Two-Dimensional Normalized Variable-Order Time-Fractional Diffusion

초록

In this article, we present a normalized variable-order time-fractional diffusion equation in two spatial dimensions, motivated by previous one-dimensional work. The normalized Caputo-type derivative provides a consistent basis for comparing different fractional orders because the total memory weight is independent of time and fractional order. A finite difference method with operator splitting is developed for the numerical solution. Numerical results show that the fractional order strongly affects the diffusion process. Smaller orders produce slower decay and stronger memory effects, while larger orders lead to faster diffusion. For time-dependent fractional orders, different temporal distributions also produce distinct diffusion behaviors, even when the average fractional order is the same. These results demonstrate that the proposed model can describe diffusion processes with time-varying memory effects.

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