list of publications based on the...

100

LIST OF PUBLICATIONS BASED ON THE THESIS

1. A. Eswari, S. Usha, L. Rajendran, Approximate solution of non-linear reaction

diffusion equations in homogeneous processes coupled to electrode reactions

for CE mechanism at a spherical electrode. American Journal of Analytical

Chemistry, 2, (2011), 103-112.

2. S. Usha, A. Eswari, L. Rajendran, Mathematical modeling of predator-prey

models: Lotka-Voterra Systems. Global Journal of Theoretical and Applied

Mathematics Sciences, Vol. 1, No. 2, (2011), 103-113.

3. S. Usha, V. Abinaya, S. Loghambal, L. Rajendran, Non-linear mathematical

model of the interaction between tumor and oncolytic viruses. Applied

Mathematics, 3, (2012), 1089- 1096.

4. S. Usha, S. Anitha, L. Rajendran, Approximate analytical solution of non-

linear reaction diffusion equation in fluidized bed biofilm reactor. Natural

Science, Vol. 4, No. 12, (2012), 983-991.

101

PAPER PRESENTATION IN CONFERENCES/ SEMINARS/

WORKSHOPS

Participated in UGC & DST sponsored national conference on “Mathematical

Models and Methods” at The Madura College, Madurai on March 1 and

2-2012.

Delivered a lecture on the topic “Non-linear mathematical model of the

interaction between tumor and oncolytic viruses” at The Madura College,

Madurai on November 17-2012.

Presented the paper “Approximate analytical solution of non-linear reaction

diffusion equation in fluidized bed biofilm reactor” in CSIR and DRDO

sponsored national seminar on Mathematical Modeling, Approximate

Analytical and Numerical Methods” at K.L.N. College of Engineering,

Pottapalayam on December 27 and 28-2012.

102

CURRICULAM VITAE

S. Usha received her M.Sc in Mathematics from Alagappa University,

Karaikudi, Tamil Nadu, India during 1993. Also she has received her M.Phil., (2008 )

in Mathematics from Alagappa University, Karaikudi, Tamil Nadu, India. Also she is

doing her Ph.D. (part time) in Mathematical modelling at Manonmaniam Sundaranar

University, Tirunelveli under the guidance of Dr. L. Rajendran, Department of

Mathematics, The Madura College, Madurai. She has published four articles in peer-

reviewed journals. She is working as a B. T. Assistant in Rajah’s Hr. Sec. School,

Sivaganga, Tamil Nadu, India from 2007.

Her current research interests include mathematical modelling based on

differential equations and asymptotic approximations, analysis of non-linear reaction

diffusion processes in electrochemical systems. Also, she has participated and presented

technical papers in national conferences.

Reprints of Publications

American Journal of Analytical Chemistry, 2011, 2, 103-112 doi:10.4236/ajac.2011.22011 Published Online April 2011 (http://www.SciRP.org/journal/ajac)

Copyright © 2011 SciRes. AJAC

Approximate Solution of Non-linear Reaction Diffusion

Equations in Homogeneous Processes Coupled to Electrode

Reactions for CE Mechanism at a Spherical Electrode

A. Eswari, S. Usha, L. Rajendran*

Department of Mathematics, The Madura College (Autonomous), Madurai, Tamilnadu, India

E-mail: *[email protected]

Received 2011

Abstract

A mathematical model of CE reaction schemes under first or pseudo-first order conditions with different

diffusion coefficients at a spherical electrode under non-steady-state conditions is described. The model is

based on non-stationary diffusion equation containing a non-linear reaction term. This paper presents the

complex numerical method (Homotopy perturbation method) to solve the system of non-linear differential

equation that describe the homogeneous processes coupled to electrode reaction. In this paper the approxi-

mate analytical expressions of the non-steady-state concentrations and current at spherical electrodes for

homogeneous reactions mechanisms are derived for all values of the reaction diffusion parameters. These

approximate results are compared with the available analytical results and are found to be in good agreement.

Keywords: Non-Linear Reaction/Diffusion Equation, Homotopy Perturbation Method, CE Mechanism,

Reduction of Order, Spherical Electrodes

1. Introduction Microelectrodes are of great practical interest for quanti-

tative in vivo measurements, e.g. of oxygen tension in

living tissues [1-3], because electrodes employed in vivo

should be smaller than the unit size of the tissue of inter-

est. Microelectrodes having the geometry of a hemi-

sphere resting on an insulating plane are difficult to fab-

ricate, but their behavior is easily predicted [4]. They

also have advantages in electrochemical measurements

of molten salts with high temperature [5]. Microelec-

trodes of many shapes have been described [6]. Micro-

electrodes of simple shapes are experimentally preferable

because they are more easily fabricated and generally

conformed to simpler voltammetric relationships. Those

shapes with restricted size in all superficial dimensions

are of special interest because many of these reach true

steady-state under diffusion control in a semi infinite

medium [7]. Nevertheless, there is interest in microelec-

trodes of more complicated shapes, only because the

shapes of small experimental electrodes may not always

be quite as simple as their fabricators intended. Moreover,

and ironically, complex shapes may sometimes be more

easily modeled than simpler ones [8]. However, many

applications of microelectrodes of different shapes are

impeded by lack of adequate theoretical description of

their behavior.

As far back as 1984, Fleischmann et al. [9,10] used

microdisc electrodes to determine the rate constant of

coupled homogeneous reactions (CE, EC’, ECE, and

DISPI mechanisms). Fleischmann et al. [9] obtained the

steady-state analytical expression of the concentration of

the species HA and H by assuming the concentration of

the specie A is constant. Also measurement of the cur-

rent at microelectrodes is one of the easiest and yet most

powerful electrochemical methods for quantitative

mechanistic investigations. The use of microelectrodes

for kinetic studies has recently been reviewed [11] and

the feasibility demonstrated of accessing nano second

time scales through the use of fast scan cyclic voltam-

metry. However, these advantages are earned at the ex-

pense of enhanced theoretical difficulties in solving the

reaction diffusion equations at these electrodes. Thus it is

essential to have theoretical expressions for non steady

state currents at such electrodes for all mechanisms.

As far back as 1984, Fleischmann et al. [9,10] used

microdisc electrodes to determine the rate constant of

coupled homogeneous reactions (CE, EC’, ECE, and

104 A. ESWARI ET AL.

DISPI mechanisms). Fleischmann et al. [9] obtained the

steady-state analytical expression of the concentration of

the species HA and H by assuming the concentration of

the specie A is constant. Also measurement of the cur-

rent at microelectrodes is one of the easiest and yet most

powerful electrochemical methods for quantitative

mechanistic investigations. The use of microelectrodes

for kinetic studies has recently been reviewed [11] and

the feasibility demonstrated of accessing nano second

time scales through the use of fast scan cyclic voltam-

metry. However, these advantages are earned at the ex-

pense of enhanced theoretical difficulties in solving the

reaction diffusion equations at these electrodes. Thus it is

essential to have theoretical expressions for non steady

state currents at such electrodes for all mechanisms.

In general, the characterization of subsequent ho-

mogenous reactions involves the elucidations of the

mechanism of reaction, as well as the determination of

the rate constants. Earlier, The steady-state analytical

expressions of the concentrations and current at micro-

disc electrodes in the case of first order EC’ and CE re-

actions were calculated [9]. However, to the best of our

knowledge, till date there was no rigorous approximate

solutions for the kinetic of CE reaction schemes under

first or pseudo-first order conditions with different diffu-

sion coefficients at spherical electrodes under non-

steady-state conditions for all possible values of reac-

tion/diffusion parameters ,S E , , E , 1,E 2 ,S

1,! 2! and 2E have been reported. The purpose of

this communication is to derive approximate analytical

expressions for the non-steady-state concentrations and

current at spherical electrodes for all possible values of

parameters using Homotopy perturbation method.

