resulting high-dimensional linear system based on the solution of generalized eigen-problem of spatial mass matrices with respect to the corresponding stiffness matrices, hence, making the complexity of the problem optimal, i.e., O(Nd+2). We carry out several numerical test cases to examine the CPU time and convergence rate of the method. The corresponding stability and error analysis of the Petrov–Galerkin method are carried out in [2].