Numerical solution of oscillatory reaction–diffusion system of $\lambda-omega$ type was done by using two finite difference schemes . The first one is the explicit scheme and the second one is the implicit Crank– Nicholson scheme with averaging of f(u,v) and without averaging of f(u,v). The comparison showed that Crank–Nicholson scheme is better than explicit scheme and Crank–Nicholson scheme with averaging of f(u,v) is more accurate but it needs more time and double storage and double time steps than Crank–Nicholson scheme without averaging of f(u,v).

1. Numerical Solution of Oscillatory Reaction – Diffusion System
of   Type
1
Saad Manaa & 2
Mohammad Sabawi
1
Department of Mathematics, College of Computers Sciences and Mathematics, Mosul University, Mosul, Iraq
2
Department of Mathematics, College of Education for Women, Tikrit University, Tikrit, Iraq
Abstract
Numerical solution of oscillatory reaction–diffusion system of   type was done by using two finite difference
schemes . The first one is the explicit scheme and the second one is the implicit rank– Nicholson scheme with averaging
of  vuf , and without averaging of  vuf , . The comparison showed that Crank–Nicholson scheme is better than
explicit scheme and Crank–Nicholson scheme with averaging of  vuf , is more accurate but it needs more time and
double storage and double time steps than Crank–Nicholson scheme without averaging of  vuf , .
1. Introduction and Mathematical Model
  Systems are a class of simple examples of two
coupled reaction – diffusion equations whose kinetics
have a stable limit cycle:
   
   
)1( a
vrurvv
vruruu
xxt
xxt







Here u and v are functions of space x and time t , with
Rx and 0t , and 22
vur  [1].  
Systems are an important prototype for the study of
reaction – diffusion equations which have been used to
model a number of biological and chemical systems [2].
Many systems in biology and chemistry are intrinsically
oscillatory. In such cases, the stable state in the absence
of spatial variation is not a stationary equilibrium , but
rather consists of temporal oscillations in the interacting
chemical or biological species . Examples include
intracellular calcium signaling , the Belousov –
Zhabotinskii reaction and some predator – prey
interactions . These systems also exhibit spatial
interactions , which are often modeled by diffusion . This
combination of local oscillations and spatial diffusion
produces a very wide range of spatiotemporal behaviors ,
including spiral waves , target patterns and
spatiotemporal chaos . In one spatial dimension , the
equivalent of both spiral waves and target patterns are
periodic travelling waves , which are periodic functions
of space and time, moving with constant shape and
speed. Sherratt studied various features of the oscillatory
reaction-diffusion systems of   type in series of
papers. He presented numerical evidence for a complex
sequence of bifurcations in the unstable region of
parameter space [6]. He also used numerical techniques
to give suggestion that the spatiotemporal irregularities
are genuinely chaotic [3]. He also made a comparison
between the numerical solutions of the oscillatory
reaction-diffusion systems of   type [4]. He used
numerical results suggesting that when this system is
solved on semi-infinite domain subject to Dirichlet
boundary conditions in which the variables are fixed at
zero periodic travelling waves develop in the domain [5].
Borzi and Griesse presented the formulation, analysis,
and numerical solution of distributed optimal control
problems governed by lambda-omega systems [6].
Garvie and Blowey undertook the numerical analysis of a
reaction-diffusion system of   type, their results
are presented for a fully practical piecewise linear finite
element method by mimicing results in the continuous
case [7]. In this paper, the numerical solution of an
oscillatory reaction-diffusion system of   type by
the use of two finite difference schemes is done to the
following system [4].
  
  
 b
vuvu
x
v
t
v
vuvu
x
u
t
u
1
31
31
22
2
2
22
2
2


















With initial and boundary conditions 0





x
v
x
u at
0x and 01.0 vu at 0x , 0 vu , for 0x at
0t and 0 vu at Lx  where L is sufficiently
large number.
2. Discretization of the   System by Using
Explicit Scheme
By using the finite difference approximations for the
derivatives [8], we have
     


 
jijijiji
jijijijiji
vuvu
h
uuu
k
uu
,,
2
,
2
,
2
,1,,1,1,
31
2
     
     2
,,,
2
,
2
,,
3
,,,1,11,
333
21
jijijijijiji
jijijijiji
vkkvuvkukv
ukukruuru

  (2)
Where 2
/ hkr 
For the boundary condition 0xu , by using the central
difference formula, we get
jj
jj
uu
h
uu
,1,1
,1,1
0
2





