In this paper, we propose normalized time-fractional diffusion equations in two and three spatial dimensions, where the normalization guarantees that the total memory weight remains equal to one and allows a consistent interpretation of memory effects for different fractional orders. An efficient Fourier spectral method in space combined with a finite difference approximation in time is used to solve the governing equations in both two- and three-dimensional (2D and 3D) settings. A rigorous error analysis shows that the proposed scheme achieves a temporal convergence rate of order $ \mathcal{O}(\Delta t^{2-\alpha}) $, and numerical experiments confirm exponential accuracy in space. Extensive 2D and 3D numerical tests demonstrate the robustness of the method, and the numerical results show that smaller fractional orders slow down the evolution dynamics due to stronger memory effects. A comparison with the standard Caputo time-fractional diffusion model indicates that the normalized formulation provides a more uniform temporal decay behavior while preserving the essential dynamics. These computational results suggest that the proposed method offers an accurate and reliable numerical algorithm for simulating multidimensional diffusion processes with memory effects.
Citation: Ke Zhang, Seungjae Lee, Xinpei Wu, Meiyun Nan, Junseok Kim. A Fourier spectral method for normalized time-fractional diffusion equations in two and three dimensions[J]. Electronic Research Archive, 2026, 34(3): 1939-1956. doi: 10.3934/era.2026087
In this paper, we propose normalized time-fractional diffusion equations in two and three spatial dimensions, where the normalization guarantees that the total memory weight remains equal to one and allows a consistent interpretation of memory effects for different fractional orders. An efficient Fourier spectral method in space combined with a finite difference approximation in time is used to solve the governing equations in both two- and three-dimensional (2D and 3D) settings. A rigorous error analysis shows that the proposed scheme achieves a temporal convergence rate of order $ \mathcal{O}(\Delta t^{2-\alpha}) $, and numerical experiments confirm exponential accuracy in space. Extensive 2D and 3D numerical tests demonstrate the robustness of the method, and the numerical results show that smaller fractional orders slow down the evolution dynamics due to stronger memory effects. A comparison with the standard Caputo time-fractional diffusion model indicates that the normalized formulation provides a more uniform temporal decay behavior while preserving the essential dynamics. These computational results suggest that the proposed method offers an accurate and reliable numerical algorithm for simulating multidimensional diffusion processes with memory effects.
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Ke Zhang, Seungjae Lee, Xinpei Wu, Meiyun Nan, Junseok Kim. A Fourier spectral method for normalized time-fractional diffusion equations in two and three dimensions[J]. Electronic Research Archive, 2026, 34(3): 1939-1956. doi: 10.3934/era.2026087
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