Numerical Solution and Stability Analysis of Huxley Equation

Raf. J. of Comp. & Math’s. , Vol. 2, No. 1, 2005
85
Numerical Solution and Stability Analysis of Huxley Equation
Saad Manaa Mohammad Sabawi
College of Computers Sciences and Mathematics
Mosul University
Received on: 25/01/2005 Accepted on: 05/04/2005
‫ﺍﻟﻤﻠﺨﺹ‬
‫ﺘ‬‫ﻤﻌﺎﺩﻟﺔ‬ ‫ﺤل‬ ‫ﻡ‬Huxley‫ﺍﻟﻤﻨﺘﻬﻴﺔ‬ ‫ﻗﺎﺕ‬ ‫ﺍﻟﻔﺭﻭ‬ ‫ﻁﺭﺍﺌﻕ‬ ‫ﻤﻥ‬ ‫ﻁﺭﻴﻘﺘﻴﻥ‬ ‫ﺒﺎﺴﺘﺨﺩﺍﻡ‬:‫ﻫﻲ‬ ‫ﺍﻷﻭﻟﻰ‬
‫ﻁﺭﻴﻘﺔ‬ ‫ﻫﻲ‬ ‫ﻭﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﺼﺭﻴﺤﺔ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬Crank-Nicholson‫ﻜﻠﺘـﺎ‬ ‫ﻨﺘـﺎﺌﺞ‬ ‫ﺒﻴﻥ‬ ‫ﻤﻘﺎﺭﻨﺔ‬ ‫ﻋﻤل‬ ‫ﺘﻡ‬ ‫ﺇﺫ‬
‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬ ‫ﻜﺎﻨﺕ‬ ‫ﺤﻴﻥ‬ ‫ﻓﻲ‬ ‫ﹰ‬‫ﺎ‬‫ﺘﻘﺎﺭﺒ‬ ‫ﻭﺍﻷﺴﺭﻉ‬ ‫ﺍﻷﺴﻬل‬ ‫ﻫﻲ‬ ‫ﺍﻷﻭﻟﻰ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬ ‫ﺇﻥ‬ ‫ﺘﺒﻴﻥ‬ ‫ﻭﻗﺩ‬ ‫ﺍﻟﻁﺭﻴﻘﺘﻴﻥ‬
‫ﺍﻷﺩﻕ‬ ‫ﻫﻲ‬.‫ﻁﺭﻴﻘﺔ‬ ‫ﺒﺎﺴﺘﺨﺩﺍﻡ‬ ‫ﺍﻟﻁﺭﻴﻘﺘﻴﻥ‬ ‫ﻜﻠﺘﺎ‬ ‫ﺍﺴﺘﻘﺭﺍﺭﻴﺔ‬ ‫ﺩﺭﺍﺴﺔ‬ ‫ﻜﺫﻟﻙ‬ ‫ﺘﻤﺕ‬ ‫ﻭﻟﻘﺩ‬Fourier
( vonn Neumann)‫ـﺎﻥ‬‫ﻜـ‬ ‫ﺇﺫﺍ‬ ‫ـﺸﺭﻭﻁ‬‫ﻤـ‬ ‫ـﻭ‬‫ﻨﺤـ‬ ‫ـﻰ‬‫ﻋﻠـ‬ ‫ـﺴﺘﻘﺭﺓ‬‫ﻤـ‬ ‫ـﻰ‬‫ﺍﻷﻭﻟـ‬ ‫ـﺔ‬‫ﺍﻟﻁﺭﻴﻘـ‬ ‫ﺇﻥ‬ ‫ـﻴﻥ‬‫ﺘﺒـ‬ ‫ﺇﺫ‬
( )
( )2
2
4
2
,
4
2
xa
x
t
ta
r
∆−
∆
≤∆
∆−
≤
β
β
‫ﻏﻴﺭ‬ ‫ﻨﺤﻭ‬ ‫ﻋﻠﻰ‬ ‫ﻤﺴﺘﻘﺭﺓ‬ ‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬ ‫ﻜﺎﻨﺕ‬ ‫ﺤﻴﻥ‬ ‫ﻓﻲ‬
‫ﻤﺸﺭﻭﻁ‬.
ABSTRACT
The numerical solution of Huxley equation by the use of two finite
difference methods is done. The first one is the explicit scheme and the
second one is the Crank-Nicholson scheme. The comparison between the
two methods showed that the explicit scheme is easier and has faster
convergence while the Crank-Nicholson scheme is more accurate. In
addition, the stability analysis using Fourier (von Neumann) method of two
schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, ( )
( )2
2
4
2
,
4
2
xa
x
t
ta
r
∆−
∆
≤∆
∆−
≤
β
β and the second
scheme is unconditionally stable.

