A Unified Spectral Method for FPDEs with Two-sided Derivatives; 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 ${}_0{\mathcal{D}}_t^{2\tau }{u}^{}+{\sum }_{i=1}^d$ $[c_{l_i}$ ${}_{a_i}{\mathcal{D}}_{x_i}^{2{\mu }_i}{u}^{}+c_{r_i}$ ${}_{x_i}{\mathcal{D}}_{b_i}^{2{\mu }_i}$ $u^{}]+$ $\gamma$ $u^{}={\sum }_{j=1}^d[{\kappa }_{l_j}$ ${}_{a_j}{\mathcal{D}}_{x_j}^{2{\nu }_j}{u}^{}$ $+{\kappa }_{r_j}$ ${}_{x_j}{\mathcal{D}}_{b_j}^{2{\nu }_j}$ $u^{}]$ $+f$, where $2\tau \in (0,2)$, $2{\mu }_i\in (0,1)$ and $2{\nu }_j\in (1,2)$, in a ($1+d$)-dimensional $\mathit{\text{space-time}}$ hypercube, $d=1,2,3,\cdots$, subject to homogeneous Dirichlet initial/boundary conditions. We employ the eigenfunctions of the fractional Sturm-Liouville eigen-problems of the first kind in \cite{zayernouri2013fractional}, called $\mathit{\text{Jacobi}}\mathit{\text{poly-fractonomial}}$s, 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 (${\mu }_i,{\nu }_j,i,j=1,2,\cdots ,d$). 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., $\mathcal{O}(N^{d+2})$. 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 \cite{samiee2016Unified2}.