2. Mathematical Formulation of the

Problems At a range of Pt microelectrodes, the electroreduction of

acetic acid, a weak acid, is strutinized by as in a usual

CE reaction scheme. This reaction is known to proceed

via the following reaction sequence [9]

1

2

2

HA H A1

H H2

k

k

e

" #

" #

"

"

!"

!"

(1)

where 1 and 2 are the rate constants for the forward

and back reactiuons respectively and are related to an-

other by the known equilibrium constant for the acid

dissociation [9]. The initial boundary value problems for

different diffusion coefficients ( ) can be

written in the following forms [9]:

k k

HA H A, ,D D D

2

HA HA HA HAHA 1 HA 2 H A2

2c c D cD k c

t r rr

$ $ $% " # "

$ $$

2

H H H HH 1 HA2

2c c D cD k c

t r rr

$ $ $% " " #

$ $$ 2 H Ak c c (3)

2

A A A AA 1 HA2

2c c D cD k c

t r rr

$ $ $% " " #

$ $$ 2 A Hk c c

c

(4)

where HA are the diffusion coefficient of

the species , 1 and 2 are the rate

constant for the forward and back reactions respectively

and HA H A are the concentration of the species

HA, H and A. These equations are solved for the follow-

ing initial and boundary conditions:

H A, and D D D

HA, H and A

, and c c c

k k

H H HA HA A A0 ; , , t c c c c c& & &% % % % (5)

H HA A; 0, 0, 0Sr r c dc dr dc dr % % % %

Ac

(6)

H H HA HA A ; , , r c c c c c& & &% & % % % (7)

where S is the radius of the spherical electrode. We

introduce the following set of dimensionless variables:

r

HA H A

HA H A

HA H A1 22

HA HA

2 2

1 2 H A

HA HA HA

2 2

1 HA 2 A1 S1

HAHA H

2 2

1 HA 2 H2 S2

HAHA A

, , , ,

, , ,

,

, ,

,

S

S

S SE S

S SE

S SE

c c c ru v w

rc c c

D t D D

D Dr

k r k c c r

D D c

k c r k c r

DD c

k c r k c r

DD c

'

( ! !

& & &

& &

&

& &

&

& &

&

% % % %

% % %

% %

% %

% %

(8)

where , u ,v ,w ' and ( represent the dimen-

sionless concentrations and dimensionless radial distance

and dimensionless time parameters respectively.

2

2

2E S

u u uu v

( ' ''$ $ $

% " # "$ $$

w (9)

2

11 12

2E S

v v vu v

!!

( ' ''$ $ $

% " " #$ $$ 1 w (10)

2

22 22

2E S

w w wu v2 w

!!

( ' ''$ $ $

% " " #$ $$

(11)

where E , S , 1E , 1S , 2E and 2S are the di-

mensionless reaction/diffusion parameters and 1! , 2!

are dimensionless diffusion coefficients. The initial and

boundary conditions are represented as follows:

0, 1; 1; 1u v w( % % % % (12)

) * ) *1, 0; 0; 0v u w' ' '% % $ $ % $ $ % (13)

k c c (2) , 1; 1; 1u v w' + & % % % (14)

The dimensionless current at the microdisc electrode


A. ESWARI ET AL.

105

can be given as follows:

) *H 1=S SI nFAD r dv d

''

%# (15)

3. Analytical Expression of Concentrations

and Current Using HPM Recently, many authors have applied the HPM to various

problems and demonstrated the efficiency of the HPM

for handling non-linear structures and solving various

physics and engineering problems [20-25]. This method

is a combination of homotopy in topology and classic

perturbation techniques. The set of expressions presented

in Equations (9)-(14) defines the initial and boundary

value problem. The Homotopy perturbation method

[26-32] is used to give the approximate solutions of cou-

pled non-linear reaction/diffusion Equations (9) to (11).

The dimensionless reaction diffusion parameters E ,

S , 1E , 1S , 2E and 2S are related to one another,

since the bulk solution is at equilibrium in the non-steady

state. Using HPM (see Appendix A and B), we can ob-

tain the following solutions to the Equations(9) to (11).

) *) *

) * ) *

) *) * ) *

) *

2

1

1

1

1

2

1

11( , ) 1 exp

1 4

1 exp 1 exp

2

1 + exp 1 exp

1 2

11 exp

4

E

E

u

erfc

erfc

! '' (

! ' (,(

'' ( (

(

! '' ( (

! ' (

'

( !,(

- . /#0 1 2% " #

1 2# 0 3 45

6#. /# # " 71 2

3 48

- #.#0 1# 35

6. /#71 2# #

1 273 48

/" 2

4

(16)

) *

) *

1

2

1

11

1 1, 1

2

11 1 1 exp

4 2

E

v erfc'

' (' ! (

' ' (! ,(

. /#% # 1 2

1 23 4

- 6. /#. /0 71 2# # #1 2 1 20 73 4 3 45 8

!

(17)

) *) *

) *

) *

) *) * ) *

) *

2

21 2

1 2 2

2

1

1

22

1 2

2 1 2 1

2

11, 1 exp

4

11 exp

4

+ exp 1 exp

1

2

E

E

w

erfc

' ' ( ! !

' ! ! ( !,(

'!

( !,(

' ! (

' ! !

'! ( ! ! ! !

! (

- . /#0 1 2% # #

1 2# 0 3 45

6. /#71 2# #

1 273 48

-#0# 5

6. /#79 " #1 2

1 273 48

The Equations (16)-(18) satisfies the boundary condi-

tions (12) to (14). These equations represent the new

approximate dimensionless solution for the concentration

profiles for all possible values of parameters E , S ,

1E , 1S , 2E , 2S , 1! and 2! . From Equations (15)

and (17), we can obtain the dimensionless current, which

is as follows:

1H H

1 1

0.28217 0.564191 E

S SI r nFD AC

:! ( ! (

&% % " " (19)

Equation (19) represents the new approximate expres-

sion for the current for all values of parameters.

4. Comparison with Fleischmann Work [9] Fleischmann et al. [9] have derived the analytical ex-

pressions of dimensionless steady- state concentrations u

and v as follows:

) *

) *

1 1

1

1 1

1 1

1 1

1 1 exp 1

EE

E S

E S

u !

' !!

' !'

"-% " #0

"056- 6- 69 # # # " 70 75 85 88

(20)

) * ) *1 1

1 1

1 1

11 exp 1E

E

E S

E S

v !

' '! '

" !

- 6- 6% # # # "0 75 8" 5 8

(21)

Fleischmann assumed that the concentration profiles

of w is constant. So the definite solution for concentra-

tion profiles of w is not arrived upon in the third specie A.

The normalized current is given by

) * )) *

*H H

1 1 1 1

1 1

1

S S

E E E S

E S

I r nFD AC:

! !

!

&%

" " "%

"

(22)

When 1 1E S % the above equation becomes

) * ) *1 1

11 exp 1 E Eu 1' ! ' !

'- 6- 6% " # # "0 75 85 8

(23)

) * ) * 1 1

11 exp 1 E Ev ' ' !

'- 6- 6% # # # "0 75 85 8

(24)

The normalized current is given by

H H 1 11S S E EI r nFD AC: !&% % " " (25)

2

(18) Previously, mathematical expressions pertaining to

steady-state analytical expressions of the concentrations

and current at microdisc electrodes were calculated by

Fleischmann et al. [9]. In addition, we have also pre-

sented an approximate solution for the non-steady state

concentrations and current.


A. ESWARI ET AL.


106

5. Discussions Fleischmann et al. is noticed when ( is large. (Figures

3a-d) show the normalized dimensional concentration

profile of w in ' space calculated using Equation (18).

The plot was constructed for various values 2 0.1, 1E %

and 1 1! ; . From these figures it is confirmed that the

value of the concentration profile of w increases when

( and 2E increases. Also from the (Figures 1a-d and

2a-d ), it is evident that the concentration of species HA

and H increases when the radius of the electrode ( S )

decreases. Therefore, the use of the electrode of the small

radius is clearly advangeous for the study of CE reaction

mechanism. The concentration of specie A decreases

when the radius of the electrode decreases. It reaches the

steady state value when

r

1( < . The dimensionless cur-

rent log versus ! for various values of 1E" is given

in (Figure 4). From these figure, it is evident that the

value of the current decreases abruptly and reaches

the steady-state value when the values of 1 0.1E" # .

