Journal
ENGINEERING WITH COMPUTERS
Volume 38, Issue 1, Pages 155-173Publisher
SPRINGER
DOI: 10.1007/s00366-020-01033-8
Keywords
Riemann-Liouville fractional derivative; Time fractional cable model; RBF-FD; Stability; Convergence
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The paper proposes a novel numerical method, the RBF-FD, to approximate the time-fractional cable model involving two fractional temporal derivatives. The method combines time discretization using the Grunwald-Letnikov expansion and spatial discretization using the RBF-FD. The proposed method is efficient and the numerical results confirm the theoretical formulation.
The cable equation is one useful description for modeling phenomena such as neuronal dynamics and electrophysiology. The time-fractional cable model (TFCM) generalizes the classical cable equation by considering the anomalous diffusion that occurs in the ionic motion present for example in the neuronal system. This paper proposes a novel meshless numerical procedure, the radial basis function-generated finite difference (RBF-FD), to approximate the TFCM involving two fractional temporal derivatives. The time discretization of the TFCM is performed based on the Grunwald-Letnikov expansion. The spatial derivatives are discretized using the RBF-FD. The pattern of data distribution in the support domain is assumed as having a fixed number of points. The RBF-FD is based on the local support domain that leads to a sparsity system and also tackles the ill-conditioning problem caused by global collocation method. The theoretical stability and the convergence analysis of the scheme are also discussed in detail. It is shown that the proposed method is efficient and that the numerical results confirm the theoretical formulation.
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