SaadManaa &Mohammad Sabawi
86
1. Introduction
It is probably not an overstatement to say that almost all partial
differential equations (PDEs) that arise in a practical setting are solved
numerically on a computer. Since the development of high-speed computing
devices the numerical solution of PDEs has been in active state with the
invention of new algorithms and the examination of the underlying theory.
This is one of the most active areas in applied mathematics and it has
a great impact on science and engineering because of the ease and efficiency
it has shown in solving even the most complicated problems. The basic idea
of the method of finite differences is to cast the continuous problem
described by the PDE and auxiliary conditions into a discrete problem that
can be solved by a computer in finitely many steps. The discretization is
accomplished by restricting the problem to a set of discrete points. By
systematic procedure, we then calculate the unknown function at those
discrete points. Consequently, a finite difference technique yields a solution
only at discrete points in the domain of interest rather than, as we expect for
an analytical calculation, a formula or closed-form solution valid at all
points of the domain [11]. Manoranjan et al [12] obtained estimates for the
critical lengths of the domain at which bifurcation occurs in the cases
,2/10,,0 ≤<= aab and 1.
Manoranjan [13] studied in detail the solutions bifurcating from the
equilibrium state au = . Eilbeck and Manoranjan [3] considered different
types of basis functions for the pseudo-spectral method applied to the
nonlinear reaction-diffusion equation in 1- and 2- space dimensions. Eilbeck
[4] extended the pseudo-spectral method to follow steady state solutions as a
function of the problem parameter, using path-following techniques. Fath
and Domanski [6] studied the cellular differentiation in a developing
organism via a discrete bistable reaction-diffusion model and they used the
numerical simulation to support their expectations of the qualitative
behavior of the system. Lewis and Keener [10] studied the propagation
failure using the one –dimensional scalar bistable equation by a passive gap
and they used the numerical simulation in their study. Binczak et al [1]
compared the numerical predictions of the simple myelinated nerve fibers
with the theoretical results in the continuum and discrete limits. Broadbridge
et al [2] re-examined the derivation of the gene- transport equations and
used the Gaussian clump of alleles by the use of a numerical method-of-
lines by using PDETWO program. Lefantzi et al [9] presented their findings
for various orders of spatial discretizations as applied to SAMR (Structured
Adaptive Mesh Refinement) simulations.

Numerical Solution and Stability…
87
In this paper, the numerical solution of Huxley equation by using two
finite difference methods and stability analysis of these two methods are
analyzed.
2. The Mathematical Model
One of the famous non-linear reaction-diffusion equations is the
generalized Burgers-Huxley (gBH) equation:
( )( ) ( )
( )0,1aand,0,0,0
112
2
∈>≥≥
−−=
∂
∂
−
∂
∂
+
∂
∂
δβα
βα δδδ
auuu
x
u
x
u
u
t
u
If we take 0,0,1 ≠≠= βαδ and , equation (1) becomes the following
Burgers-Huxley (BH) equation:
( )( ) ( )212
2
auuu
x
u
x
u
u
t
u
−−=
∂
∂
−
∂
∂
+
∂
∂
βα
Equation (2) shows a prototype model for describing the interaction
mechanism, convection transport. When 1,0 == δβ and , equation (1) is
reduced to Burgers equation which describes the far field of wave
propagation in nonlinear dissipative systems
( )302
2
=
∂
∂
−
∂
∂
+
∂
∂
x
u
x
u
u
t
u
α
When 1,0 == δα and , equation (1) is reduced to the Huxley equation
which describes nerve pulse propagation in nerve fibers and wall motion in
liquid crystal
( )( ) ( )412
2
auuu
x
u
t
u
−−=
∂
∂
−
∂
∂
β
It is known that nonlinear diffusion equations (3) and (4) play
important roles in nonlinear physics. They are of special significance for
studying nonlinear phenomena [19]. Zeldovich and Frank- Kamenetsky
formulated the equation (4) in 1938 as a model for flame front propagation
and for this reason this equation sometimes named Zeldovich-Frank-
Kamenetsky (ZF) equation, which has been extensively studied as a simple
nerve model [1]. In 1952 Hodgkin and Huxley proposed their famous
Hodgkin-Huxley model for nerve propagation. Because of the mathematical
complexity of this model, it led to the introduction of the simpler Fitzhugh-
Nagumo system. The simplified model of the Fitzhugh-Nagumo system is
Huxley equation [18]. Because Huxley equation is a special case of
Fitzhugh-Nagumo system, it is sometimes named Fitzhugh-Nagumo (FN)
equation [5] or the reduced Nagumo equation or Nagumo equation [15]. In
sixties, Fitzhugh used equation (4) as an approximate equation for the