Also, the value of the current increases when the

reaction diffusion parameter 1E" increases.

Equations (16)-(18) are the new and simple approximate

solution of the concentrations of the isomers calculated

using Homotopy perturbation method for the initial and

boundary conditions Equations (12)-(14). The closed

approximate solution of current is represented by the Eq.

(19). The dimensionless concentration profiles of u ver-

sus dimensionless distance $ are expressed in (Figure

1a-d). From these figures, we can infer that the value of

the concentration decreases when ! and distance $

increases when 1E" % . Moreover when 1E" % and

1! & , the concentration attains the steady- state value. In

(Figure 2a-d), the normalized concentration profiles of

isomers v for various values of parameters are plotted.

From these figures, it is inferred that the concentration v

increases abruptly and reaches the steady-state value

when 5$ # . In (Figures 1a-d and 2a-d), the values of

dimensionless concentrations u and v for various values

of E" , E" and ! and for 1 1' ( are reported and a

satisfactory agreement with the available [9] estimates of

Dimensionless distance

Dim

ensi

onle

ss c

once

ntr

atio

n u

1 1.5 2 2.5 3 3.5 4 4.5 5

1.005

1

0.995

0.99

0.985

0.98

0.975

0.97

! = 1

! = 10, 100

! = 0.5

! = 0.1

E = 0.1, !1 = 0.01


Dim

ensi

onle

ss c

once

ntr

atio

n u

1 1.5 2 2.5 3 3.5 4 4.5 5

1

0.96

0.94

! = 1

! = 10, 100

! = 0.5

! = 0.1

E = 0.1, !1 = 0.5

0.98

0.92

0.9

0.88

0.86

(a) (b)


Dim

ensi

onle

ss c

once

ntr

atio

n u

1 1.5 2 2.5 3 3.5 4 4.5 5

1

0.9

0.95

! = 1

! = 10, 50

! = 0.5 ! = 0.1

E = 1, !1 = 0.010.85

0.8

0.75

0.7


Dim

ensi

onle

ss c

once

ntr

atio

n u

1 1.5 2 2.5 3 3.5 4 4.5 5

1

0.9

0.95

! = 1

! = 10, 100

! = 0.5

! = 0.1

E = 1, !1 = 0.5

0.85

0.8

0.75

0.7

0.65

0.6

0.55

(c) (d)

Figure 1. Normalized concentration u at microelectrode. The concentrations were computed using Equation (16) for various

values of ! and for some fixed small value of 1

"E

when the reaction/diffusion parameter and dimensionless diffusion coef-

ficient (a) , 1

0.1 0.01" 'E) ) (b) ,

10.1 0.5" '

E) ) (c) ,

11 0.01" '

E) ) (d) ,

11 0.5" '

E) ) . The key to the graph: ( __ )

represents Equation (16) and (+) represents Equation (23) [9].

A. ESWARI ET AL.

107


Dim

ensi

onle

ss c

once

ntr

atio

n v

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

1

! = 1

! = 10, 50

! = 0.5

! = 0.1

E1 = 0.1, !1 = 0.01

0.8

0.6

0.4

0.2

0


Dim

ensi

onle

ss c

once

ntr

atio

n v

1 1.5 2 2.5 3 3.5 4 4.5 5

1

! = 1

! = 10, 50

! = 0.5

! = 0.1

E1 = 0.1, !1 = 0.5

0.8

0.6

0.4

0.2

0

(a) (b)


Dim

ensi

onle

ss c

once

ntr

atio

n v

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

1

! = 1

! = 10, 100

! = 0.5

! = 0.1

E1 = 1, !1 = 0.01

0.8

0.6

0.4

0.2

0


Dim

ensi

onle

ss c

once

ntr

atio

n v

1 1.5 2 2.5 3 3.5 4

1

! = 1

! = 10, 100

! = 0.5

! = 0.1

E1 = 1, !1 = 0.5

0.8

0.6

0.4

0.2

0

(c) (d)

Figure 2. Normalized concentration v at microelectrode. The concentrations were computed using Equation (17) for various

values of ! and for some fixed small value of "E

when the reaction/diffusion parameter and dimensionless diffusion coef-

ficient (a) , 1 1

0.1 0.01" 'E) ) (b) ,

10.1 .5

10" '

E) ) (c) ,

1 11 0.01" '

E) ) (d) ,

1 11 0.5" '

E) ) . The key to the graph: ( __ )

represents Equation (17) and (+) represents Equation (24) [9].


Dim

ensi

onle

ss c

once

ntr

atio

n w

1 1.5 2 2.5 3 3.5 4 4.5 5

1

! = 1

! = 0.01

! = 0.5

! = 0.1

E2 = 0.1, !1 = 0.01

1.035

1.03

1.025

1.02

1.015

1.01

1.005

0.995


Dim

ensi

onle

ss c

once

ntr

atio

n w

1 1.5 2 2.5 3 3.5 4 4.5 5

1

! = 1

! = 0.01

! = 0.5

! = 0.1

E2 = 0.1, !1 = 0.5

1.003

1.0025

0.999

1.002

1.0015

1.001

1.0005

0.9995

(a) (b)


A. ESWARI ET AL.


108


Dim

ensi

onle

ss c

once

ntr

atio

n w

1 1.5 2 2.5 3 3.5 4 4.5 5

1

! = 1

! = 0.01

! = 0.1

E2 = 1, !1 = 0.01

1.08

0.99

1.07

1.06

1.05

1.04

1.03

1.02

1.01


Dim

ensi

onle

ss c

once

ntr

atio

n w

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

1

! = 1

! = 0.01

! = 0.1

E2 = 1, !1 = 0.5

1.045

1.04

1.005

0.995

1.035

1.03

1.025

1.02

1.015

1.01

(c) (d)

Figure 3. Normalized concentration w at microelectrode. The concentrations were computed using Equation (18) for various

values of ! and for some fixed value of the reaction/diffusion parameter and dimensionless diffusion coefficient (a) ,

2 10.1 0.01" '

E) ) (b) ,

2 10.1 0.5" '

E) ) (c) ,

2 11 0.01" '

E) ) (d) ,

2 11 0.5" '

E) ) .

Dimensionless time !

Dim

ensi

onle

ss C

urr

ent

log "

0 1 2 3 4 5 6 7 8 9 10

!1 = 0.01

104

#E1 = 10

103

102

101

100

#E1 = 1

#E1 = 0.5#E1 = 0.1

Dimensionless time !

Dim

ensi

onle

ss C

urr

ent

log "

0 1 2 3 4 5 6 7 8 9 10

!1 = 0.5

#E1 = 10

103

102

101

100

#E1 = 1 #E1 = 0.5

#E1 = 0.1

(a) (b)

Figure 4. Variation of normalized non-steady-state current response log as a function of the dimensionless time ! for

various values of 1

"E

and for the fixed values of (a) 1

0.01' ) (b) 1

0.5' ) . The curves were computed using Equation (19).

The key to the graph: ( __ ) represents Equation (19) and (+) represents Equation (25) [9].

6. Conclusions The time dependent non-linear reaction/diffusion equa-

tions for spherical microelectrodes for CE mechanism

has been formulated and solved using HPM. The primary

result of this work is simple approximate calculation of

concentration profiles and current for all values of fun-

damental parameters. We have presented approximate

solutions corresponding to the species HA, H and A in

terms of the parameters of E" , S" , 1E" , 1S" , 2E" ,

2S" , 1' , 2' and ! based on the Homotopy perturba-

tion method. This method can be easily extended to find

the concentrations and current for all mechanism for all

microelectrodes for various complex boundary condi-

tions.

