American Journal of Computational Mathematics, 2011, 1, 152-158
doi:10.4236/ajcm.2011.13017 Published Online September 2011 (http://www.SciRP.org/journal/ajcm)
Copyright © 2011 SciRes. AJCM
A Fourth Order Improved Numerical Scheme for the
Generalized BurgersHuxley Equation
Athanassios G. Bratsos
Department of Mathematics, Technological Educational Institution of Athens, Athens, Greece
E-mail: bratsos@teiath.gr
Received March 30, 2011; revised May 5, 2011; accepted May 22, 2011
Abstract
A fourth order finite-difference scheme in a two-time level recurrence relation is proposed for the numerical
solution of the generalized Burgers-Huxley equation. The resulting nonlinear system, which is analyzed for
stability, is solved using an improved predictor-corrector method. The efficiency of the proposed method is
tested to the kink wave using both appropriate boundary values and conditions. The results arising from the
experiments are compared with the relevant ones known in the available bibliography.
Keywords: Burgers-Huxley; Finite-Difference Method; Modified Predictor-Corrector
1. Introduction
A. Hodgkin and A. Huxley [1] proposed a model, known
henceforth as the Huxley equation, in order to explain
the ionic mechanisms underlying the initiation and pro-
pagation of action potentials in the squid giant axon. The
most general form of the Huxley equation, known as the
generalized Burgers-Huxley equation (BgH) [2,3], has
the form [4]
 
=1;0 1,>0
txxx
uuuu uuuxt

 
 ,
(1.1)
where is a sufficiently often differentiable
function,

=,uuxt
a real parameter, 0
, and

0,1
>0
. Equation (1.1), which models the interaction be-
tween reaction mechanisms, convection effects and dif-
fusion transport, is the modified Burgers equation for
=0
(see [5] and the references therein), is also the
Huxley equation [1] for =0
, =1
and is the Fitz-
hugh-Nagoma equation [6] for =0
.
Many researchers have used various methods to solve
the BgH equation. A theoretical study of the BgH equa-
tion was found in Wang et al. [4], while analytical solu-
tions using various techniques in [7-11], etc., have been
proposed. As far as the numerical methods are concerned
among others the Adomian decomposition method was
used by Ismail et al. [12] for the BgH and the Bur-
gers-Fisher equation, and by Hashim et al. [13] for the
BgH equation. Javidi [14] used the pseudospectral me-
thod, while Javidi [15], Javidi and Golbabai [16] the
spectral collocation method. Batiha et al. [17] used the
variational iteration method and Khattak [18] the collo-
cation method with radial basis functions. Babolian and
Saeidian [19] used the homotopy analysis method, etc.
The initial condition associated with Equation (1.1)
will be
,0=; 01.uxf xx (1.2)
Theoretical Solution
It is known [4] that Equation (1.1) has the following kink
wave solution
 
1
,= tanh
22
uxtkx ct




(1.3)
in which


241
=41
k
 
 
and
 

2
14
=121
c
  



 



1
are the wave number and the velocity respectively.
2. The numerical Method
2.1. Development of the Method
2.1.1. Grid and Solution Vecto r
To obtain numerical solutions the region
=,Rxt
0< <10,
x
T with its boundary consisting of
R
A. G. BRATSOS
153
the lines , and is covered with a
rectangular mesh of points, , with co-ordinates
=0x=1x=0t
G
,
x
t

=,
mn
x
t

n=,mh

,
mn
with . The
theoretical solution of Equation (1.1) at the typical mesh
point
=0,1, ,1mN
x
t

Ut
will be denoted by and the relevant
of an approximating difference scheme by .
n
m
u
n
m
U
=n
tt
1
, .
n
N
U
Let the solution vector at time level be
01
,,
n
U
==U
nn
U
(2.1)
2.1.2. Boundaries
The following were used:
1) The space derivatives at the left boundary
were replaced with second order finite-difference replace-
ments of the form ([20] p. 17)
=0x

2
n
UU

2
has
01
4
nn

=0
xx
1
=3
2
uU
h0,h
(2.2)