88
description of dynamics of the giant axon. This equation was among the first
models of excited media [8].
In this paper, we shall take the Huxley equation as a model problem [12]:
( )( )
[ ]
( ) ( ) ( )
( ) ( ) ( )6,,
50,0,0,
0,LL,-x
1
2
2
2
btLutLu
HbHxHbxu
t
auuu
x
u
t
u
==−
>≥+−=
≥∈
−−=
∂
∂
−
∂
∂
β
For the purpose of numerical calculations, we shall take:
( ) .30,10,10,1,1,0,1 ≤≤≤<≤≤=∈= tHandbLaβ
3. Derivation of the Explicit Scheme Formula of Huxley Equation
[14] is ( ){ }ctLxLtxR ≤≤≤≤−= 0,:, Assume that the rectangle
subdivided into ( )1−n by( )1−m rectangles with sides kthx =∆=∆ , .
Start at the bottom row, where ,01 == tt and the initial condition is [12]:
( ) ( ) ( ) .1,...,3,2,, 2
1 −=+−== niHxHbxftxu iii
A method for computing the approximations to ( )txu , at grid points in
successive rows will be developed
( ){ } ( )7,...,4,3,2,1,...,4,3,2:, mjnitxu ji =−=
The difference formulas used for ( )txut , and ( )txuxx , are:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )9
,,2,
,
8
,,
,
2
2
hO
h
thxutxuthxu
txu
kO
k
txuktxu
txu
xx
t
+
−+−+
=
+
−+
=
Where the grid points are:
kttktthxxhxx jjjjiiii −=+=−=+= −+−+ 1111 ,,,
Neglecting the terms ( )kΟ and ( )2
hΟ , and use approximation jiu ,
for ( )ji txu , in equations (8) and (9), which are in turn substituted in
equation (4), we get
( )( ) ( )101
2u
,,,2
,1,,1,1ji,
auuu
h
uuu
k
u
jijiji
jijijiji
−−=
+−
−
− −++
β
From equation (10), we have
( ) ( )( ) ( )1112 ,,,,1,,1,1, auuukuuuruu jijijijijijijiji −−++−=− −++ β
Where 2
/ hkr =

89
After some mathematical manipulation, we obtain
( ) ( ) ( ) ( ) ( )12121 ,
2
,,,1,11, jijijijijiji uaukuakruuru −++−−++= +−+ ββ
Equation (12) represents the explicit finite difference formula for
equation (4). Equation (12) is employed to create ( )thj 1+ row across the
grid, assuming that approximations in the jth row are known.
Notice that this formula explicitly gives the value 1, +jiu in terms of
jiji uu ,,1 ,− , and jiu ,1+ .
4. Stability Analysis of the Explicit Scheme Using Fourier
(von Neumann) Method
The basic idea of this method is to replace the solution of the finite
difference method mnu , at time t by ( ) xi
et γ
ψ , where 0,1 >−= γi [16].
To apply von Neumann method to equation (4), we resort to the linearized
stability analysis [7], we have
( )132
2
ua
x
u
t
u
β−
∂
∂
=
∂
∂
The finite difference explicit formula for (13) is:
( )
( )14
2
,2
,1,,1,1,
mn
mnmnmnmnmn
ua
x
uuu
t
uu
β−
∆
+−
=
∆
− −++
Substituting ( ) xi
mn etu γ
ψ=, in (14), we have
( ) ( ) ( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
[ ] ( )
( ) ( ) ( )[ ] ( )ttaeetrttt
etaeee
x
t
e
t
ttt
eta
x
etetet
t
etett
xixi
xixixixixi
xi
xxixixxixixi
ψβψψψ
βψ
ψψψ
βψ
ψψψψψ
γγ
γγγγγ
γ
γγγγγ
∆−−+=−∆+
⇒−−+
∆
=



