New Publications from FMATH Group
- Aug 27, 2018
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 high-dimensional fractional partial differential equations with two-sided derivatives and constant coefficients subject to homogeneous Dirichlet initial/boundary conditions. We employ the eigen-functions of the fractional Sturm–Liouville eigen-problems of the first kind, 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. 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. 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 the second part of this paper as a separate publication.
A Unified Spectral Method for FPDEs with Two-Sided Derivatives; Part II: Stability, and Error Analysis
We present the stability and error analysis of the unified Petrov–Galerkin spectral method, developed in Part-I for high-dimensional fractional partial differential equations with two-sided derivatives and constant coefficients in any (1+d)-dimensional space-time hyper-cube, subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method. Finally, we perform several numerical simulations to compare the theoretical and computational rates of convergence.