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A novel computational method to nonlinear reaction-diffusion system using
shifted Legendre polynomials
M.Mahalakshmi ,R.Seethalakshmi G.Hariharan
Department of Mathematics, School of Humanities & Sciences,
SASTRA Deemed University, India
Abstract
The mathematical model of Praveen et al. (Biochemical Engineering Journal, 91
(2014):129-39) in immobilized enzymes is discussed. In this paper, an efficient shifted Legendre
computational matrix method (LCMM) is applied for solving the nonlinear differential equations
with Michaelis-Menten kinetics. To the best our knowledge, until there is no rigorous Legendre
operational matrix based solutions has been reported for the above model. Theoretical results are
obtained to analyze the effect of different parameter values. Several examples are given to
demonstrate the validity and applicability of the proposed spectral method. The numerical results
are compared with MADM results. Satisfactory agreement with the ADM results is noticed.
Keywords: Mathematical modeling; immobilized enzymes; Legendre operation matrices;
Adomian decomposition method; steady state conditions.
1. Introduction
In recent years, many orthogonal functions and polynomials have been implemented to
obtain the quality solutions of differential equations[1]. The reaction-diffusion model has been
developed in the spherical catalyst particles for Michaelis-Menten type kinetics and the steady
state equation to determine the concentration profiles of substrate inside catalyst particles to
assess the design and performance of such reactors [2]. Mohammadi and Hosseini[3] used a new
Legendre wavelet approach for solving the singular ordinary differential equation [3]. Khellat
and Yousefi [4] derived the linear Legendre mother wavelets matrices of integration. In recent
years, wavelets have found their place in many applications such as image processing, signal
processing and solving differential and integral equations. Yousefi [5] applied the Legendre
wavelets method (LWM) for solving the differential equations of Lane – Emder type. Razzaghi
and Yousefi [6, 7] had used the Legendre wavelets for constrained optimal control problems and
variational problems. Recently, Hariharan [8] used an efficient Legendre wavelet based
approximation method for a few-Newell and Allen-Cahn equations. Hariharan and Kannan [9]
reviewed the wavelet solutions for the solutions of reaction-diffusion equations arising in science
and engineering. Excellent references there in [9]. Jafari et al. [10] solved fractional differential
equations by using Legendre wavelets in collocation method. Praveen et al.[11] have studied
the theoretical analysis of intrinsic reaction kinetics and the behavior of immbolized enzymes
International Journal of Pure and Applied MathematicsVolume 119 No. 15 2018, 387-403ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/
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system for steady state conditions. Zaser Aghazadeh et al.[12] developed the Legendre wavelet
method for solving singular Intergo-differential equations. Recently, Zhijun et al. [13] used
Legendre wavelet method for solving fractional integro-differential equations. Fukang Yin et at.
[14] presented the Spectral methods using Legendre wavelets for nonlinear Kelin/sine Gordon
equations. Fukang Yin et al.[15] established a coupled method of Laplace transforms and
Legendre wavelets for nonlinear Klein-Gordon equations. Masood Roodaki et al.[16] applied
two dimensional Legendre wavelets and their Applications to integral equations. Sahu et al.[17]
studied the Legendre wavelet operational method for the numerical solutions of nonlinear
Volterra integro-differential equations system.
In this paper, we have applied Legendre Computational Matrix Method for solving
differential equations with Michaelis-Menten kinetics.
The organization of this paper as follows: In section 2, some relevant properties of shifted
Legendre polynomials are presented. In section 3, Mathematical model of the Michaelis-Menten
kinetics is discussed. Method of solution by the Legendre computational matrix method
presented in section 4. Illustrative examples are given in section 5. Results and discussions are
given in section 6. Concluding remarks are given in section 7.