2
4
n
U
nd 1,u

0ash
2
3
n
U
 
01
=tgt
01
5
nn
=g
2
1
=2uU
h
0,ut
=0
xx xhU
a
(2.3)
and with analogous replacements to the right boundary
.
=1x
2) The boundary conditions

t (2.4)
=0,1=0;u
n
>
xx0,t
=1
(2.5)
were used, while at the other interior points of the grid G
the well-known approximants based on the central-dif-
ference formulas.
2.2. The Proposed Method
Applying Equation (1.1) at each point of the grid G at
time level ; leads to a first-order
initial-value problem, which is written in a matrix-vector
form as
==
n
tt ,2,n
 


1
,N
tt
fx


U

0=0 , =
Dt t 
UU

=
A
fx
UU

01
,
Bt

U=
fx
;>0
f
(2.6)
in which
=diagd dDt,

== =d 

g
nn
t
 
=diag
m
,
n
m
Uia (2.7)


g
nn
m
nn
mm
UU

;
== =d
=diag1
tia
 
 




(2.8)
for are diagonal matrices,
=0,1, ,1mN
2
3 41
10 1
...
1
=,
1 01
2
14 3
2541
12 1
.. .
1
=121
14 52
Ah
Bh




(2.9)
or
2
2 2
10 1
.. .
1
=,
1 01
2
22
22
12 1
...
1
=12 1
22
Ah
Bh
(2.10)
tridiagonal matrices arising from the use of the boundary
values (2.2) - (2.3) or the boundary condition (2.5) re-
spectively and f the vector of the initial condition, all of
order 2N
.
Relation (2.6) gives
=DAB

(2.11)
hence can easily be obtained.
2
D
Using the recurrence relation

=exp ; =0,,,tDttUU  (2.12)
where
DtU is given by (2.6) and replacing the ma-
trix-exponential term with the fourth order rational ap-
proximant ([21] p. 134) gives


22
22
11
212
11
=.
212
IDD t
ID Dt








U
U


(2.13)
Equation (2.13) using the notations (2.7) - (2.8) and
Equation (2.11) leads to the following nonlinear system
Copyright © 2011 SciRes. AJCM
154 A. G. BRATSOS








 
11
22
21 211
11 11
11 1
22
22
1
2
1
12
1
=2
1
12
nn
nn
nn nn
nn n
nn
nnn
nn nn
tABt
AB AB
ABAB
ABt
tABt
AB AB
ABAB


 
 


 


 
 
 

 




U
UU
 



U

.
nn n
ABt
 

U

n
U
(2.14)
Let
1=4rh
, 2
2=2rh, 3=2r
,
22 2
4=48rh
, 24
5=12rh, 22
6=1r
2,
23
7=24rh
, 2
8=2rh

4 and 22
9=12rh
.
Equation (2.14), when applied to the general mesh point
of the grid G, gives



1111 11
11121
1111 11 1
341212
11111 1
1152 1
2
11 11
126 7
2
46
4
nnnnnnn
mmmm mmm
nnnn nn n
mmmm mm m
nnnnn n
mmmmm m
nn nn
mm mm
UrUU rU UU
rU rUU
UrU UU
UU rUr
 
 
  
 
 
 
 

 

 
 



11
2
111 111
1128 11
11111 11
117121 1
1111 111
12119 11
11
22
2
2
nn
mm
nnn nnn
mmm mmm
nnnnn nn
mmmmm mm
nnnn nnn
mmmm mmm
nn
mm
U
UUU rU
Ur UUU
UUrU
U

 
 
 

 



 
 
 



111111
1181 1
111 1
91 1
2
nnnnnn
mmmm m m
nnn n
mmm m
Ur UU
rUUU

 
 

 
 
1
1


11121 1
34121211
2
521126
72112
811
=2
464
22
nnn nnnn
mmmmm mm
nnn nnnnnn
mmm mmmmmm
nnnnnn n
mmmmmm m
nnnn n
mmmm m
nnn
mmm
UrU UrUUU
rU rUU
UrUUUUUrU
rUUUU
rU
 
 
 


 
 
 

 

 


117 12
11 1211
9111181 1
91 1
2
2
2;
nn nn
mm mm
nnnnnnnn
mmmm mmmm
nnnnnnnnnn
mmmm mmmmmm
nnn n
mmm m
Ur U
UU UU
rUU UrUU
rU UU
 
 
 


