High-order finite difference method for the two-dimensional variable-order fractional cable equation in complex systems and neuronal dynamics

In this paper, the two-dimensional (2-D) fractional cable equation (FCE) with the Caputo variable-order (V-O) derivative was utilized for simulating systems with memory and hereditary characteristics that vary across time and space. This variable-order fractional model is particularly well suited for the description of neuronal dynamics in biological systems. The accurate modeling of dynamic, memory-dependent behaviors that vary over space and time, which are essential for applications such as neuronal dynamics, presents a challenge for conventional numerical methods. Furthermore, there is a lack of stable and effective numerical techniques for 2-D V-O systems, highlighting the need for improved computational approaches. In order to solve the cable equation numerically with high accuracy and computing efficiency, this work primarily focused on using a higher-order finite difference method. The proposed method's robustness was confirmed by stability and convergence analyses, while its efficacy was demonstrated by numerical simulations, which were presented in tabular and graphical formats. These findings demonstrate its precision and efficiency when dealing with the intricate dynamics of V-O fractional equations. The study concludes that the higher-order finite difference method offers an accurate and effective framework for solving fractional partial differential equations (FPDEs), particularly in applications that necessitate precision modeling, such as biological and physical systems. It also creates opportunities for future research, such as the application of the method to multivariate problems, the integration of machine learning techniques, or the adaptation of the method to systems with variable coefficients.