2. Some properties of the shifted Legendre polynomials [1]
The well known Legendre polynomials Pn(z), defined on the interval [-1,1], have the
following properties:
Pn(z) = (-1)n Pn(z), Pn(-1) = (-1)
n, Pn(1) = 1 (1)
It is well known that the weight function is ω (z) =1 and the weighted space )1,1(2 L is
equipped with the following inner product and norm;
2
11
1
),(,)()()(),( uuudzzzvzuvu
. (2)
The set of Legendre polynomials forms a complete orthogonal system )1,1(2 L and;
,12
2)(
2
nhzP nn (3)
is obtained. In order to use these polynomials on the interval [0, L] the so-called shifted
Legendre polynomials are defined by introducing the change of variable z = .12
L
x
The shifted Legendre polynomials are defined as;
()(*
nn PxP )12
L
xwhere ,)1()0(
* n
nP (4)
The analytic form of the shifted Legendre polynomial )(*
xPn of degree n is given by;
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n
k
k
k
kn
n xLkkn
knxP
02
*
)!()!(
)!()1()( . (5)
Let ,1)( xL and the weighted space ),0(2 LL
Lis defined with the following inner product and
norm;
.,,)()()(, 21
0
LLLuuudxxxvxuvu L
L
(6)
The set of the shifted Legendre polynomials forms a complete ),0(2 LLL
orthogonal system and
122)(
2*
n
Lh
LxP nn
L is obtained. The function u(x) which is square integrable in [0,L], may
be expressed in terms of shifted Legendre polynomials as;
0
* ),()(i
ii xPcxu (7)
where the coefficients ic are given by;
,.....2,1,0,)()()()(
1
0
*
2*
idxxxPxuxP
c L
L
i
i
i
L
(8)
2.1 Fundamental relations
It is suggested the solution LCxu m ,0)( can be approximated in terms of the first (m+1) terms of shifted Legendre polynomials given by
m
i
ii xPcxu0
* ).()( (9)
3. Formulation of the problem and analysis [11]
The mass balance differential equation for substrate and product are
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)11(02
)10(02
2
2
2
2
Vdr
dp
rdr
pdD
Vdr
ds
rdr
sdD
p
s
where s and p are the concentration of substrate and product. Here sD and pD is the diffusion
coefficient for substrate and product within the support. The enzymatic reaction rate V for
various kinetic mechanisms is given as follows
(i) Simple Michaelis-Menten
sK
sVV
s
m
(12)
(ii) Uncompetitive substrate inhibition
sI
m
KKss
sVV
)/1( (13)
(iii)Total competitive product inhibition
sIs
m
KKpK
sVV
)/1( (14)
(iv) Total non- competitive product inhibition
V))(/1( sKKp
sV
sIs
m
(15)
(v) Reversible Michaelis - Menten reaction
)(
)('
es
em
ssK
ssVV
(16)
where sK - Intrinsic half saturation constant.
IK - Intrinsic inhibition constant.
mV - Maximum reaction rate
mV ' - Maximum reaction rate for reversible reaction.
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es - Equilibrium substrate concentration (reversible reaction)
Using the following dimensionless variables
II
s
Iss K
p
K
K
U
Rt
K
p
K
s
K
s 00
0
0 ,,,,,
2
1
2
1
33
sp
m
p
ss
m
sKD
VU
KD
VU (17)
Eq.(10) can be written in dimensionless form for simple Michaelis-Menten kinetics,
uncompetitive substrate inhibition, reversible Michaelis-Menten reaction [12]
)(92 2
2
2
fdx
d
xdx
ds (18)
where the nonlinear reaction term )(f representse
e
1,
)1)1(,
1 for
simple Michaelis- Menten, uncompetitive substrate inhibition and reversible Michaelis-Menten
reaction respectively.
e - dimensionless equilibrium substrate concentration for reversible reaction.
For total competitive and non-competitive product inhibition systems the mass balance Eq.(10 )
and
Eq. (11) becomes
;),(92 2
2
2
fdx
d
xdx
ds ),(9
2 22
2
fdx
d
xdx
dp (19)
where ),( f represents )1)(1(
,1
for total competitive and non-
competitive product inhibition respectively.
The boundary conditions for Eq. (9) and (10) are
1,
,00
00
xwhen
xwhendx
d
dx
d
(20)
where
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-Dimensionless substrate concentration.
