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（PS）方法，用于求解一类含变系数与周期解的一维时间分数阶偏微分方程（FPDE）初值问题（IVP）。本研究的关键要素在于采用Bourafa等[1]近期发展的具有滑动正固定记忆长度的周期型RL/Caputo分数阶导数（FD）算子，或Elgindy [2]推导的简化形式，作为精确建模含周期解的FPDEs的自然FD算子。所提方法通过基于傅里叶配置与盖根鲍尔正交的PS技术，将IVP转化为良态线性方程组。降阶线性系统具有简单特殊结构，可借助标准线性系统求解器快速精确求解。本文对所提方法的计算存储需求、误差与收敛性进行了严格分析。预期本文思想与结果将为未来解决涉及含周期解FPDE的广义问题提供有益参考。