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In this paper, we present a novel pseudospectral (PS) method for solving anew class of initial-value problems (IVPs) of time-dependent one-dimensionalfractional partial differential equations (FPDEs) with variable coefficientsand periodic solutions. A main ingredient of our work is the use of therecently developed periodic RL/Caputo fractional derivative (FD) operators withsliding positive fixed memory length of Bourafa et al. [1] or their reducedforms obtained by Elgindy [2] as the natural FD operators to accurately modelFPDEs with periodic solutions. The proposed method converts the IVP into awell-conditioned linear system of equations using the PS method based onFourier collocations and Gegenbauer quadratures. The reduced linear system hasa simple special structure and can be solved accurately and rapidly by usingstandard linear system solvers. A rigorous study of the computational storagerequirements as well as the error and convergence of the proposed method ispresented. The idea and results presented in this paper are expected to beuseful in the future to address more general problems involving FPDEs withperiodic solutions.

Fourier-Gegenbauer Pseudospectral Method for Solving Time-Dependent  One-Dimensional Fractional Partial Differential Equations with Variable  Coefficients and Periodic Solutions