0 - Dimensionless substrate concentration in the catalyst surface and outside the support.
- Dimensionless product concentration.
0 - Dimensionless product concentration outside the support.
x - Dimensionless radial distance.
- Ration of intrinsic half saturation constant over intrinsic inhibition constant.
s - Thiele modulus for substrate
p - Thiele modulus for product.
3.1 First order catalytic kinetics
We consider the situation where the dimensionless substrate concentration is less than 1.
Hence Eq. (18) reduces to
92 2
2
2
sdx
d
xdx
d (21)
3.2 Zero order catalytic kinetics
We consider the second major limiting situation found in practice, when the dimensionless
substrate concentration is very much greater than 1. Eq.(18) reduces to the simple linear
equation
2
2
2
92
sdx
d
xdx
d
(22)
4. Method of Solution
Consider the Eq.(21)
)(9)(2
)(22 xxD
xxD s (23)
with the initial conditions 0)0(',)1( 0
The proposed method is applied with m = 2 and the solution )(x is approximated as follows;
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)()(*
2
0
xPcx ii
i
(24)
For m = 2, a system of linear equations is obtained, two of them from the initial conditions and
other from the main equation using the collocation point x =0.5 which is the root of )(*
xPi = 0
Eq.(24) can be written in the following matrix form;
AxPx )()(
where )()()()( *2*1*0 xPxPxPxP , A Tccc 210
Here 166)(,12)(,1)( 2*
2
*
1
*
0 xxxPxxPxP
5. Illustrative Examples:
Limiting Cases
Case: (i)
Consider s =1, 0 =100 in Eq.(23)
The matrix equation for this problem is;
FAPPDx
PD )2
( 2 (25)
where 5.001P ,
000
000
12002D
000
600
020
D
The main matrices for the initial conditions are
0)0(,100)1( DAPAP (26)
From the Eq. (25) and (26), we get 76.1128.3596.52 210 ccc
Using the aforesaid Legendre spectral method, one can obtain
44.2956.70)( 2 xx
Case: (ii)
Consider s = 0.1, 0 =10
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The matrix equation for this problem is;
FAPPDx
PD )2
( 2 (27)
The initial conditions are
0)0(,10)1( DAPAP (28)
From the Eq. (27) – (28) we get 025.0075.09.9 210 ccc
Using the aforesaid Legendre spectral method, one can obtain
85.915.0)( 2 xx
Case:(iii)
Consider s =0.3, 0 =10
The matrix equation for this problem is;
FAPPDx
PD )2
( 2 (29)
The initial conditions are
0)0(,10)1( DAPAP (30)
From the Eq. (29) – (30) we get 184.9612.0204.0 210 ccc
Using the aforesaid Legendre spectral method, one can obtain
776.8224.1)( 2 xx
Case. (iv) Consider the equation (22)
22 9)(
2)( sxD
xxD (31)
with the initial conditions 0)0(',)1( 0 .
