Fourier spectral exponential time differencing methods for multi-dimensional space-fractional reaction-diffusion equations

Abstract In this paper, we propose two numerical methods for solving the space-fractional reaction–diffusion equations, which are based on a Fourier spectral approach in space and exponential time differencing schemes in time. The advantages of the approaches are that they attain spectral convergence, produce a full diagonal representation of the fractional operator, and the extension to multiple spatial dimensions is the same as the one-dimensional space. That can overcome the constraints associated with many of the numerical schemes for these equations such as the computational efficiency caused by the non-locality of the fractional operator, which results in full, dense matrices. Moreover, the proposed schemes are second-order convergent, unconditionally stable, and highly efficient due to the predictor–corrector feature when comparing them with the existing method. It is observed that the scheme based on using Pade(1,1) approximations to the matrix exponential function introduces oscillations with non-smooth initial data for some time steps due to high frequency components present in the solution which diminish as the fractional order decreases. However, the scheme based on Pade(0,2) approximations to the matrix exponential function is oscillation-free for any time step. Numerical experiments for well-known models from the literature are performed to show the reliability and effectiveness of the proposed methods.

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