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.

翻译：本文提出一种新型伪谱（PS）方法，用于求解一类含变系数周期解的时变一维分数阶偏微分方程（FPDEs）初值问题（IVPs）的新类别。本研究的关键要素在于采用Bourafa等人[1]近期提出的带滑动正固定记忆长度的周期RL/Caputo分数阶导数（FD）算子，或Elgindy[2]推导的简化形式，作为精确刻画周期解FPDEs的自然分数阶导数算子。该方法通过基于Fourier配点与Gegenbauer求积的伪谱技术，将初值问题转化为良态线性方程组。简化后的线性系统具有特殊结构，可利用标准线性求解器快速精确求解。本文对所提方法进行了严格的误差与收敛性分析。文中提出的思想与成果有望为未来解决涉及周期解FPDEs的更一般性问题提供帮助。