3 research outputs found
Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
In this paper, we propose a fast spectral-Galerkin method for solving PDEs
involving integral fractional Laplacian in , which is built upon
two essential components: (i) the Dunford-Taylor formulation of the fractional
Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions
(MCFs) as basis functions. As a result, the fractional Laplacian can be fully
diagonalised, and the complexity of solving an elliptic fractional PDE is
quasi-optimal, i.e., with being the number of modes in
each spatial direction. Ample numerical tests for various decaying exact
solutions show that the convergence of the fast solver perfectly matches the
order of theoretical error estimates. With a suitable time-discretization, the
fast solver can be directly applied to a large class of nonlinear fractional
PDEs. As an example, we solve the fractional nonlinear Schr{\"o}dinger equation
by using the fourth-order time-splitting method together with the proposed
MCF-spectral-Galerkin method.Comment: This article has a total of 24 pages and including 22 figure
Variable-Order Fractional Partial Differential Equations: Analysis, Approximation and Inverse Problem
Variable-order fractional partial differential equations provide a competitive means in modeling challenging phenomena such as the anomalous diffusion and the memory effects and thus attract widely attentions. However, variable-order fractional models exhibit salient features compared with their constant-order counterparts and introduce mathematical and numerical difficulties that are not common in the context of integer-order and constant-order fractional partial differential equations.
This dissertation intends to carry out a comprehensive investigation on the mathematical analysis and numerical approximations to variable-order fractional derivative problems, including variable-order time-fractional, space-fractional, and space-time fractional partial differential equations, as well as the corresponding inverse problems. Novel techniques are developed to accommodate the impact of the variable fractional order and the proposed mathematical and numerical methods provide potential tools to analyze and compute the variable-order fractional problems