Limiting Cases
Case: (i)
Consider s =1, 0 =100
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The matrix equation for this problem is;
FAPDx
PD )2
( 2 (32)
The initial conditions are
0)0(,100)1( DAPAP (33)
From the Eq. (32) – (33) we get 25.075.099 210 ccc
Using the aforesaid Legendre spectral method, one can obtain
5.985.1)( 2 xx
Case:(ii)
Consider s =0.1, 0 =10
The matrix equation for this problem is;
FAPDx
PD )2
( 2 (34)
The initial conditions are
0)0(,10)1( DAPAP (35)
From the Eq. (34) – (35) we get 0025.00075.099.9 210 ccc
Using the aforesaid Legendre spectral method, one can obtain
985.9015.0)( 2 xx
Case:( iii)
Consider s =0.3, 0 =10
The matrix equation for this problem is;
FAPDx
PD )2
( 2 (36)
The initial conditions are
0)0(,10)1( DAPAP (37)
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From the Eq. (36) – (37) we get 028.0084.0888.9 210 ccc
Table: 1 Comparison of dimensionless substrate concentration (Zero and First order reaction)for s =0.1 and 0 = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.84
9.86
9.88
9.9
9.92
9.94
9.96
9.98
10
10.02
MADM
LCMM
Eq.21
LCMM
Eq.22
x
MADM LWM
(Eq.21)
LWM
(Eq.22)
0
9.9900 9.9901 9.9850
0.2
9.9906 9.9905 9.9856
0.4
9.9922 9.9920 9.9874
0.6
9.9951 9.9940 9.9941
0.8
9.9989 9.9946 9.9946
1
10.004 10.000 10.000
100
1.0
s
X
X
)(x
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Fig1. Comparion between the MADM and LCMM for s and 0
Table: 2 Comparison of dimensionless substrate concentration (Zero and First order
reaction) for s =0.3 and 0 = 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 18.5
9
9.5
10
MADMLCMM
Eq.22
LCMM
Eq.21
Fig2. Comparion between the
and LCMM for s and 0 MADM
Table: 3 Comparison of dimensionless
substrate concentration (Zero and First
X
MADM LCMM
(Eq2.2)
0
9.8700 9.8702
0.2
9.8749 9.8747
0.4
9.8896 9.8994
0.6
9.9147 9.9145
0.8
9.9483 9.9438
1
9.9927 10.000
100
3.0
s
X
)(x
x
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order reaction) for s =1and 0 = 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 198.5
99
99.5
100
MADM
LCMM
Eq.22
Fig3. Comparion between the MADM and LCMM for s and 0
X MADM
LCMM
(Eq.22)
0
98.500 98.500
0.2
98.574 98.560
0.4
98.7525 98.740
0.6
99.0495 99.040
0.8
99.465 99.460
1
100.00 100.000
100,10 s
)(x
x
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6. Results and Discussion
The numerical simulation of dimensionless concentration )(x for various values of
parameters 0
, ands
are given in figures (1-3). Tables ( 1-3) show the comparison between
the proposed LCMM results and the modified Adomian decomposition method (MADM) for
various values0
ands
.Our results have been compared with MADM results[11]. For larger
M, we get the results closer to the analytical solution.
7. Conclusion
We have applied a new Legendre spectral algorithm combined with the associated
operational matrices of derivatives. This proposed algorithm was employed for solving nonlinear
differential equations model with Michaelis-Menten enzyme kinetics. According to the
numerical simulations, it has been concluded that the proposed spectral method may be extended
to solve other types of linear and nonlinear enzyme kinetics models.
References:
[1] Mohamed meabad Khader , Ahmed saied Hendy, A Legendre Computational matrix
Method for solving High-Order Fractional Differential Equations, Walailak Journal of
Science and Technology, 11(4) (2014)289-305.
[2] W.Hartmeier, Immoblized biocatalyst, from simple to comples systems, Trends in
Biotechnology ,(1985) 111-127.
[3] F.Mohammadi , MM Hosseini , A new legendre wavelet operational matrix of derivative
and its applications in solving the singulat ordinary differential equations. Journal of the
Franklin Institute ,348(2011)1787-1796.
[4] F Khellat, SA. Yousefi, The linear Legendre mother wavelets operational matrix of
intergration and its application. Journal of the Franklin Institute 343(2006) 181-190.
[5] SA.Yousefi, Legendre eavelets methods for solving differential equations of Lane-Emden
type, Applied Mathematics and Computations, 181(2006) 1417-1442.
[6] M.Razzaghi , S. Yousefi, Legendre wavelets direct method for constrained optimal
control problems. Mathematical methods in the Applied Sciences, 25(7)(2002)529-539.
[7] M.Razzaghi , S. Yousefi , Legendre wavelet direct method for variational problems,
International Journal of Pure and Applied Mathematics Special Issue
399
-
14
Mathematics and Computer in simulation, 53(3)(2000) 185-192.
[8] G.Hariharan, An effiecient Legendre wavelet based approximation method for a few-Newell
and Allen-Cahn equations, Journal of Membrane Biology ,247(5)(2014) 371-380.