 
  
 

= 0,1,...,1.mN

n
m
(2.15)
Stability Analysis
Following the Fourier method of analysing stability ([21]
p. 142) if =e
is the amplification factor and
the numerical value of actually obtained, an error
of the form
n
m
U
n
m
U
=
nnnim
mm
UU e
h
; =1i with
a
complex number and
real is considered. Then Equa-
tion (2.15) leads to the following stability equation


2
10 320
224
10603 905
22
40027 80
7
142sin2
sin
=1 2816
sin sin
4242sin
sin
sin4
rr ir
rr rrr
rirrr
r

2



 
 
 
(2.16)
where 0 a typical value of , ; U1n
m
Un
m
U=0,1, ,m
1N
used for the linearization of the nonlinear terms,
00
=U
,
000
=1
 
and =2h
with
0,
π2. Equation (2.16) is of the form
=;,ABAB


(2.17)
with the set of the complex numbers, so the von
Neumann necessary criterion for stability
1
will
always be satisfied when
.BA
(2.18)
Inequality (2.18) for =0
leads to
2
10 6010
121 ,rr r
 
 
(2.19)
which for =0
holds, while for =0
will be satis-
fied when
1. (2.20)
If =π2
, inequality (2.18) leads to

2
51060390
310
116 28
14 ,
rr rrr
rr
 
 
 
which subject to (2.20) holds.
2.3. The Modified Predictor-Corrector Scheme
To avoid solving the nonlinear system (2.14) the follow-
ing Modified Predictor-Corrector (MPC) scheme is pro-
posed.
2.3.1. Predictor
ˆt
U is evaluated from the reccurence relation (2.12)
replacing the matrix-exponential term with the following
explicit second order rational approximant
 

22 2
1
ˆ=0
2
tIDDtas

 


UU .
(2.21)
Then Equation (2.21) subject to Equation (2.11) using
Copyright © 2011 SciRes. AJCM
A. G. BRATSOS
155
again the notations (2.7) - (2.8) leads to

 
 

222
2
ˆ
=
2
.
nn
nn
nn nn
nn n
t
tABt
n
A
BA
ABAB
ABt


 
 

 




U
UU
U


B
(2.22)
Let 2
1=ph, 2=2ph
, 22 2
3=8ph
,
22
4=2p
, 24
5=2ph, 23
6=4ph
,
2
7=ph

4 and 23
8=4ph
.
Equation (2.22), when applied to the general mesh point
of the grid G, gives

 
1
11 1
211312
2
11 124
5211 2
62
ˆ=2
464
2
nn nnn nn
mmmmmmm
nnnnn n
mmmmm m
nnnnnn
mmmmm m
nnnnn
mmmmm
nn
mm
UUUpUUU
pUUpU
UUp
pUUU UU
pU

 
 
 
  
 
 

 



112
71111
612 1 1
11 12
811 11
7
2
22
2
nnn
mmm
nnn nn
mmm mm
n nnnnn
m mmmmm
nnnnn
mmmmm
nn nnnn
mmmm mm
nn
mm
UUU
pU UU
pUU U
UU
pUU U
p

 
 
 
 



 
 





11
81 1
2;
= 0,1,...,1.
nn
mm
nnn n
mmm m
UU
pU UU
mN



n
m
U
(2.23)
Stability Analysis
Following again the Fourier method of analysing stabil-
ity Equation (2.23) leads to the following stability equa-
tion


22
0401 80
2
42
530 0
2670 6
=14 2sin
16422
sin sin
42 sin24sin4
ppp
pp i
ppp p
 

,


 
 
(2.24)
which is of the form (2.17) with =1
A
. Then condition
(2.18) for =0
leads to
222
00
1
11
2

 

 .
(2.25)
which for =0
is obvious, while for =0
is satis-
fied when the condition (2.20) holds. When =π2
condition (2.18) leads to
22 2
000
242
4841
11
2
hhh

 
,

 
 


(2.26)
which again subject to (2.20) is always satisfied.
2.3.2. Corrector
The corrector arises from Equation (2.13) as follows
 

22
22
11 ˆ
=212
11
.
212
tDDt
I
DD





 


 