7. References [1] J. Koryta, M. Brezina, J. Pradacova in A. J. Bard (Ed.),

“Electroanalytical Chemistry”, Vol. 11, Marcel Dekker,

New York, 1972.

[2] D. B. Cater and I. A. Silver in D. J. G. Ives and G. J. Jane

(Eds.), “Reference Electrodes”, Academic Press, New

York, 1961, p. 464.

A. ESWARI ET AL.

109

[3] R. S. Pickard, “A review of printed circuit microelec-

trodes and their production”, Journal of Neuroscience

Method, Vol. 1, 1979, pp. 301.

[4] K. B. Oldham, “Comparison of voltammetric steady

states at hemispherical and disc microelectrodes”, Jour-

nal of Electroanalytical Chemistry, Vol. 256, 1988, pp.

11-19.

[5] G. J. Hills, D. Inman and J. E. Oxley in I. S. Longmuir

(Ed.), “Advances in Polarography”, Vol. 3, Pergamon

PressOxford, 1960, pp. 982.

[6] R. M. Wightman, D. O. Wipf, in “Electroanalytical

Chemistry”, Bard A. J (ed.), Marcel Decker, New York,

1989, pp. 26.

[7] R. M. Wightman, D. O. Wipf, in “Electroanalytical

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[29] A. Eswari, L. Rajendran, “Analytical solution of steady

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A. ESWARI ET AL.


110

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Appendix A: Solution of the Equations (9) to (11) Using Homotopy Perturbation Method

Substituting Equation (A10) into Equations (A1) and

(A2) and (A3) and arranging the coefficients of powers

, we can obtain the following differential equations p In this Appendix, we indicate how Equations (16) to (18)

in this paper are derived. To find the solution of Equa-

tions (9) to (11) we first construct a Homotopy as fol-

lows:

20 0 0 0

2

2:

d u du dup

d dd $ $ !$0* + ) (A11)

, -2

2

2

2

21

20E S

d u du dup

d dd

d u du dup u

d dd

$ $ !$

" "$ $ !$

. /+ * +0 1

2 3

. /* * + + *0 1

2 3

21 1 1 1

0 0 02

2: 0E S

d u du dup u

d dd" "

$ $ !$v w* + + * ) (A12)

vw )

(A1) and

20 0 0 01

1 2

2:

d v dv dvp

d dd

''

$ $ !$0* + ) (A13)

, -2

11 2

2

11 12

21

20E S

d v dv dvp

d dd

d v dv dvp u

d dd

''

$ $ !$

'' "

$ $ !$

. /+ * +0 1

2 3

. /* * + * +0 1

2 31vw" )

(A2)

21 1 1 1 1

1 1 02

2: 0E S

d v dv dvp u

d dd1 0 0v w

'' "

$ $ !$"* + * + ) (A14)

and 2

0 0 0 022 2

2: 0

d w dw dwp

d dd

''

$ $ !$* + ) (A15)

, -2

22 2

2

22 22

21

20E S

d w dw dwp

d dd

d w dw dwp

d dd

''

$ $ !$

'' "

$ $ !$

. /+ * +0 1

2 3

.

21 1 2 1 1

2 2 02

2: 0E S

d w dw dwp u

d dd2 0 0v w

'' "

$ $ !$"* + * + ) (A16)

2u vw"* * + * + )0 12 3

/ (A3)

Subjecting Equations (A11) to (A16) to Laplace trans-

formation with respect to ! results in

2

0 002

21 0

d u dusu

dd $ $$* + * ) (A17)

and the initial approximations are as follows

0 0 00; 1; 0; 1u v w! ) ) ) ) (A4)

, -0 0 01; 0; 0, 0v du d dw d$ $) ) ) )$ (A5) 2

0 002

1 1

20

d v dv sv

dd $ $ ' '$1

* + * ) (A18)

0 0 0; 1; 0; 1u v w$ 45 ) ) ) (A6)

0; 0; 0; 0i i iu v w! ) ) ) ) (A7)

, - , -1; 0; 0, 0i i iv du d dw d$ $) ) ) )$ (A8)

2

0 002

2 2

20

d w d w sw

dd $ $ ' '$1

* + * ) (A19)

, -, -1 12

1 112

2 10

s

ES

d u du esu

d s s sd

' $"

"$ $ $$

+ +6 78 9* + + * + )8 9: ;

(A20) ; 0; 0; 0 1, 2,i i iu v w i$ 45 ) ) ) < ) (A9)

and

, -, -1 12

11 1 112

1 1 1

2 1

0

s

SEd v dv s ev

d s s sd

' $""

$ $ ' ' ' $$

+ +6 78 9* + * + +8 9: ;

)

(A21)

2 3

0 1 2 3

2 3

0 1 2 3

2 3

0 1 2 3

u u pu p u p u

v v pv p v p v

w w pw p w p w

= ) * * * *>>) * * * *?

> ) * * * *>@

(A10)

A. ESWARI ET AL.

111

, -, -1 12

21 1 212

2 2 2

2 1

0

s

SEd w dw s ew

d s s sd

' $""

$ $ ' ' ' $$

+ +6 78 9* + * + +8 9: ;

)

(A22)

Now the initial and boundary conditions become

0 0 00; 1; 0; 1u v w! ) ) ) ) (A23)

, - , -0 0 01; 0; 0, 0v du d dw d$ $) ) ) )$ (A24)

0 0 0; 1 ; 1 ; 1u s v s w s$ 45 ) ) ) (A25)

0; 0; 0; 0i i iu v w! ) ) ) ) (A26)

, - , -1; 0; 0, 0i i iv du d dw d$ $) ) ) )$ (A27)

; 0; 0; 0 1, 2,i i iu v w i$ 45 ) ) ) < ) (A28)

where s is the Laplace variable and an overbar indicates

a Laplace-transformed quantity. Solving equations (A17)

to (A22) using reduction of order (see Appendix-B) for

solving the Equation (A20), and using the initial and

boundary conditions (A26) to (A28), we can find the

following results

, -0 1 u $ ) s (A29)

, -, -, -

, - , -, -, -

, -, -, -

, - , -

11 1

1 11 2 2

11

1

1

1

11 1

1 1

s s

S S SE

s

S

e eu

s s ss

e

s s

$ ' $

$

" ' " " '"$

$ '' $

" '

' $

+ + + +

+ +

) + * +++ *

*+ *

(A30)

and

, -, -, -1 1

0

1s

ev

s s

' $

$$

+ +

) + (A31)

, -, -, -

, -, -, -

1

1

1

1 11 11 2 2 2

1

1

1

1 12

s

S SE E

s

S

ev

2s s s s

e

s

' $

' $

" "" "$

$

"$

'

+ +

+ +

6 7) + *8 9

: ;

+ +

+

(A32)

and

, -0 1w $ ) s (A33)

, -, -, -

, -, -, -

, -, -, -, -

, - , -

1

2

2

1

2 2 121 2 2

2 1

1

2 1 2

2 2 1

1

2 1 2

2 2 1

s

S SE

s

S

s

S

ew

s s s

e

s

e

s s

' $

' $

' $

" " '"$

$ ' '

" ' '

$ ' ' '

" ' '

$ ' ' '

+ +

+ +

+ +

) + ++

** +

** +

(A34)

According to the HPM, we can conclude that

, - , - 0 11

limp

u u u u$ $4

) ) * * (A35)

, - , - 0 11

limp

v v v v$ $4

) ) * * (A36)

, - , - 0 11

limp

w w w w$ $4

) ) * * (A37)

After putting Equations (A29) and (A30) into Equa-

tion (A35) and Equations (A31) and (A32) into Equation

(A36) and Equations (A33) and (A34) into Equation

(A37). Using inverse Laplace transform [33], the final

results can be described in Equations (16) to (18) in the

text. The remaining components of and , -nu x , -nv x

be completely determined such that each term is deter-

mined by the previous term.