∆
−∆+
⇒−
∆
+−
=
∆
−∆+
∆−∆
∆−∆
∆−∆+
2
2
2
2
2
)()(
Where 2
/ hkr =

90
( ) ( ) ( )[ ] ( )
( )[ ] ( )
( )[ ] ( )
( )[ ] ( )
( ) ( )[ ] ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( )[ ] ( )
( ) ( ) ( ) ξβγψψ
ψβγ
ψβγψψψ
ψβγψ
ψβγψ
ψβγψ
ψβγψ
ψβγψ
ψβγγγγψψψ
=∆−∆−=∆+
⇒∆−∆−=
∆−∆−=∆+
⇒∆−∆−=
∆−∆−−−=
∆−∆−−=
∆−−∆=
∆−−∆=
∆−−∆−∆+∆+∆=−∆+
taxrttt
ttaxr
ttaxtrttt
ttaxtr
ttaxtr
ttaxtr
ttaxtr
ttaxtr
ttaxixxixtrttt
2/sin41/
2/sin41
2/sin4
2/sin4
)2/sin21(12
cos12
1cos2
2cos2
2sincossincos
2
2
2
2
2
Where ξ can be visualized as the amplification factor and we get
( ) ( ) ( )15/ ξψψ =∆+ ttt
As we advance the solution from a particular plane ( )tψ to the next
plane ( )tt ∆+ψ , ( ) ( )ttt ψψ −∆+ must start decreasing or alternatively ( )tψ
must be bounded function, i.e. ( )tψ should not tend infinity for large t .
From equation (15), for boundedness of (15), we need
( ) ( )
( ) ( )1612/sin41
1
1/1
2
≤∆−∆−
⇒≤
⇒≤+
taxr
tt
βγ
ξ
ψψ
In the above inequality, the right-side inequality is:
( ) 12/sin41 2
≤∆−∆− taxr βγ
Implies 0>r and this is always true.
Hence, in order that (16) is to be satisfied, we need
( )
( )
( )
( )2/sin
42
1
2/sin42
2/sin42
2/sin411
2
2
2
2
xr
ta
taxr
taxr
taxr
∆≥
∆
−
⇒∆+∆≥
⇒∆−∆−≤−
⇒∆−∆−≤−
γ
β
βγ
βγ
βγ
For some β , ( )2/sin2
x∆γ is unity and hence the above condition
reduces to

91
( )
( )
( )
( )
( )18
4
2
,(17),xt /r
17
4
2
2
2
2
haveweinequalityfromSince
xa
x
t
ta
r
∆+
∆
≤∆
∆∆=
∆−
≤
β
β
This precisely the conditions imposed on the explicit scheme to be
stable.
5. Derivation of the Crank-Nicholson Scheme Formula of Huxley
Equation
The is diffusion term xxu in this method is represented by central
differences, with their values at the current and previous time steps averaged
[17]:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )20
,,
u
(19)
,,2,,,2,
2
1
t
2
k
txuktxu
h
thxutxuthxukthxuktxukthxu
uxx
−+
=