UU
Ut
(2.27)
Instead of the classical substitution of

t
a m
1n
m
U
U the
right-hand side of (2.27) by
ˆtUodified pre-
dictor-corrector method (MPC) was applied [5]. The
MPC method, which is explicit and is applied once, con-
sists of considering (2.27) component-wise and using an
updated component in the corrector vector as soon as it
becomes available. Hence, in computing
in
,
the cor-
rected value 1
1
n
m
U
instead of the predicted value 1
1
ˆn
m
U
is used. The stability analysis of the corrector is given in
Section 2.2.1.
3. Numerical Results
For the linearization was given.
Let the error at time level ; be

0
=0,1,...,1
0=maxmN
m
Uu
=tn=1n,2,
=0,1, ,1
===
max
eetLnn
m
u U
mN
m
and e
x
the x-co-
ordinate at which e occurs. Then e(2.2) - (2.3) denotes the
error arising when using the boundary values (2.2) - (2.3),
while analogous notations for the other boundary condi-
tions are used. In all experiments the initial condition
(1.2) was given by the value

=
 
,0
f
xux
=0.1h
with u the
theoretical solution (1.3). Experiments proved that the
most accurate results are obtained for and
4
=10 .
For reasons of comparison with the correspond-
ing works in [12,13,16,17] the same parameter values
were used.
3.1. Problem [12]
From the experiments the following are deduced:
1) when =0
(Table 1) using:
i) the boundary values (2.2) - (2.3) the method intro-
duced gives more accurate results for all time levels used
than the corresponding results in [12] and marginally
more accurate than those in [13,17],
ii) the boundary condition (2.5) gives more accurate
results than those in [12] and approximately equivalent
to those in [13,17].
Copyright © 2011 SciRes. AJCM
A. G. BRATSOS
Copyright © 2011 SciRes. AJCM
156
From (i) - (ii) it is deduced that the boundary values
(2.2) - (2.3) give more accurate results than the boundary
condition (2.5).
2) when =0
(Table 2) using the boundary values
(2.2) - (2.3) the method introduced has given
-for =1
more accurate results for all time levels
used than the corresponding in [12], and
-for >1
results with marginally inferior accuracy
to those in [12].
In Figure 1(a) the solution u for
0,1t and
44
10 ,10x

