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- W2765447624 abstract "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}^{2tau}u^{} + sum_{i=1}^{d}$ $[c_{l_i}$ $_{a_i}{mathcal{D}}_{x_i}^{2mu_i} u^{} +c_{r_i}$ $_{x_i}{mathcal{D}}_{b_i}^{2mu_i}$ $u^{} ] +$ $gamma$ $u^{} = sum_{j=1}^{d} [ kappa_{l_j}$ $_{a_j}{mathcal{D}}_{x_j}^{2nu_j} u^{}$ $+kappa_{r_j}$ $_{x_j}{mathcal{D}}_{b_j}^{2nu_j}$ $u^{} ]$ $+ f$, where $2tau in (0,2)$, $2mu_i in (0,1)$ and $2nu_j in (1,2)$, in a ($1+d$)-dimensional textit{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 textit{Jacobi 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}." @default.
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- W2765447624 date "2019-05-01" @default.
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- W2765447624 title "A unified spectral method for FPDEs with two-sided derivatives; part I: A fast solver" @default.
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- W2765447624 doi "https://doi.org/10.1016/j.jcp.2018.02.014" @default.
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