A Unified Spectral Method for FPDEs with Two-sided Derivatives; A Fast Solver
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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 _0D_t^2τu^ + ∑_i=1^d [c_l_i _a_iD_x_i^2μ_i u^ +c_r_i _x_iD_b_i^2μ_i u^ ] + γ u^ = ∑_j=1^d [ κ_l_j _a_jD_x_j^2ν_j u^ +κ_r_j _x_jD_b_j^2ν_j u^ ] + f, where 2τ∈ (0,2), 2μ_i ∈ (0,1) and 2ν_j ∈ (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 zayernouri2013fractional, 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 (μ_i, ν_j, 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(N^d+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 samiee2016Unified2.
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