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：本文提出了一种新颖的伪谱方法，用于求解一类新的含变系数和周期解的时间依赖一维分数阶偏微分方程初值问题。本研究的关键要素是采用Bourafa等人[1]近期提出的具有滑动正固定记忆长度的周期RL/Caputo分数阶导数算子，或其由Elgindy[2]推导的简化形式，作为精确建模具有周期解的分数阶偏微分方程的自然分数阶导数算子。所提方法通过基于傅里叶配置和盖根鲍尔求积的伪谱方法，将初值问题转化为一个良态的线性方程组。该简化线性系统具有简单的特殊结构，可利用标准线性系统求解器精确快速地求解。本文对所提方法的误差与收敛性进行了严谨分析。文中提出的思想与结果有望为未来处理涉及周期解的分数阶偏微分方程的更一般问题提供参考。