Efficient numerical simulation of variable-order fractional diffusion processes with a memory kernel

Abstract

Diffusion equations are fundamental in modeling the transport of heat, mass, or contaminants in porous media. However, classical models often fail to capture the anomalous diffusion behavior inherent in heterogeneous and memory-dependent materials. To address this, we investigate a fractional diffusion integro-differential equation involving variable-order derivatives in both time and space, subject to suitable conditions. The solutions are shown to exist and be unique through the rigorous application of fixed-point theorems. A finite difference-based numerical scheme is formulated to handle the variable-order fractional operators and convolution-type integral terms efficiently. Stability analysis confirms the accuracy and robustness of the method. In addition, approximate solutions are computed for three representative cases:(i) constant-order fractional diffusion ($\alpha =\text{constant}$), (ii) time-dependent order $\alpha \left(t\right)$, and (iii) fully variable-order $\alpha (x,t)$. By incorporating variable order dynamics and integro-differential structures, this work extends conventional models and provides a unified framework for simulating complex transport processes in porous media.