Appendix B In this Appendix, we derive the solution of Equation

(A20) by using reduction of order. To illustrate the basic

concepts of reduction of order, we consider the equation 2

2

d d

dd

c cP Qc$$

* * ) R (B1)

where P, Q, R are function of r. Equation (A20) can be

simplified to

, -, -1 12

1 112

d d2 10

dd

s

ES

u u esu

s s s

' $"

"$ $ $$

+ +6 78 9* + + * + )8 9: ;

(B2)

Using reduction of order, we have

2; P Q

$s) ) +

and

, -, -1 11

s

ES

eR

s s s

' $"

"$

+ +6 78) + +8 9: ;

9 (B3)

Let u cv) (B4)

Substitute (B4) in (B1), if is so chosen that u

d2

d

cPc

$0* ) (B5)


112 A. ESWARI ET AL.

Substituting the value of P in the above Equation (A7)

becom

1c $) (B6)

The given Equation (B3) reduces to

' '

1v Q v R* ) 1 (B7)

where 2 '

1 0, 4 2

P P RQ Q R

c) + + ) )1 (B8)

Substituting (B8) in (B7) we obtain,

, -, -1 1

' '

s

SES

ev sv

s s s

' $" $" $

"+ +6 7

8 9+ ) + *8 9: ;

(B9)

Integrating Equation (B9) twice, we obtain

2

SS Ev A e B es

$$ " $+

) * +

, -, -

, -

1 1

1

2

1

1

s

SS

e

s s

' $" $ '

"'

+ +6 78* +8 9+: ;

9 (B10)

Substituting (B6) and (B10) in (B4) we have,

, -, -

, -

1

2

1

1

2

1

1

SSE

s

SS

A e B eu

s

e

s s

$$

' $

"$ $

" '"

$ '

+

+ +

) * +

68 9* +8 9+: ;

7 (B11)

Using the boundary conditions Equations (A27) and

(A28), we can obtain the value of the constants A and B.

Substituting the value of the constants A and B in the

Equation (B11) we obtain the equation (A30). Similarly

we can solve the other differential Equations (A17),

(A18), (A19), (A21) and (A22) using the reduction of

order method.

Appendix C

Nomenclature

Symbols

HAc Concentration of the species HA (mole cm–3)

Hc Concentration of the species H (mole cm–3)

Ac Concentration of the species A (mole cm–3)

HAc5 Bulk concentration of the species HA

(mole cm–3)

Hc5 Bulk concentration of the species H

(mole cm–3)

Ac5 Bulk concentration of the species A

(mole cm–3)

HAD Diffusion coefficient of the species HA

(cm2sec–1)

HD Diffusion coefficient of the species H

(cm2sec–1)

AD Diffusion coefficient of the species A

(cm2sec–1)

D Diffusion coefficient (cm2sec–1)

R Radial distance(cm)

T Time (s)

1k

Rate constant for the forward reactions

(cm3/mole sec)

2k

Rate constant for the backward reactions

(cm3/mole sec)

Sr Radius of spherical electrode (cm)

r Distance in the radial direction (cm)

u, v, w Dimensionless concentrations (dimensionless)

$ Dimensionless radial distance (dimensionless)

! Dimensionless time (dimensionless)

SI Current density at a sphere (ampere/cm2)

A Area of the spherical electrode (cm2)

F Faraday constant (C mole–1)

n Number of the electron (dimensionless)

Greek symbols

1' Dimensionless diffusion coefficient

(dimensionless)

2' Dimensionless diffusion coefficient

(dimensionless)

1 1

2 2

, ,

, ,

,

E S

E S

E S

" "

" "

" "

Dimensionless reaction/diffusion parameters

(dimensionless)


Global Journal of Theoretical and Applied Mathematics Sciences.

ISSN 2248-9916 Volume 1, Number 2 (2011), pp. 103-113

© Research India Publications

http://www.ripublication.com

Mathematical Modeling of Predator-prey Models:

Lotka-Volterra Systems

S. Usha, A. Eswari and L. Rajendran*

Department of Mathematics,

The Madura College, Madurai-625011, Tamilnadu, India.

Corresponding Author Email: [email protected]

Abstract

The boundary value problem in predator-prey system is formulated and

approximate expressions for predator and prey populations are presented. He’s

Homotopy perturbation method is used to give approximate and analytical

solutions of non-linear reaction equations containing a non-linear term related

to predator-prey model. The relevant analytical solutions for the predator and

prey population profiles are presented in terms of dimensionless parameters !, ml and . Our analytical results are compared with simulation results and

satisfactory agreement is noted.

Keywords: Non-linear reaction equations; Homotopy perturbation method;

Lotka-Volterra model; Predator-Prey model.

Introduction Mathematical modeling will always be an important field of mathematics because of

its applications to the real world. While no model is perfet, if a close enough

approximation can be obtained, then scientists can see how certain factors will affect a

situation by merely working out equations on a piece of paper as opposed to actually

running an experiment. One mathematical model that is frequently examined is the

Lotka-Volterra predator-prey model. This refers to a system in which there are two

populations known as the predator and the prey. The model states that the prey will

grow at a certain rate but will also be eaten at a certain rate because of predators. The

predators will die at a certain rate but will then grow by eating prey.

In the recent decades, considerable work on the permanence, the extinction and the global asymptotic stability of autonomous or nonautonomous Lotka-Volterra type

predator–prey systems have been studied extensively, for example [1-10]. In addition

to these, the book by Takeuchi [11] is a good source for dynamical behavior of Lotka-

104 S. Usha et al

Volterra systems. The predator–prey problem attempts to model the relationship in the

populations of different species that share the same environment where some of the

species (predators) prey on the others. The prey is assumed to exhibit linear growth

given by a positive parameter. Predator species consume preys with a nonlinear

interaction with another set of parameters that determine the rate of competition

between predators. The natural death rate of the predator is assumed to be linear and

given by a negative parameter. One of the earliest implementations, the Lotka–

Volterra model serves as a starting point of more advanced models in the analysis of

population dynamics. Because of its unrealistic stability characteristics [12], stability

analysis of the model and its generalizations has recently gained much attention.

To understand the behavior of a nonlinear system one can analyze the existence

and stability of equilibrium points. As parameters are varied changes in the number

and stability of equilibrium points lead to bifurcation. Numerical methods are usually

employed to perform this analysis [13]. Approximate techniques near equilibrium

points, such as the normal form method [14] provides a complementary approach for

our study. The well-known generalizations of the Lotka–Volterra model include the

addition of polynomial interactions [15], non-monotonic response functions [16], time

delayed [17] and diffusion effected, time delayed [18] non-monotonic interactions.

Nutku has proposed a generalization where an additional cubic rather than a quadratic

interaction is involved.

In pest management, insecticides are useful because they quickly kill a significant

proportion of an insect population. Integrated pest management (involves combining

biological, mechanical, and chemical tactics) has been proved to be more effective

than the classic methods (such as biological control or chemical control) both

experimentally (e.g. [19-21]) and theoretically (e.g. [22, 23]). Recently, Nie et al. [9]

proposed a predator-prey state-dependent impulsive system by releasing natural

enemies and spraying pesticide at different thresholds. However, to the best of our

knowledge, till date no general analytical results for the prey and the predator for the

Lotka-Volterra system for all values of the parameters have been reported [24, 25].

The purpose of this communication is to derive analytical expressions for prey and the

predator for the Lotka-Volterra system.

Construction of the Predator-Prey Model This initial section is dedicated towards seeing the evolution of the pred-prey model.

This will lead to the understanding of each part of the model and give a better feel of

what the model is exactly trying to represent.

