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

Muhammad Asim Khan; Majid Khan Majahar Ali; and Saratha Sathasivam; Muhammad Asim Khan; Majid Khan Majahar Ali; and Saratha Sathasivam

doi:10.3934/math.2025396

AIMS Mathematics

2025, Volume 10, Issue 4: 8647-8672. doi: 10.3934/math.2025396

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

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Gelugor, Pulau Pinang, Malaysia

Received: 10 January 2025 Revised: 10 February 2025 Accepted: 14 February 2025 Published: 15 April 2025
MSC : 35R11, 65N06

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.
- fractional cable equation,
- variable-order derivative,
- higher-order finite difference method,
- neuronal dynamics,
- numerical simulations
Citation: Muhammad Asim Khan, Majid Khan Majahar Ali, and Saratha Sathasivam. High-order finite difference method for the two-dimensional variable-order fractional cable equation in complex systems and neuronal dynamics[J]. AIMS Mathematics, 2025, 10(4): 8647-8672. doi: 10.3934/math.2025396

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Abstract

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.

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