 ++−−+++++−+−
=
By using the approximation jiu , for ( )ji txu , in equations (19) and
(20), which are in turn substituted into equation (4), we have
( )( ) ( )211
2
2
2
2
,,,2
,1,,1
2
1,11,1,1,1,
auuu
h
uuu
h
uuu
k
uu
jijiji
jijijijijijijiji
−−=
+−
−
+−
−
− −++−++++
β
From (21), we get
( )
( ) ( )( )( )auuukuuur
uuuruu
jijijijijiji
jijijijiji
−−++−=
+−−−
−+
+−++++
,
2
,,,1,,1
1,11,1,1,1,
22
222
β
2
/ hkr = Where
After some mathematical manipulation, we get
( ) ( ) ( ) ( )
( ) ( ) ( )221,...,4,3,2,12
22222
,
2
,
,,1,11,1,11,1
−=−++
−−++=+++− +−++++−
niuauk
uakruururuur
jiji
jijijijijiji
β
β
Equation (22) represents the Crank-Nicholson formula for equation (4).
The terms on the right hand side of equation (22) are all known.
Hence, the equations in (22) form a tridiagonal linear algebraic system
BAX = .
The boundary conditions are used in the first and last equations only
i.e. .,, 1,,1,1,1 jbuuandbuu jnjnjj ∀==== ++
Equations in (22) are especially pleasing to view in their tridiagonal
matrix form BAX = , where A is the coefficient matrix, X is the unknown
vector and B is the known vector as shown below:

92
=








































+−
−+−
−+−
−+−
−+
+
+
+
+
+
1j1,-n
1j2,-n
1,
1,3
1,2
u
u
:
:
22
22
22
22
22
jp
j
j
u
u
u
rr
rrr
rrr
rrr
rr
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) 























+−−+
−++−−++
−++−−++
−++−−++
+−++−−+
−−
−−−−−
+−
rbukarru
uaukukaruu
uaukukaruu
uaukukaruur
ruuaukukarrb
jnjn
jnjnjnjnjn
jpjpjpjpjp
jjjjj
jjjj
2222
12222r
:
12222r
:
12222
122222
,1,2
,2
2
,2,2,1,3
,
2
,,,1,1
,3
2
,3,3,4,2
,3,2
2
,2,2
β
ββ
ββ
ββ
ββ
When the Crank-Nicholson scheme is implemented with a computer,
the linear system BAX = can be solved by either direct means or by
iteration.In this paper, the Gaussian elimination method (direct method) has
been used to solve the algebraic system AX = B.
6. Stability Analysis of the Crank- Nicholson Scheme Using Fourier
(von Neumann) Method
The finite difference Crank-Nicholson formula for (13) is:
( ) ( )
( )23u-
2
2
2
2
mn,2
,1,,1
2
1,11,1,1,1,
βa
x
uuu
x
uuu
t
uu mnmnmnmnmnmnmnmn
∆
+−
+
∆
+−
=
∆
− −++−++++
Substituting ( ) xi
mn etu γ
ψ=, in equation (23), we have

93
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
[ ] ( )
( )
[ ]
( )
( ) ( ) ( )[ ] ( )[ ] ( )ttaee
tr
ee
ttr
ttt
eta
eee
x
t
eee
x
tt
e
t
ttt
eta
x
etetet
x
ettettett
t
etett
xixixixi
xi
xixixixixixixi
xi
xxixixxi
xxixixxixixi
ψβ
ψψ
ψψ
βψ
ψψψψ
βψ
ψψψ
ψψψψψ
γγγγ
γ
γγγγγγγ
γ
γγγ
γγγγγ
∆−−++−+
∆+
=−∆+
⇒
−−+
∆
+−+
∆
∆+
=