(3)
from (2) , we have , for 0i
     
     3
,0,0,0
2
,0
2
,0,0
3
,0,0,1,11,0
333
21
jjjjjj
jjjjj
vkkvuvkukv
ukukruuru

  (4)
Substitute (3) in (4) , we obtain
   
     2
,0,0,0
2
,0
2
,0,0
3
,0,0,11,0
333
212
jjjjjj
jjjj
vkkvuvkukv
ukukrruu

 (5)
Equation (5) represents the right boundary condition,
with respect to equation (1b)

2.      jijijiji
jijijijiji
vuvu
h
vvv
k
vv
,,
2
,
2
,
2
,1,,1,,
31
21



 
     
     3
,,,
2
,
2
,,
3
,,,1,11,
333
21
jijijijijiji
jijijijiji
ukkuuvkukv
vkvkrvvrv

  (6)
The boundary condition 0xv , by using the central
difference formula, we have:
jj
jj
vv
h
vv
,1,1
,1,1
0
2




(7)
From (6) we have , for 0i
     
     3
,0,0,0
2
,0
2
,0,0
3
,0,0,1,11,0
333
21
jjjjjj
jjjjj
ukkuuvkukv
vkvkrvvrv

  (8)
Substitute (7) in (8), we obtain
   
     3
,0,0,0
2
,0
2
,0,0
3
,0,0,11,0
333
212
jjjjjj
jjjj
ukkuuvkukv
vkvkrrvv

 (9)
which represents the right boundary condition, with
respect to equation (1b).
3. Discretization of the   System by Using of
Crank – Nicholson Scheme without Averaging of
 vuf ,
The diffusion term xxu in this method is represented
with central differences, with their values at the current
and previous time steps averaged [8].
     




 
jijijiji
jijijijijijijiji
vuvu
h
uuu
h
uuu
k
uu
,,
2
,
2
,
2
,1,,1
2
1,1,1,1,1,
31
2
2
2
2
     
         3
,,,
2
,
2
,,
3
,,
,1,11,1,11,1
33322
22
jijijijijijijiji
jijijijiji
vkkvuvkukvukukr
uururuur

 
(10)
For the boundary condition 0xu , from central
difference formula , we have
jj
jj
uu
h
uu
,1,1
,1,1
0
2



 (11)
from (10) , we have , for 0i
     
         3
,0,0,0
2
,0
2
,0,0
3
,0,0
,1,11,01,11,1
33322
22
jjjjjjjj
jjjjj
vkkvuvkukvukukr
uururuur

 
from (11) , we get
1,11, 
 jji
uu (12)
substitute (11) and (12) in (10), we obtain the equation
for the left boundary condition which is
     
     3
,0,0,0
2
,0
2
,0,0
3
,0,0,11,01,1
333
222222
jjjjjj
jjjjj
vkkvuvkukv
ukukrruurru

  (13)
for (b) , we have
     




 
jijijiji
jijijijijijijiji
vuvu
h
vvv
h
vvv
k
vv
,,
2
,
2
,
2
,1,,1
2
1,11,1,1,1,
31
2
2
2
2
     
         3
,,,
2
,
2
,,
3
,0,
,1,11,1,11,1
33322
22
jijijijijijijii
jijijijiji
ukkuuvkukvvkvkr
vvrvrvvr

 
(14)
For the boundary condition 0xv , from central
difference formula, we have
jj
jj
vv
h
vv
,1,1
,1,1
0
2



 (15)
from (14), we have
     
      
   3
,0,0,0
2
,0
2
,0,0
3
,0,0
,1,11,01,11,1
333
22
22
jjjj
jjjj
jjjjj
ukkuuvk
uvkvkvkr
vvrvrvvr


 
from (15), we have
1,11,1 
 jj
vv (16)
Substitute (15) and (16) in (14), we have the equation for
the left boundary condition, with respect to equation
(1b)-
     
      3
,0,0,0
2
,0
2
,0,0
3
,0,0,11,01,1
333
222222
jjjjjj
jjjjj
ukkuuvkuvk
vkvkrrvvrrv

  (17)
The equations in above are especially pleasing to view in
their tridiagonal matrix form 111 BXA  , where 1A is the
coefficient matrix, X is the unknown vector and B is the
known vector as shown below:

3. 


































































1,
1j1,-n
1j2,-n
1,
1,2
1,1
u
u
:
:
1
22
22
22
22
222
jn
jp
j
j
u
u
u
u
rr
rrr
rrr
rr
r
r
r
=
         
         
         
         
         




