is shown, while in Figure 1(b) the rele-
vant solution U when
0,1x.
Table 1. Problem [12]. Comparisons of the proposed method for various values of x, t with α = 1, β = 1, γ = 0.001 and δ = 1 (h
= 0.1, = 10–4).
t x Exact e(2.2) - (2.3) e(2.5) e [12] e [13] e [17]
0.1 0.5000187E03 1.87406E08 1.26463E09 1.93715E07 1.87406E08 1.87405E08
0.5 0.5000687E03 1.87399E08 1.97698E08 1.93730E07 1.87406E08 1.87405E08
0.05
0.9 0.5001187E03 1.87250E08 4.60177E08 1.93745E07 1.87406E08 1.87405E08
0.1 0.5000250E03 3.74813E08 6.39532E09 3.87434E07 3.74812E08 3.74813E08
0.5 0.5000750E03 3.74736E08 3.99558E08 3.87464E07 3.74812E08 1.37481E08
0.1
0.9 0.5001250E03 3.74186E08 7.66328E08 3.87494E07 3.74812E08 3.74813E08
0.1 0.5001374E03 3.74814E07 3.29223E07 3.87501E06 3.74812E07 3.74812E07
0.5 0.5001874E03 3.72103E07 3.79222E07 3.87531E06 3.74812E07 3.74813E07 1
0.9 0.5002374E03 3.68427E07 4.29222E07 3.87561E06 3.74812E07 3.74813E07
Table 2. Problem [12]. Comparisons of the proposed method for various values of x, t and δ with α = 0, β = 1 and γ = 0.001 (h
= 0.1, = 10–4).
δ = 1 δ = 2 δ = 3
t x
e(2.2) - (2.3) e [12] e
(2.2) - (2.3) e [12] e
(2.2) - (2.3) e [12]
0.1 2.49875E08 1.87465E07 1.11763E06 5.58901E07 3.96731E06 1.9841E06
0.5 2.49875E08 1.87486E07 1.11750E06 5.58836E07 3.96652E06 1.98371E06
0.05
0.9 2.49874E08 1.87508E07 1.11737E06 5.58772E07 3.96572E06 1.98331E06
0.1 4.99750E08 3.74934E07 2.23526E06 1.11779E06 7.93462E06 3.96811E06
0.5 4.99750E08 3.74977E07 2.23500E06 1.11766E06 7.93304E06 3.96731E06 0.1
0.9 4.99749E08 3.75019E07 2.23474E06 1.11753E06 7.93144E06 3.96652E06
0.1 4.99750E07 3.75002E06 2.23526E05 1.11754E05 7.93462E05 3.96632E05
0.5 4.99749E07 3.75044E06 2.23500E05 1.00741E05 7.93303E05 3.96553E05 1
0.9 4.99749E07 3.75086E06 2.23474E05 1.11728E05 7.93143E05 3.96473E05
(a) (b)
Figure 1. Problem [12] with δ = 1, α = 1, β = 1, γ = 0.001 when t [0,1]: In (a) the surface shows u(x,t) for x [–104, 104],
while in (b) the numerical solution U whe n x [0,1].
A. G. BRATSOS
157
Table 3. Problem [17]. Comparisons of the proposed method for various values of δ and γ when α = β = 1 (h = 0.1, = 10–4).
t = 1 δ = 1 γ = 10–3 t = 0.5 δ = 2 γ = 10–2 t = 0.5 δ = 4 γ = 10–2
x e(2.2) - (2.3) e [17] x e(2.2) - (2.3) e [17] x e(2.2) - (2.3) e [17]
0.1 3.74814E07 3.74812E07 0.1 3.89463E05 2.75734E04 0.1 5.69322E05 1.08762E03
0.5 3.72103E07 3.74814E07 0.3 3.89656E05 2.75614E04 0.3 5.69778E05 1.08644E03
0.9 3.68427E07 3.74813E07 0.5 3.89844E05 2.75493E04 0.5 5.70134E05 1.08527E03
Table 4. Problem [16]. Boundary conditions (2.4 ) – (2.5). Comparisons of the propose d method for various values of t withα =
5 and δ = 1 (h = 0.1, = 10–4).
γ = 10–3 γ = 10–4 γ = 10–5
t β
Method e [16] Method e [16] e e [16]
1 3.1570E08 3.1616E08 3.1584E10 3.1630E10 3.3410E12 3.1632E12
10 3.9684E07 3.9742E07 3.9702E09 3.9760E09 3.9704E11 3.9762E11
0.3
100 5.0291E06 5.0365E06 5.0316E08 5.0389E08 5.0318E10 5.0392E10
1 3.3393E08 3.3394E08 3.3408E10 3.3409E10 3.3410E12 3.3411E12
10 4.1976E07 4.1977E07 4.1995E09 4.1996E09 4.1997E11 4.1998E11
0.9
100 5.3165E06 5.3166E06 5.3221E08 5.3223E08 5.3224E10 5.3225E10
3.2. Problem [17]
From Table 3 it is deduced that the method introduced
using the boundary values (2.2) - (2.3) has given more
accurate results for all time levels and parameters used
than the relevant method in [17].
3.3. Problem [16]
For reasons of comparison with the relevant results in
[18] the boundary conditions (2.4) - (2.5) with
0=
g
t
and

0,ut
 
1=1,
g
tut were used. From Table 4 it
is deduced that the proposed method:
-has given marginally more accurate results to those in
[16] for all time levels and
,
used,
-for fixed
,
and
, the accuracy increases and U tends to identify
with u at long time level as
is refined,
, as
increases, the accuracy decreases.
4. Conclusions
An implicit finite difference scheme based on fourth-
order rational approximants to the matrix exponential
term was proposed for the numerical solution of the Bur-
gers-Huxley equation. The resulting nonlinear scheme
was solved using an improved predictor-corrector method.
The computational efficiency of the proposed method
given in detail in Section 3 was tested by comparing the
numerical results to selected ones in [12,13,16,17] using
both appropriate boundary values and conditions. Con-
clusions for the boundaries used were derived.
Since the real world problems lead to the numerical
solution of nonlinear equations or systems of equations,
the introduced low cost and easy-to-handle method en-
ables us to obtain accurate solutions.
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