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 computational storage requirements as well as 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方法，用于求解一类含变系数和周期解的时变一维分数阶偏微分方程初值问题。我们的核心创新在于采用Bourafa等人[1]近期发展的具有滑动正定记忆长度的周期RL/Caputo分数阶导数算子，或Elgindy[2]推导的简化形式，作为描述周期解分数阶偏微分方程的自然导数算子。该方法通过傅里叶配点结合盖根鲍尔求积的伪谱技术，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，可使用标准线性方程组求解器快速精确求解。本文对计算存储需求、误差及收敛性进行了严格分析。文中提出的思想与结果预期将为未来处理更复杂的周期解分数阶偏微分方程问题提供重要参考。