Numerical Approach for a Stable Solution of Fractional Order Calcium Diffusion Model

S. Ruban Raj¹ , Sheena Mathew² , Tessy Tom³

Section:Research Paper, Product Type: Isroset-Journal
Vol.5 , Issue.4 , pp.380-383, Aug-2018

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.380383

Online published on Aug 31, 2018

Copyright © S. Ruban Raj, Sheena Mathew, Tessy Tom . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

View this paper at   Google Scholar | DPI Digital Library

XML View     PDF Download

How to Cite this Paper

394 Views    127 Downloads    80 Downloads

Abstract :
Diffusion is one of the main carrying phenomena which is seen in signaling mechanisms of ions and molecules in living cells, such as neurons. Here we present a one dimensional fractional diffusion model of calcium dynamics in cylindrical dendrites with Dirichlet boundary conditions. Based on shifted Grunwald‐Letnikov definition of fractional derivatives, implicit finite difference scheme is used to approximate the solution. The numerical solutions are obtained by using MATLAB programme.

Key-Words / Index Term :
Fractional Fick’s law, Fractional Differential equation, Grunwald‐Letnikov Derivative, Finite Difference Method, Calcium Diffusion.

References :
[1] R.S. Zucker, N.Stockbridge , “Presynaptic calcium diffusion and the time courses of transmitter release and synaptic facilitation at the squid giant synapse”, Journal of Neuroscience, Vol.3, No.6, pp.1263‐1269 , 1983.
[2]M. M. Meerschaert and C.Tadjeran, “Finite difference approximations for two‐ sided space‐fractional partial differential equations”, Appl. Math. Model, vol.56, pp.80‐90, 2006 .
[3] Yu, B.Jiang, Xiaoyun, “A fractional anomalous diffusion model and numerical simulation for sodium ion transport in the intestinal wall”, Advances in Mathematical Physics, vol.2013, 2013 .
[4] X. Y. Jiang, M. Y. Xu and H. T. Qi, “The fractional diffusion model with an absorption term and modified Fick’s law for non‐local transport processes”, Nonlinear Analysis: RealWorld Applications, vol.11, no.1, pp. 262‐269, 2010 .
[5] M.ZecovÃ, J.TerpÃik, “Heat conduction modeling by using fractional‐ order derivatives”, Applied Mathematics and Computation, vol. 29, 2015 .
[6] I.Podlubny. “Fractional differential equations”, Academic Press, San Diego, CA, 1999.
[7]M.M.Meerschaert and C.Tadjeran, “Finite difference approximations for fractional advection‐dispersion flow equations”, Journal of Computational and Applied Mathematics, vol.172, pp.65‐77, 2004.
[8]A.Biess, E.Korkotian, D.Holcman, “Barriers to diffusion in dendrites and estimation of calcium spread following synaptic inputs”, PLoS Computational Biology, vol.7, 2011.