0
33322
33322
33322
33322
333222
3
,1,1,1
2
,1
2
,1,1
3
,1,1,2
3
,2,2,2
2
,2
2
,2,2
3
,2,1,2,3
3
,,,
2
,
2
,,
3
,,1,,1
3
,2,2,2
2
,2
2
,2,2
3
,2,3,2,1
3
,1,1,1
2
,1
2
,1,1
3
,1,2,1
jnjnjnjnjnjnjnjnjn
jnjnjnjnjnjnjnjnjnjn
jpjpjpjpjpjpjpjpjpjp
jjjjjjjjjj
jjjjjjjjj
vkkvuvkukvukukrru
vkkvuvkukvukruukrru
vkkvuvkukvukruukrru
vkkvuvkukvukruukrru
vkkvuvkukvukruukr



































































1,
1j1,-n
1j2,-n
1,
1,2
1,1
v
v
:
:
1
22
22
22
22
222
jn
jp
j
j
v
v
v
v
rr
rrr
rrr
rr
r
r
r
=
         
         
         
         
         




































0
33322
33322
33322
33322
3333222
3
,1,1,1
2
,1
2
,1,1
3
,1,1,2
3
,2,2,2
2
,2
2
,2,2
3
,2,1,2,3
3
,,,
2
,
2
,,
3
,,1,,1
3
,2,2,2
2
,2
2
,2,2
3
,2,3,2,1
3
,1,1,1
2
,1
2
,1,1
3
,1,2,1
jnjnjnjnjnjnjnjnjn
jnjnjnjnjnjnjnjnjnjn
jpjpjpjpjpjpjpjpjpjp
jjjjjjjjjj
jjjjjjjjj
ukkuuvkukvvkvkrrv
ukkuvukvkuvkrvvkrrv
ukkuvukvkuvkrvvkrrv
ukkuvukvkuvkrvvkrrv
ukkuvukvkuvkrvvkr
4. Discretization of the   System by Using of Crank – Nicholson Scheme with Averaging of  vuf ,
In this scheme the functions u and v in system (1b)will be replaced by their values at the current and previous steps
averaged.

4. 












 





 













 





 









2
3
2
22
1
2
2
2
2
,1,,1,
2
,1,
2
,1,
2
,1,,1
2
1,11,1,1,1,
jijijiji
jijijiji
jijijijijijijiji
vvuu
vvuu
h
uuu
h
uuu
k
uu
(18)
    jijiji
jijiji
jijijijiji
vuu
h
uuu
h
uuu
k
uu
,,
2
,2
,1,,1
2
2/1,12/1,2/1,1,2/1,
31
2
2
2
2
2/








(19a)
    
 










2/1,2/1,
2
2/1,
2
2/1,2
2/1,12/1,2/1,1
2
1,11,1,12/1,1,
3
1
2
2
2
2
2/
jiji
jiji
jijiji
jijijijiji
vu
vu
h
uuu
h
uuu
k
uu
(19b)
     
     2
2/1,2/1,1
3
2/1,12/1,11
2/1,12/1,111,11,11,11
322
22




jijijiji
jijijijiji
uvkukukr
uururuur
(20a)
  2/1,12/1,
2
2/1,1 3   jijiji vkuvk (20b)
Where
2
1 2/ hkr 
By the same manner we have followed in (4), with repeat to right boundary condition 0xu , we get
   
       3
,01,01,0
2
,01
2
,0,01
3
,01
,011,112/1,012/1,11
333
222222
jjjjjjj
jjjj
vkvkuvkuvkuk
ukrururur

 
(21a)
   
     
 3
2/1,012/1,01
2/1,0
2
2/1,01
2
2/1,02/1,01
3
2/1,01
2/1,0112/1,111,011,11
33
3
222222






jj
jjjjj
jjjj
vkvk
uvkuvkuk
ukrururur
(21b)
with repeat to (1b) , we have













 





 













 





 









22
3
22
1
2
2
2
2
,1,,1,
2
,1,
2
,1,
2
,1,,1
2
1,11,1,1,1,
jijijiji
jijijiji
jijijijijijijiji
vvuu
vvuu
h
vvv
h
vvv
k
vv
(22)
     jijijiji
jijiji
jijijijiji
vuvu
h
vvv
h
vvv
k
vv
,,
2
,
2
,2
,1,,1
2
2/1,12/1,2/1,1,2/1,
31
2
2
2
2
2/








(23a)
    
 










2/1,2/1,
2
2/1,
2
2/1,2
2/1,12/1,2/1,1
2
1,11,1,12/1,1,
3
1
2
2
2
2
2/
jiji
jiji
jijiji
jijijijiji
vu
vu
h
vvv
h
vvv
k
vv
(23b)
     
