Following the fractional cable equation established in the letter [B.I. Henry, T.A.M. Langlands, and S.L. Wearne, Phys. Rev. Lett. 100 (2008) 128103], we present the time-space fractional cable equation which describes the anomalous transport of electrodiffusion in nerve cells. The derivation is based on the generalized fractional Ohm's law; and the temporal memory effects and spatial-nonlocality are involved in the time-space fractional model. With the help of integral transform method we derive the analytical solutions expressed by the Green's function; the corresponding fractional moments are calculated; and their asymptotic behaviors are discussed. In addition, the explicit solutions of the considered model with two different external current injections are also presented.
Abstract
Following the fractional cable equation established in the letter [B.I. Henry, T.A.M. Langlands, and S.L. Wearne, Phys. Rev. Lett. 100 (2008) 128103], we present the time-space fractional cable equation which describes the anomalous transport of electrodiffusion in nerve cells. The derivation is based on the generalized fractional Ohm's law; and the temporal memory effects and spatial-nonlocality are involved in the time-space fractional model. With the help of integral transform method we derive the analytical solutions expressed by the Green's function; the corresponding fractional moments are calculated; and their asymptotic behaviors are discussed. In addition, the explicit solutions of the considered model with two different external current injections are also presented.
关键词
anomalous diffusion /
time-space fractional Cable equation /
Green’s function /
Fox-H function
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Key words
anomalous diffusion /
time-space fractional Cable equation /
Green’s function /
Fox-H function
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中图分类号：
05.10.Gg
87.10.-e
87.15.Vv
87.19.L-
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基金
Supported by the Program for New Century Excellent Talents in University under Grant No. NCET-09-0438, the National Natural Science Foundation of China under Grant No. 11271173, the training Program of the Major Research Plan of the National Natural Science Foundation of China under Grant No. 91120014, the Starting Research Foundation from the Xi'an University of Technology under Grant No. 108-211206, and the Scientific Research Foundation of the Education Department of Shaanxi Province under Grant No. 2013JK0581
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