Mathematical formulation of the problem

Volterra first proposed a simple model for the predation of one species by another to

explain the oscillatory levels of certain fish catches in the Adriatic. If N(t) is the prey

population and )(tP that of the predator at time t then Volterra’s model is [24, 25]

" #)()()(

tbPatNdt

tdN$% (1)

Mathematical Modeling of Predator-prey Models 105

" #dtcNtPdt

tdP$% )()(

)( (2)

The parameters cba ,, , 0 d are interpreted as follows: a represents the natural

growth rate of the prey in the absence of predators, b represents the effect of

predation on the prey, c represents the efficiency and propagation rate of the predator

in the presence of prey and d represents the natural death rate of the predator in the

absence of prey. As a first step in analyzing the Lotka-Volterra model we

nondimensionalise the system by writing [24, 25]

d

tcNu

)()( %& ,

a

tbPv

)()( %& , at%& ,

a

d%! (3)

and it becomes [24, 25]

" #)(1)()(

&&&&

vud

du$% (4)

" #1)()()(

$% &&!&&

uvd

dv (5)

Initial conditions are

lu %% ,0& (6)

mv %% ,0& (7)

Solution of boundary value problem using Homotopy perturbation

method (HPM) In recent days, Homotopy perturbation method is often employed to solve several

analytical problems. In addition, several groups demonstrated the efficiency and

suitability of the HPM for solving non-linear equations and other electrochemical

problems [26-29]. Recently, many authors have applied the HPM to various problems

and demonstrated the efficiency of the HPM for handling non-linear structures and solving various physics and engineering problems [26-29]. This method is a

combination of homotopy in topology and classic perturbation techniques. Ji-Huan He

used the HPM to solve the Lighthill equation [30], the Duffing equation [31] and the

Blasius equation [32]. The idea has been used to solve non-linear boundary value

problems [33], integral equations [34-36], Klein–Gordon and Sine–Gordon equations

[37], Emden –Flower type equations [38] and many other problems. This wide variety

of applications shows the power of the HPM to solve functional equations. The HPM

is unique in its applicability, accuracy and efficiency. The HPM [39] uses the

imbedding parameter p as a small parameter, and only a few iterations are needed to

search for an asymptotic solution. Recently, Rajendran et al. [40] reported an

approximate analytical method (He’s Homotopy perturbation method) to solve the

non-linear differential equations that describe the diffusion coupled with Michaelis-

Menten kinetics. Furthermore, Rajendran et al. [41] obtained an approximate

106 S. Usha et al

analytical solutions (He’s Homotopy perturbation method) corresponding to coupled

non-linear reaction diffusion equations. Using Homotopy perturbation method (refer

Appendix A), we obtained the approximate solutions of the eqns. (4) and (5) as

follows:

''''

(

)

****

+

,

$$-$--

-$$--

$$-$

$%

$$$$$$

$$$$$$$$

$$$$

2)12(2)1(2)1(2

)12(2)1(222)1()1(2

22)2(3)1(33

2

222

2222

22222

)1(

5.0)(

meememmememe

ememlmemeeml

meeemlemlel

u

&!&!&!&&&

!&!&&!&!&

&&!&!&&

!!!

!!!

!!!!!

!!& (8)

'''

(

)

***

+

,

$$-

-$--$$$

--$-$

$%

$$$$$

$$$$$$$$$

$$$$$$$$$$$

!&!&!&

!&!&!&!&!&&!&!

!&!&!&!&!&!&

!!!!!!!

!!!!!!

!&

elmelme

lmelmeeleleleel

elelelelellem

v

222

222222

222

)1(

5.0)(

)12()1(

)1()1(222

)1(22)1(2)2(22)2(32)1(3223

(9)

Numerical Simulation The non-linear differential equations (8-9) are solved by numerical methods. The

function pdex4 in SCILAB software which is a function of solving the boundary

value problems for differential equation is used to solve this equation. Its numerical

solution is compared with Homotopy perturbation method in Figures 1-3 and it gives

a satisfactory result for various values of ! and ml, . The SCILAB program is also

given in Appendix B.

Figure 1: Profile of the normalized populations of the Prey u and Predator v were

computed using equations (8) and (9) when the parameter 1%! and .1.0,5.0 %% ml

The key to the graph: (__) represents the Eqs. (8-9) and (+) represents the numerical

results.



computed using equations (8) and (9) when the parameter 1%! and

.66.0,25.1 %% ml The key to the graph: (__) represents the Eqs. (8-9) and (+)

represents the numerical results.


computed using equations (8) and (9) when the parameter 1%! , .5.0and1 %% ml

The key to the graph: (__) represents the Eqs. (8-9) and (+) represents the numerical

results.

108 S. Usha et al

Result and Discussion Equations (8) and (9) represent the most general new analytical expressions for the

prey and predator population profiles for all values of ! and ml, . It satisfies the

initial conditions. In Figures (1-3) we present the series of normalized population

profiles for a prey and predator as a function of the dimensionless parameters for

various values of ! and ml, . From these figures it is inferred that the maximum and

minimum value of Homotopy perturbation method prey u and predator v populations

are equal when the value of the parameter l.! . The typical periodic solution of prey

u and predator v are illustrated in Figures 1-3. The maximum and minimum value of

u and v depends upon the value of l,! and .m

Conclusions The time dependent non-linear reaction-diffusion equation in predator–prey models in

Lotka- Volterra system has been formulated and solved analytically and numerically.

Analytical expressions for the prey and predator populations are derived by using the

HPM. The primary result of this work is simple approximate calculations of prey and

predator populations for all values of dimensionless parameter ! and the constant

value ml and . The HPM is an extremely simple method and it is also a promising

method to solve other non-linear equations. This method can be easily extended to

find the solution of all other non-linear equations.

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Appendix A. Solution of the equations (4) and (5) using Homotopy perturbation method.

In this Appendix, we indicate how Eqs. (8) and (9) in this paper are derived. To find

the solution of Eqs. (4) and (5), we first construct a Homotopy as follows:

0)1( %'(

)*+

, -$-'(

)*+

, $$ uvud

dupu

d

dup

&& (A1)

0)1( %'(

)*+

, $--'(

)*+

, -$ uvvd

dvpv

d

dvp !!

&!

& (A2)

The initial approximations are as follows:

mvlu %%% 00 ;;0& (A3)

0; 0; 0i iu v& % % % ......,2,1%/ i (A4)

and

.......

.......

3

3

2

2

10

3

3

2

2

10

01

023

----%

----%

vpvppvvv

upuppuuu

(A5)

Substituting Eq. (A5) into Eqs. (A1) and (A2) and arranging the coefficients of

like powers of p , we can obtain the following differential equations.

0: 000 %$ u

d

dup

& (A6)

0: 00111 %-$ vuu

d

dup

& (A7)

0: 0110222 %--$ vuvuu

d

dup

& (A8)

and

0: 000 %- v

d

dvp !

& (A9)

0: 00111 %$- vuv

d

dvp !!

& (A10)

112 S. Usha et al

0: 0110222 %$$- vuvuv

d

dvp !!!

& (A11)

Now the initial condition become

mvlu %%% 00 ;;0& (A12)

0; 0; 0i iu v& % % % ......,2,1%/ i (A13)

The solution of the above equations are as follows:

)(0&& leu % (A14)

" #&!&

!& ee

lmu $% $ )1(

1 )( (A15)

''(

)

**+

,

$$---

$-$-$

$%

$$$$$$

$$$$$$$$

&!&!&&!&

!&!&&!&!&

!!

!!!!!

!!&

mememeemem

emelelelellmu

)]12([)]1([)]12([

)]1([)]1([22)]2([3)]1([3

2212

22222

)1(

5.0)( (A16)

and

)(0

!&& $% mev (A17)

" #!&&!!& $$ $% eelmv )1(

1 )( (A18)

''(

)

**+

,

$-$-$

$---$

$%

$$$$$$$$$

$$$$$$$$

)]12([)]1([)]1([)]2([2

2)]1([2)]2([33)]1([3

22222

22

)1(

5.0)(

!&!&!&!&!&

!&!&!&!&!&

!!!

!!!!!