     
   3
,,1,
2
,1
2
,,1
3
,1,11
,1,112/1,12/1,12/1,11
333
22
22
jijijiji
jijijiji
jijijijiji
ukukuvk
uvkukvkr
vvrvrvvr


 
(24a)
     
     
   3
2/1,12/1,
2
2/1,1
2
2/1,2/1,1
3
2/1,12/1,11
12/1,111,11,11,11
3
22
22






jijiji
jijijiji
ijijijiji
ukuvk
uvkukukr
uururuur
(24b)
By the same manner we have already followed in (4) , with repeat to the right boundary condition , we get
   
       3
,01,01,0
2
,01
2
,0,01
3
,01
,011,112/1,012/1,11
333
222222
jjjjjjj
jjjj
ukukuvkuvkvk
vkrvrvrvr

  (25a)

5. Conclusions
In this study, we concluded that the Crank – Nicholson
Schemes without averaging and with averaging of  vuf ,
are more accurate than the explicit scheme which is
simpler and needs less time and storage . Crank –
Nicholson scheme with averaging of  vuf , is more
accurate than Crank – Nicholson scheme without
averaging of  vuf , but it doubles the time steps and
needs more storage as shown in table (1) below. The
figures (1), (2) and (3) show a comparison among
explicit, Crank-Nicholson without averaging and Crank-
Nicholson with averaging schemes. The figures (4) and
(5) show a comparison between explicit scheme and
Crank-Nicholson without averaging scheme when
4.0,01.0,2.0,1001,101,10,10  rkhmnba
. From the figures we concluded that the solutions
converge to the steady state solution 0,0  vu when
gets large.
Acknowledgement
The authors would like to express their gratitude and indebtedness to their colleague A. F. Kassim (Mosul University,
Iraq), for some references.
Table (1)
Crank-
Nicholson
with averaging
Crank –
Nicholson
without
averaging
Explicit
Scheme
0.010000.010000.01000
0.006890.004880.00192
0.005150.003340.00357
0.004120.002660.00191
0.003480.002270.00225
0.003040.002010.00161
0.002730.001820.00172
0.002410.001670.00149
0.002310.001550.00144
0.002170.001450.00131
0.002040.001210.00126
0.001640.001230.00117
0.001850.001180.00112
0.001770.001130.00106
0.001610.001060.00101
0.001640.001040.00965
0.001580.001010.00923
0.001530.000970.00883
0.001480.000640.00847
0.001440.000610.00812
0.001310.000860.00780
0.001360.000860.00749
0.001330.000840.00720
0.001210.000810.00693
0.000130.000790.00667
Table (1) shows a comparison among explicit, Crank-Nicholson without averaging and Crank-Nicholson
schemes with averaging of u when 1.0,004.0,4.0,251,11,1,1  hkrmnba .

6. Figure (1)
explains the solution function u of the system by using explicit scheme when
4.0,01.0,2.0,1001,101,10,10  rkhmnba
Figure(2)
explains the solution function u of the system by using Crank- Nicholson scheme without averaging when
4.0,01.0,2.0,1001,101,10,10  rkhmnba .
0 20 40 60 80 100 120
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
u
0 20 40 60 80 100 120
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
x
u
x

7. Figure(3)
explains the solution function u of the system by using Crank-Nicholson with averaging when
4.0,01.0,2.0,1001,101,10,10  rkhmnba .
Figure (4)
explains the solution function v of the system by using explicit scheme when
4.0,01.0,2.0,1001,101,10,10  rkhmnba .
0 20 40 60 80 100 120
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
x
u
0 20 40 60 80 100 120
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
x
v

8. Figure (5)
explains thee solution function v of the system by using Crank-Nicholson scheme without averaging when
4.0,01.0,2.0,1001,101,10,10  rkhmnba .
Figure (6)
explains thee solution function v of the system by using Crank-Nicholson scheme with averaging when
4.0,01.0,2.0,1001,101,10,10  rkhmnba .
0 20 40 60 80 100 120
-0.01
0
0.01
0.02
0.03
0.04
0.05
0 20 40 60 80 100 120
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03

