L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations |
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Authors: | Rumeng Zheng Xiaoyun Jiang Hui Zhang |
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Affiliation: | School of Mathematics, Shandong University, Jinan 250100, PR China |
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Abstract: | In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy. |
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Keywords: | Two-dimensional time-fractional nonlinear equations Caputo derivative L1 Fourier spectral methods Stability and convergence FFT2 |
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