!&

memeemmeel

leelelleellmv (A19)

According to the HPM, we can conclude that

.....................)(lim)( 101

--%%4

uuuup

5& (A20)

.............................)(lim)( 101

--%%4

vvvvp

5& (A21)

After putting Eqns. (A14) to (A16) into Eq. (A20) and Eqs. (A17) to (A19) into

Eq. (A21) we can obtain the final results which can be described in Eqs. (8) and (9) in

the text. The remaining components of )(xun and )(xvn be completely determined

such that each term is determined by the previous term.


Appendix B

SCILAB Programs to find the solution of the Eqs. (4) - (7)

function main1

options= odeset('RelTol',1e-6,'Stats','on');

%initial conditions

Xo = [0.5; 0.1];

tspan = [0,10];

tic

[t,X] = ode45(@TestFunction,tspan,Xo,options);

toc

figure

hold on

plot(t, X(:,1))

plot(t, X(:,2),':')

legend('x1','x2')

ylabel('x')

xlabel('t')

return

function [dx_dt]= TestFunction(t,x)

a=1.5;

dx_dt(1) =x(1)*(1-x(2));

dx_dt(2) =a*x(2)*(x(1)-1);

dx_dt = dx_dt';

return

Applied Mathematics, 2012, 3, 1089-1096

doi:10.4236/am.2012.39160 Published Online September 2012 (http://www.SciRP.org/journal/am)

Non-Linear Mathematical Model of the Interaction

between Tumor and Oncolytic Viruses

Seetharaman Usha1, Vairamani Abinaya

1, Shunmugham Loghambal

2,

Lakshmanan Rajendran1*

1Department of Mathematics, The Madura College, Madurai, India

2Department of Mathematics, V V College of Engineering, Tisaiyanvilai, India

Email: *[email protected]

Received June 23, 2012; revised July 25, 2012; accepted August 2, 2012

ABSTRACT

A mathematical modeling of tumor therapy with oncolytic viruses is discussed. The model consists of two coupled, de-

terministic differential equations allowing for cell reproduction and death, and cell infection. The model is one of the

conceptual mathematical models of tumor growth that treat a tumor as a dynamic society of interacting cells. In this

paper, we obtain an approximate analytical expression of uninfected and infected cell population by solving the non-

linear equations using Homotopy analysis method (HAM). Furthermore, the results are compared with the numerical

simulation of the problem using Matlab program. The obtained results are valid for the whole solution domain.

Keywords: Mathematical Modeling; Non-Linear Differential Equations; Numerical Simulation; Homotopy Analysis

Method; Tumor Cells; Oncolytic Viruses

1. Introduction

Oncolytic viruses are viruses that infect and kill cancer

cells but not normal cells [1-4]. Oncolytic virus therapy

originated early in the last century upon the observation

of occasional tumor regressions in cancer patients suf-

fering from virus infections or those receiving vaccina-

tions. Many types of oncolytic viruses have been studied

as candidate therapeutic agents including adenoviruses,

herpes viruses, reoviruses, paramyxoviruses, retroviruses,

and others [2,4]. Probably, the best-characterized onco-

lytic virus, that has drawn a lot of attention, is ONYX-

015, an attenuated adenovirus that selectively infects

tumor cells with a defect in the p53 gene [3]. This virus

has been shown to possess significant antitumor activity

and has proven relatively effective at reducing or elimi-

nating tumors in clinical trials [5-7]. Furthermore, a

small number of patients who were treated with the on-

colytic virus showed regression of metastases [2]. Al-

though safety and efficacy remain substantial concerns,

several other oncolytic viruses acting on different princi-

ples, including tumor-specific transcription of the viral

genome, have been developed, and some of these viruses

have entered in trials [2,8-10].

The oncolytic effect has several possible mechanisms

that yield complex results. The first such mechanism is

the result of viral replication within the cell and rupture

out of the cell [11,12]. The third mechanism involves

virus infection of cancer cells that induces antitumoral

immunity. Surviving mice acquired a resistance to re-

challenge with tumor cells [13]. Host immune response

maximizes antitumor immunity but also interferes with

virus propagation. Wakimoto et al. [14] studied the limi-

tation of virus propagation caused by host immune re-

sponse in the central nervous system. Ikeda et al. [15]

showed that the viral survival term was prolonged and

that virus propagation was increased by the anti-immune

drug, cyclophosphamide.

Several mathematical models that describe the evolu-

tion of tumors under viral injection were recently devel-

oped. Wodarz [13,16] presented a mathematical model

that describes interaction between two types of tumor

cells (the cells that are infected by the virus and the cells

that are not infected by the virus) and the immune system.

However, to the best of our knowledge, till date no gen-

eral analytical expressions for the mathematical model-

ing of two populations of cells namely uninfected tumor

cells and infected cells [17]. The purpose of this paper is

to derive approximate analytical expression of two types

of cells growing namely uninfected and infected tumor

cells by solving the non-linear differential equations us-

ing Homotopy analysis method [18-20].

2. Mathematical Models

The model, which considers two types of cells growing *Corresponding author.

Copyright © 2012 SciRes. AM

S. USHA ET AL.1090

in the logistic fashion, has the following form [17]:

1

d1

d

X X Y bXYr X

t K

! "# $ $% & ' ( X Y, (1)

2

d1

d

Y X Y bXYr Y aY

t K X Y

! "# $ $% & ' ( (2)

The equation can be solved subject to the following

initial conditions:

) * 00X a# + 0

0

(3)

) * 00Y b# + (4)

where X is the size of the uninfected cell population; Y is

the size of the infected cell population; 1 and 2 are

the maximum per capita growth rates of uninfected and

infected cells; K is the carrying capacity; b is the trans-

mission coefficient; and a is the rate of infected cell kill-

ing by virus. We introduce the following set of dimen-

sionless variables as follows [17]:

r r

* *

1

1 2 1 0 0

*

0 0 1

, ,

, ,

and .

,

,

X X K Y Y K a r

b r r r A a K

B b K t r t

,

- .

# # #

# # #

# #

(5)

The governing non-linear differential equations (Equa-

tions (1) and (2)) expressed in the following non-dimen-

sionless format:

) ** *

* * *

* *

d1

d

*

*

X X YX X Y

t X

-! "# $ $' ( Y (6)

) ** *

* * *

* *

d1

d

Y XY X Y

t X

- **

*

YY

Y. ,! "# $ $' (

(7)

An appropriate set of boundary condition are given by:

) **

00 0X A# + , (8) ) **

00Y B# + 0

3. Solution of Boundary Value Problem Using Homotopy Analysis Method

The Homotopy analysis method (HAM) is a powerful

and easy-to-use analytic tool for nonlinear problems [21-

23]. It contains the auxiliary parameter h, which provides

us with a simple way to adjust and control the conver-

gence region of solution series. Furthermore, the ob-

tained result is of high accuracy. Solving the Equations

(6) and (7) using HAM (see Appendix A) and simulta-

neous equation method (see Appendix B and C), the

steady state and transient contributions to the model are

given by:

) * ) ** * * * *

SS TRX t X X t# (9)

where

* 1SSX # (10)

) * ) * ) *) * ) * ) *

) *

) * ) * ) *

) *

) * ) * ) *

) *

) * ) * ) *

) *

* *

*

*

*

*

2 2 1 1

01* *

0

1 1 *

0 0

3 3 4 1 *

0

0

2 1 2 3*

0

0

2 11

1

1

1

2 3

1

1 2

t t

t

TR

t t

t t

t t

h A e eX t A e

B h A e e

h A e e

B

A h e e

B

- -

-

. , -

. - , -

- - . ,

-

. - ,

- . ,

-

- . ,

$ $

$

$ $

$ $ $

$ $ $

! "$ $% &' (# $ $

! "$ $% &' ( $

! "$ $% &' ( $ $

! "$ $% &' ( $ $

(11)

Similarly we can obtain as follows: ) ** *Y t

) * * * ) ** * *

SS TRY t Y Y t# (12)

where

* 0SSY # (13)

) * ) *) * ) *

) *) * ) *

) *) * ) *

) *) *) * ) *

) * ) *

* *

1 * *

0* * *

0

22 * *

0

2 3 1 22 * *

0

0

3 1 4 33

0

0

1

2

1 2 1

1 2 3 1 2

t t

t

TR

t t

t t

t t

hB e eY t B e

hB e e

B h e e

A

hB e e

A

. , . - ,

. - ,

. - , . - ,

- , . - ,

. - , . - ,

.

