Time-fractional cable equation for anomalous electrodiffusion on a metric star graph with existence, uniqueness and numerical solution

Yash Vats, Mani Mehra and Dietmar Oelz

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 248, issue C, 108-124

Abstract: In this study we consider the pre-synaptic and its post-synaptic neurons together as a metric star graph. Electric currents in the neuron are driven by electrodiffusion of ions across trans-membrane channels. This research paper explores the time-fractional cable equation representing the topology of a single pre-synaptic neuron cell and its post-synaptic neurons on a metric star graph. The existence and uniqueness of solutions is established through the eigenfunction expansion method. Subsequently, a finite difference approximation for the time-fractional cable equation is proposed in which the Caputo time-fractional derivative is approximated by the L1 scheme. The existence of a unique solution of the proposed numerical scheme is established, and its stability and convergence are analyzed through the discrete energy method, exhibiting convergence of order 2−α in time and order 2 in space. Theoretical findings are validated through numerical results for two test problems. Finally, a real application is included to show the effect of the fractional derivative on the model.

Keywords: Time-fractional cable equation; Star graph; Stability; Convergence; Finite difference method; Solution (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:248:y:2026:i:c:p:108-124

DOI: 10.1016/j.matcom.2026.04.011

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