∆
−∆+
⇒−
∆
+−
+
∆
∆++∆+−∆+
=
∆
−∆+
∆−∆∆−∆
∆−∆∆−∆
∆−∆+
∆−∆+
2
2
2
2
2
2
2
2
2
2
2
2
22
2
)()(
2
)()(
Where 2
/ hkr =
( ) ( ) ( )[ ]
( )[ ] ( )
( )[ ] ( )[ ] ( )
( )[ ] ( )[ ] ( )
( )[ ] ( )[ ] ( )
( ) ( )[ ] ( ) ( )[ ] ( )ttaxtrxttr
ttaxtrxttr
ttaxtrxttr
ttax
tr
x
ttr
ttaxixxix
tr
xixxix
ttr
ttt
ψβγψγψ
ψβγψγψ
ψβγψγψ
ψβγ
ψ
γ
ψ
ψβγγγγ
ψ
γγγγ
ψ
ψψ
∆−∆−−−∆−−∆+−=
∆−∆−−∆−∆+−=
∆−−∆+−∆∆+=
∆−−∆+−∆
∆+
=
∆−−∆−∆+∆+∆+
−∆−∆+∆+∆
∆+
=−∆+
)2/sin21(1)2/sin21(1
cos1cos1
1cos1cos
2cos2
2
2cos2
2
2sincossincos
2
2sincossincos
2
22
( ) ( ) ( ) ( ) ( )⇒∆−∆−∆∆+−= ttaxtrxttr ψβγψγψ 2/sin22/sin2 22
( ) ( ) ( ) ( ) ( ) ( ) ( )⇒∆−∆−=∆∆++∆+ ttaxtrtxttrtt ψβγψψγψψ 2/sin22/sin2 22
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )[ ]
( )
( ) ( ) ξψψξ
γ
βγ
ψψ
γ
βγ
ψψ
ψβγψγ
=∆+⇒=
∆+
∆+∆−
=∆+
⇒
∆+
∆−∆−
=∆+
⇒∆−∆−=∆+∆+
ttt
xr
taxr
ttt
xr
taxr
ttt
ttaxrttxr
/
2/sin21
2/sin21
/
2/sin21
2/sin21
/
)2/sin21()2/sin21(
2
2
2
2
22
For stability, we need
( ) ( ) ⇒≤∆+ 1/ ttt ψψ
( )[ ]
( )
tar
xr
taxr
∆∀≤
∆+
∆+∆−
⇒≤ ,,,,1
2/sin21
2/sin21
1 2
2
β
γ
βγ
ξ
Hence, the Crank-Nicholson scheme is unconditionally stable.
7. Conclusions
We concluded from the comparison between the two schemes that the
explicit scheme is easier and has faster convergence than the Crank-

94
Nicholson scheme which is more accurate than the explicit scheme and the
results of this paper are affirming the analytical results which obtained by
Manoranjan et al [12] as shown below:
.( ) ( )aLiftastxu −<∞→→ 1/0, π(1) If0=bthen
.( ) π<∞→→ Liftasatxu ,(2) Ifab =then
.( ) aLiftastxu /1, π<∞→→(3) If1=bthen
as shown in figure (1) and table (1). In addition, from stability analysis, we
concluded that the explicit scheme is conditionally stable if
( )
( )2
2
4
2
,
4
2
xa
x
t
ta
r
∆−
∆
≤∆
∆−
≤
β
β
while the Crank-Nicholson scheme is
unconditionally stable.
Figure (1)
Figure(1)Explains the solution of the Huxley equation by the use of
Crank-Nicholson scheme for various values of H at a=b=0.8.
The figure shows that the solution of the problem converges to the
steady state solution u = a = 0.8 as t gets large at specific boundary
condition b = 0.8.

Table (1)
Explicit
b=0.25, a=0.25,
H=0.1
Crank-Nicholson
b=0.25, a=0.25,
H=0.1
Explicit
b=0.25,
a=0.25, H=0.3
Crank-Nicholson
b=0.25, a=0.25,
H=0.3
0.1000 0.1000 0.3000 0.3000
0.1287 0.1274 0.2910 0.2914
0.1573 0.1513 0.2817 0.2836
0.1771 0.1706 0.2751 0.2772
0.1933 0.1862 0.2696 0.2720
0.2056 0.1987 0.2654 0.2678
0.2152 0.2087 0.2621 0.2644
0.2228 0.2167 0.2595 0.2616
0.2287 0.2232 0.2575 0.2594
0.2333 0.2284 0.2559 0.2576
0.2369 0.2326 0.2546 0.2561
0.2397 0.2359 0.2536 0.2549
0.2420 0.2387 0.2528 0.2540
0.2437 0.2409 0.2522 0.2532
0.2451 0.2426 0.2517 0.2526
0.2461 0.2440 0.2514 0.2521
0.2470 0.2452 0.2511 0.2517
0.2476 0.2461 0.2508 0.2514
Table (1) shows the solution of Huxley equation by the use of
Crank-Nicholson scheme and explicit scheme for some values of a, b, and H.
The table above explains that the solution of the two schemes
converges to the steady state solution u = a = 0.25 and the number of steps
which are needed to reach the solution u = a = 0.25 in the explicit scheme is
less than the number of steps in the Crank-Nicholson scheme at specific
boundary condition b = 0.25 and H = 0.1, 0.3.
Acknowledgement
The authors would like to express their gratitude and indebtedness to
every person helped in this paper.
95