[9] G.Hariharan and K.Kannan, Review of wavelet method for the solution of reaction diffusion
equation problem in science and Engineering, Applied Mathematical modeling, 38(2013)
199-813.
[10] Jafari, H,Soleymanivaraki M, Firoozjaee, Legendre wavelets for solving fractional
differential equations , Journal of Applied Mathematics, 7(4)(2011)65-70.
[11] T.Praveen ,Pedro Valencia, L.Rajendran, Theoretical analysis of intrinsic reaction and
the behavior of immboilized enzymes system for steady state conditions, Biochemical
Engineering Journal, 91 (2014)129-139.
[12] Zaser Aghazadeh , Y.Gholizade Atani , P.Noras, The Legendre wavelet method for
solving singular Intergo-differential equations ,Computational methods for differential
equations, 2(2) (2014) 62-68.
[13] Zhijun Meng ,Lifeng Wang, Haoli , Wei zhang, Legendre wavelet method for solving
fractional integro-differential equations, International Journal of Computer Mathematics
92 (2015) 1275-1291.
[14] Fukang yin, Tian Tian, Junqiang song , Minzhu, Spectral methods using Legendre
wavelets for nonlinear Kelin/sine Gordon equations, Applied mathematics 275(2015)
321- 334.
[15] Fukang Yin, Junqiang song and Fengshun Lu, A Couple method of Laplace transform and
Legendre wavelets for nonlinear Kelin-Gordon equations, Mathematical Methods in the
Applied Scineces, 37(2014) 781-792.
[16] Masood Roodaki , Zahra Jafari Behbahani, Two dimensional Legendre wavelets and
their Applications to integral equations, Indian journal of Science and Technology, 8
(2015) 157-164.
[17] P.K.Sahu, S.SahaRay , Legendre wavelets operaitonal method the numerical solutions
of nonlinear Volterra integro-differential equations system, Applied Mathematics and
computation,256 (2015)715-723.
[18] D.Jeison, Simulation of the effect of intrinsic reaction kinetics and particle size on the
behavior of immobilized enzymes under internal diffusional restrictions and steady state
operation, Process Biochemistry,39(2003)393-399.
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[19] P.Valencia, Effect of particle size distribution on the simulation of immobilized enzyme
reactor performance, Biochemical Engineering Journal ,49 (2) (2010)256-263.
[20] BY Guo, JP Yan.Legendre- Gauss collocation method for initial value problems of
second order ordinary differential equations,Applied Numerical Mathematics,(2009)
1386- 13408.
[21] Razzaghi M , S.Yousefi, Legendre wavelets method for the solution of non linear
problems problems in the calculus of variations, Mathematical and Computer Modeling,
4(12)(2001)45-54.
[22] Mehmet Sezer, Mustafa Gulsu, Solving high-order linear differential equations by a
Legendre matrix method based on hybrid Legendre and Taylor polynomials, Numerical
Methods fir Partial Differential Equations, 26(3)(2010)641-661.
[23] R.Baronas , F.Ivanauskas, J.Kulys, M.Sapagovas, Modeling of amperometric biosensors
with rough surface of the enzyme membrane. Journal of Mathematical Chemistry,
34(2003)227-242.
[24] P.Manimozhi, A.Subbiah, L.Rajendran, Solution of steady –state substrate concentration in
the action of biosensor response at mixed enzyme kinetics, Sensors Actuators B, 147(2010)
290- 297.
[25] C.V.Pao, Ma thematical analysis of enzyme –substrate reaction diffusion in some
biochemical systems, Non-linear Anal,Theor,4(2)(1979)369-392.
[26] Naresh Berwal ,D. Panchal, C.l. Parihar ,Haar wavelet method for Numerical solution of
Telegraph equations, Italian journal of pure applied mathematics N-30(2013)317-328.
[27] F.Mohammadi ,M.Hosseini, A new Legendre wavelet operational matrix of derivative
and its applications in solving the singular ordinary differential equation with non analytic
solution, Applied Mathematical Sciences, 5(2011) 2537-2548.
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