The time fractional cable equation is a fundamental equation used for modeling neuronal dynamics. The equation is obtained by substituting the first order time derivative in the standard equation with a Caputo fractional derivative of order α, where 0 < α < 1. In this paper, two implicit difference schemes, the fully implicit (FI) and the Crank-Nicolson (C-N) difference schemes are employed in solving the 2D time fractional cable equation (TFCE). A comparative study between these two schemes will be conducted via numerical experiments. The efficiency of the schemes in terms of accuracy and computing time will be reported and discussed.

1.
R. L.
PodlubnyBagley
and
R.
Calico
,
Journal of Guidance, Control, and Dynamics
14
(
2
),
304
311
(
1991
).
2.
I.
Podlubny
,
Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
. (
Academic press
,
1998
).
3.
C. M.
Pinto
and
J. T.
Machado
,
Nonlinear Dynamics
65
(
3
),
247
254
(
2011
).
4.
D.
Baleanu
,
S. I.
Muslih
and
E. M.
Rabei
,
Nonlinear Dynamics
53
(
1-2
),
67
74
(
2008
).
5.
D.
Valério
,
J. J.
Trujillo
,
M.
Rivero
,
J. T.
Machado
and
D.
Baleanu
,
The European Physical Journal Special Topics
222
(
8
),
1827
1846
(
2013
).
6.
N.
Laskin
,
Physics Letters A
268
(
4
),
298
305
(
2000
).
7.
X.
Guo
and
M.
Xu
,
Journal of mathematical physics
47
(
8
),
082104
(
2006
).
8.
G. W.
Bohannan
,
Journal of Vibration and Control
14
(
9-10
),
1487
1498
(
2008
).
9.
T.
Langlands
,
B.
Henry
and
S. L.
Wearne
,
SIAM Journal on Applied Mathematics
71
(
4
),
1168
1203
(
2011
).
10.
B.
Henry
,
T.
Langlands
and
S.
Wearne
,
Physical review letters
100
(
12
),
128103
(
2008
).
11.
T.
Langlands
,
B.
Henry
and
S.
Wearne
,
Journal of mathematical biology
59
(
6
),
761
808
(
2009
).
12.
N.
Sweilam
,
M.
Khader
and
M.
Adel
,
Journal of advanced research
5
(
2
),
253
259
(
2014
).
13.
R.
Hilfer
,
Applications of fractional calculus in physics
. (
World Scientific
,
2000
).
14.
Y.
Luchko
and
R.
Gorenflo
,
Preprint series A08–98, Fachbereich Mathematik und Informatik
, (
Freie Universitat Berlin
,
1998
).
15.
P.
Zhuang
and
F.
Liu
,
Journal of Applied Mathematics and Computing
22
(
3
),
87
99
(
2006
).
16.
I.
Karatay
,
N.
Kale
and
S. R.
Bayramoglu
,
Fractional Calculus and Applied Analysis
16
(
4
),
892
910
(
2013
).
17.
J.
Liu
,
H.
Li
and
Y.
Liu
,
Journal of Applied Mathematics and Computing
,
1
17
(
2015
).
18.
A.
Bhrawy
and
M.
Zaky
,
Nonlinear Dynamics
80
(
1-2
),
101
116
(
2015
).
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