96
REFERENCES
[1] Binczak, S., Eilbeck, J.C. and Scott, A.C., (2001), Ephaptic
Coupling of Myelinated Nerve Fibers, Physica D 148 pp. 159-174.
[2] Broadbridge, P., Bradshaw, B.H., Fulford, G.R. and Aldis, G.K.,
(2002), Huxley and Fisher Equations for Gene Propagation : An
Exact Solution, ANZIAM J. Vol. 44, part 1, pp. 11-20.
[3] Eilbeck, J.C. and Manoranjan, V.S., (1986), A Comparison of Basis
Functions for The Pseudo-Spectral Method for a Model
Reaction-Diffusion Problem, J. Comput. Appl. Math., 15, pp. 371-
378.
[4] Eilbeck, J. C., (1986), The Pseudo-Spectral Method and Path
Following in Reaction-Diffusion Bifurcation Studies, SIAM J. Sci.
Stat. Comput., Vol. 7, No. 2, pp. 599-610.
[5] Estevez, P.G. and Gordoa, P.R., (1990), Painleve Analysis
of the Generalized Burgers-Huxley Equation, J. Phys. A:
Math. Gen., 23,pp. 4831-4837.
[6] Fath, G. and Domanski, Z., (1999), Avalanche of Bifurcations and
Hysteresis in a Model of Cellular Differentiation, Phys. Rev. E.
Vol. 60, No. 4, pp. 4604-4609.
[7] Garbey, M., Kaper, H.G., and Romanyukha, N., (2001), A Fast
Solver for Systems of Reaction-Diffusion Equations, Thirteenth
International Conference on Domain Decomposition Methods,
Editors: Debit, N., Garbey, M., Hoppe, R., Periaus, J., Keses, D. and
Knznetsov, Y.,pp. 385-392.
[8] Landa, P.S., (1996), Nonlinear Oscillations and Waves in
Dynamical Systems, Kluwer Academic Publishers.
[9] Lefantzi, S., Kennedy, C.A., Ray, J. and Najm, H. N., (2003), A
Study of the Effect of Higher Order Spatial Discretizations in
SAMR(Structured Adaptive Mesh Refinement) Simulations,
Proceedings of the Fall Meeting of the Western States Section of The
Combustion Institute, Los Angeles, AC.
[10] Lewis, T. J. and Keener, J.P., (2000), Wave-Block in Excitable
Media Due to Regions of Depressed Excitability, SIAM J. Appl.
Math., Vol. 61, No.1, pp. 293-316.
[11] Logan, J.D., (1987), Applied Mathematics, John Wiley & Sons.

97
[12] Manoranjan, V.S., Mitchell, A.R., Sleeman, B.D. and Yu, K. P.,
(1984), Bifurcation Studies in Reaction-Diffusion, J. Comput.
Appl. Math., 11, pp. 27-37.
[13] Manoranjan, V.S., (1984), Bifurcation Studies in Reaction-
Diffusion II, J. Comput. Appl. Math., 11, pp. 307-314.
[14] Mathews, J.H. and Fink, K.D., (1999), Numerical Methods Using
Matlab, Prentice-Hall, Inc.
[15] Pesin, Y. and Yurchenko, A., (2004), Some Physical Models of the
Reaction-Diffusion Equation and Coupled Map Lattices, Russian
Math. Surveys, Vol. 59, No. 3, to appear.
[16] Shanthakumar, M., (1989), Computer Based Numerical Analysis,
Khanna Publishers.
[17] Sherratt, J. A., (1996), Oscillatory and Chaotic Wakes Behind
Moving Boundaries in Reaction-Diffusion Systems, J. Dynamics
and Stability of Systems, Vol. 11, No. 4, pp. 303-324.
[18] Wang, X.Y., (1985), Nerve Propagation and Wall in Liquid
Crystals, Phys. Lett., Vol. 112 A, No. 8, pp. 402-406.
[19] Wang, X. Y., Zhu, Z. S. and Lu, Y. K., (1990) , Solitary Wave
Solutions of the Generalized Burgers-Huxley Equation, J. Phys.
A : Math. Gen., 23, pp. 271-274.

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