A Time-Fractional Diffusion Model with Normalization and Short Memory Effect

We propose a time-fractional (TF) diffusion model that incorporates normalization and a short memory effect, which allows fast and efficient computations. The normalized TF diffusion equation enables fair comparison of the influence of the fractional order on evolution dynamics by using a weighting function whose total integral is uniformly equal to one across all times and fractional orders. However, simulating long-term evolution using the TF equations typically requires substantial computational memory and processing time. To resolve these issues, the proposed model applies a short memory scheme that limits the integral to a fixed time interval, which improves computational efficiency. We redefine the normalized weighting function so that its total integral over the fixed time interval is always equal to one. To highlight the improved efficiency of the short memory normalized TF diffusion equation, we perform a series of numerical experiments. We investigate dynamics with a given initial condition under different fractional orders and truncated time intervals. We apply the equation to more complex initial conditions to examine the evolution over time under varying fractional orders and memory intervals. Furthermore, to comprehensively evaluate the performance and applicability of the proposed method in higher-dimensional settings, we conduct three-dimensional (3D) numerical experiments.