In this paper, we present a novel pseudospectral (PS) method for solving a
new class of initial-value problems (IVPs) of time-dependent one-dimensional
fractional partial differential equations (FPDEs) with variable coefficients
and periodic solutions. A main ingredient of our work is the use of the
recently developed periodic RL/Caputo fractional derivative (FD) operators with
sliding positive fixed memory length of Bourafa et al. [1] or their reduced
forms obtained by Elgindy [2] as the natural FD operators to accurately model
FPDEs with periodic solutions. The proposed method converts the IVP into a
well-conditioned linear system of equations using the PS method based on
Fourier collocations and Gegenbauer quadratures. The reduced linear system has
a simple special structure and can be solved accurately and rapidly by using
standard linear system solvers. A rigorous study of the error and convergence
of the proposed method is presented. The idea and results presented in this
paper are expected to be useful in the future to address more general problems
involving FPDEs with periodic solutions.Comment: 13 pages, 3 figures. arXiv admin note: text overlap with
arXiv:2304.0445