A novel computational method to n onlinear reaction ... · A novel computational method to n...

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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/

387

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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;

International Journal of Pure and Applied Mathematics Special Issue

388

3

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


389

4

)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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8


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


85.915.0)( 2 xx

Case:(iii)

Consider s =0.3, 0 =10


FAPPDx

PD )2

( 2 (29)


0)0(,10)1( DAPAP (30)



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


394

9


FAPDx

PD )2

( 2 (32)


0)0(,100)1( DAPAP (33)

From the Eq. (32) – (33) we get 25.075.099 210 ccc


5.985.1)( 2 xx

Case:(ii)



FAPDx

PD )2

( 2 (34)


0)0(,10)1( DAPAP (35)



985.9015.0)( 2 xx

Case:( iii)



FAPDx

PD )2

( 2 (36)


0)0(,10)1( DAPAP (37)


395

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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


397

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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


398

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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.

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