9. References
1. J. A. Sherratt, On the Speed of Amplitude
Transition Waves in Reaction-Diffusion Systems
of   Type, IMA Journal of applied mathematics,
52, (1994), pp. 79-92.
2. J. A. Sherratt, Unstable Wavetrains and Chaotic
Wakes in Reaction-Diffusion Systems of  
Type, Physica D 82, (1995), PP. 165-179.
3. J. A. Sherratt, Periodic Waves in Reaction-
Diffusion Models of Oscillatory Biological
Systems, Forma, 11, (1996), pp. 61-80.
4. J. A. Sherratt, A Comparison of Two Numerical
Methods for Oscillatory Reaction-Diffusion
Systems, Appl. Math. Lett. Vol. 10, (1997), No. 2,
pp. 1-5.
5. J. A. Sherratt, Periodic Travelling Wave Selection
by Dirichlet Boundary Conditions in Oscillatory
Reaction-Diffusion Systems, SIAM J.APPL.
MATH. Vol. 63, (2005), No. 5, pp. 1520-1583.
6. A. Borzi and R. Griesse, Distributed Optimal
Control of Lambda-Omega Systems,
www.ricam.oeaw.ac.at/people/page/griesse/publicat
ions/oclosn.pdf.
7. Garvie, M. R. and Blowey, J. F., (2002), A
Reaction-Diffusion System of Type Part I I:
Numerical Analysis, Euro. Jnl of Applied
Mathematics pp. 1-26.
8. J. H. Mathews and K. D. Fink, Numerical Methods
Using MATLAB, Fourth Edition, (2004), Pearson
Prentice Hall.
‫االنتشار‬ ‫لنظام‬ ‫العددي‬ ‫الحل‬–‫ع‬‫النو‬ ‫من‬ ‫المتذبذب‬ ‫التفاعل‬ 
1
‫سعد‬‫اهلل‬ ‫عبد‬‫مناع‬‫و‬2
‫محيميد‬ ‫عبد‬ ‫محمد‬
1
‫ياضيات‬‫ر‬‫ال‬ ‫قسم‬،‫ياضيات‬‫ر‬‫ال‬‫و‬ ‫الحاسبات‬ ‫علوم‬ ‫كلية‬،‫الموصل‬ ‫جامعة‬‫اق‬‫ر‬‫الع‬ ، ‫الموصل‬ ،
2
‫ياضيات‬‫ر‬‫ال‬ ‫قسم‬،‫للبنات‬ ‫بية‬‫ر‬‫الت‬ ‫كلية‬،‫يت‬‫ر‬‫تك‬ ‫جامعة‬‫اق‬‫ر‬‫الع‬ ، ‫يت‬‫ر‬‫تك‬ ،
‫الملخص‬
‫االنتشار‬ ‫نظام‬ ‫حل‬ ‫تم‬–‫ع‬‫النو‬ ‫من‬ ‫المتذبذب‬ ‫التفاعل‬ ‫هي‬ ‫الثانية‬‫و‬ ‫يحة‬‫ر‬‫الص‬ ‫يقة‬‫ر‬‫الط‬ ‫هي‬ ‫األولى‬ . ‫المنتهية‬ ‫الفروقات‬ ‫ائق‬‫ر‬‫ط‬ ‫من‬ ‫يقتين‬‫ر‬‫ط‬ ‫باستخدام‬
‫يقة‬‫ر‬‫ط‬Crank – Nicholson‫ـ‬‫ل‬ ‫المعدل‬ ‫بأخذ‬ ‫األولى‬ : ‫حالتين‬ ‫في‬ ‫الضمنية‬ vuf ,‫ـ‬‫ل‬ ‫المعدل‬ ‫اخذ‬ ‫بدون‬ ‫الثانية‬‫و‬ ، vuf ,‫يقة‬‫ر‬‫ط‬ ‫إن‬ ‫نة‬‫ر‬‫المقا‬ ‫بينت‬ ‫إذ‬ .
Crank – Nicholson‫يقة‬‫ر‬‫ط‬ ‫ان‬‫و‬ ‫يحة‬‫ر‬‫الص‬ ‫يقة‬‫ر‬‫الط‬ ‫من‬ ‫أفضل‬ ‫هي‬Crank – Nicholson‫ـ‬‫ل‬ ‫المعدل‬ ‫بأخذ‬ vuf ,‫إلى‬ ‫تحتاج‬ ‫ولكنها‬ ‫دقة‬ ‫أكثر‬ ‫هي‬
‫يقة‬‫ر‬‫ط‬ ‫من‬ ‫مضاعف‬ ‫وخزن‬ ‫زمنية‬ ‫ات‬‫و‬‫خط‬ ‫الى‬‫و‬ ‫أكثر‬ ‫زمن‬Crank – Nicholson‫ـ‬‫ل‬ ‫المعدل‬ ‫اخذ‬ ‫بدون‬ vuf ,.