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 (0)D(t)(2 tau)u + Sigma(d)(i=1)[Cl-i(ai) D(xi)(2 mu i)u+C-ri(xi) D(bi)(2 mu i)u ] + gamma u = Sigma(d)(j=1) [kappa(lj aj) D-xj(2 nu j) u + kappa(rj xj) D-bj(2 nu j) u] + f, where 2 tau is an element of (0, 2), 2 tau not similar or equal to 1, 2 mu(i) is an element of (0, 1) and 2 nu(j) is an element of (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 (mu(i), nu(j) , i , j = 1, 2, ..., d). Furthermore, we formulate a novel unified fast linear solver for the resulting highdimensional 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]. Published by Elsevier Inc.