-

.

. - ,

-

. - ,

.

. - ,

$ $

$

$ $

$ $ $

$ $ $

! "$' (# $

! "$' ( $

! "$' ( $ $ $

! "$% &' ( $ $ $

(14)

Here *

SSX and represent a time independent *

SSY

steady state term and ) ** *

TRX t and denote the ) ** *

TRY t

time dependent transient component. The steady state

term will be important at long times as In con- *t /0.

trast the transient term will be of important at short times

as * 0.t /

4. Numerical Simulation

The non-linear equations (Equations (6) and (7)) for the

boundary conditions (Equation (8)) are solved by nu-

merically. The function ode45 in Scilab software is used

to solve two-point boundary value problems (BVPs) for

ordinary differential equations. The Matlab program is

also given in Appendix C. The numerical results are also

compared with the obtained analytical expressions

(Equations (9) and (12)) for all values of parameters , ,

- , . , 0A and .0B

5. Results and Discussion

Equations (9) and (12) represent the simple approximate


S. USHA ET AL. 1091

analytical expressions of the uninfected and infected tu-

mor cells for all values of parameters , , - , . , 0A

and 0 . The two types of tumor cell growing using

Equations (9) and (12) are represented in Figures 1-4. In

Figures 1, 2, the dimensionless uninfected tumor cells

reach the constant value when for some fixed

value of

B

* 2t #, and different values of - and . . The

dimensionless infected tumor cells *X reaches the

Figure 1. Size of the dimensionless uninfected cell popula-

tion X*(t*) are plotted using Equation (9) for the values A0 =

10, B0 = 2, = 2, ! = 1, h = 0.1 (a) " = 5 ( ) (b) " = 10

( ) (c) " = 15 ( ) and (d) " = 20 ( ).

Figure 2. Size of the dimensionless uninfected cell popula-

tion X*(t*) are plotted using Equation (9) for the values A0 =

10, B0 = 2, ! = 1, " = 6, h = 0.21 (a) = 2 ( ) (b) = 6

( ) (c) = 10 ( ) and (d) = 15 ( ).

steady state value when .* 2t 1In Figures 3 & 4, the dimensionless infected tumor

cells reach the constant value when for some

fixed value of

* 1t #. and different values of - and , .

The dimensionless infected tumor cells reaches the

steady state value when . Figures 5 and 6 give us

the confirmation for the above discussion in three-di-

mensional graphs also.

*Y* 1t #

Figure 3. Size of the dimensionless infected cell population

Y*(t*) are plotted using Equation (12) for the values A0 = 10,

B0 = 2, " = 2, ! = 1, h = 0.05, (a) = 5 ( ) (b) = 10 ( )

(c) = 15 ( ) and (d) = 20 ( ).

Figure 4. Size of the dimensionless infected cell population

Y*(t*) are plotted using Equation (12) for the values A0 = 10,

B0 = 2, = 50, ! = 1, h = 0.75 (a) " = 1.5 ( ) (b) " = 5

( ) (c) " = 10 ( ) and (d) " = 20 ( ).


S. USHA ET AL.1092

Figure 5. The normalized three-dimensional of dimen-

sionless uninfected tumor cells X*(t*) (Equation (9)) for t* =

0 to 1.

Figure 6. The normalized three-dimensional of dimen-

sionless infected tumor cells Y*(t*) (Equation (12)) for t* = 0

to 0.1.

6. Conclusion

In this work, the coupled system of time dependent dif-

ferential equations for the two types of cells growing has

been solved analytically using the Homotopy Analysis

Method. Approximate analytical expressions for unin-

fected and infected cell population are derived for all

values of parameters. Furthermore, on the basis of the

outcome of this work, it is possible to calculate the ap-

proximate rate of the tumor cells growth. The extension

of the procedure to other systems that include interaction

between tumor cells and anticancer agents seem possible.

7. Acknowledgements

This work was supported by the Council of Scientific and

Industrial Research (CSIR No.: 01(2442)/10/EMR-II),

Government of India. The authors also thank Mr. M.S.

Meenakshisundaram, Secretary, The Madura College

Board, Dr. R. Murali, The Principal, and Prof. S. Thiga-

rajan, HOD, Department of Mathematics, The Madura

College, Madurai, Tamilnadu, India for their constant

encouragement. The authors S. Usha is very thankful to

the Manonmaniam Sundaranar University, Tirunelveli

for allowing the research work.

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S. USHA ET AL.1094

Appendix A: Basic Concept of Liao’s Homotopy Analysis Method (HAM)

Consider the following nonlinear differential equation

) * 0N u t #! "' ( (A1)

where N is a nonlinear operator, t denotes an independent

variable, is an unknown function. For simplicity,

we ignore all boundary or initial conditions, which can

be treated in the similar way. By means of generalizing

the conventional Homotopy method, Liao constructed

the so-called zero-order deformation equation as:

) *u t

) * ) * ) *) * ) *

01 ;

;

p L t p u t

phH t N t p

2

2

$ $! "'

# ! "' (

( (A2)

where 3 40,1p5 is the embedding parameter, 0h 6 is

a nonzero auxiliary parameter, is an auxiliary

function, L is an auxiliary linear operator, ) * 0H t 6

) *t

0p

0 is an

initial guess of , is an unknown function.

It is important, that one has great freedom to choose aux-

iliary unknowns in HAM. Obviously, when

u

) *u t ) :t p2 *

# and

, it holds: 1p #

) * )0;0t u2 # *t

**

and (A3) ) * ) *;1t u t2 #

respectively. Thus, as p increases from 0 to 1, the solu-

tion varies from the initial guess to the

solution . Expanding in Taylor series

with respect to p, we have:

) ;t p2) *u t

) *0u t

) ;t p2

) * ) * ) *01

; m

mm

t p u t u t p2 0

#

# 7 (A4)

where

) * ) *

0

;1

!

m

m m

p

t pu t

m p

2

#

8#

8 (A5)

If the auxiliary linear operator, the initial guess, the

auxiliary parameter h, and the auxiliary function are so

properly chosen, the series (A4) converges at p = 1 then

we have:

) * ) * ) *01

mm

u t u t u t 0

#

# 7 . (A6)

Define the vector

9 0 1, , , nu u u# u : (A7)

Differentiating Equation (A2) for m times with respect

to the embedding parameter p, and then setting 0p #and finally dividing them by m!, we will have the so-

called mth-order deformation equation as:

3 4 ) * )1 1m m m m mL u u hH t; $ $$ # < u * (A8)

where

) *) *

) *1

1 1

;1

1 !

m

m m m

N t p

m p

2$

$ $

8 !'< #$ 8

u"( (A9)

and

0, 1,

1, 1. m

m

m;

=># ?

+@ (A10)

Applying 1L$ on both side of Equation (A8), we get

) * ) * ) * ) *1

1 1m m m m mu t u t hL H t; $$ $! "# <' (u (A11)

In this way, it is easily to obtain for at mu 1,m 1thM order, we have

) * ) *0

M

mm

u t u t#

#7 (A12)

When , we get an accurate approximation

of the original Equation (A1). For the convergence of the

above method we refer the reader to Liao. If equation

(A1) admits unique solution, then this method will pro-

duce the unique solution. If equation (A1) does not pos-

sess unique solution, the HAM will give a solution

among many other (possible) solutions.

M / 0

Appendix B: Steady State Solution

For the case of steady-state condition, the Equations (6)

and (7) becomes as follows:

) ** *

* * *

* *1 0SS SS

SS SS SS

SS SS

X YX X Y

X Y

-! "$ $ #' ( (B1)

) ** *

* * * *

* *1 0SS SS

SS SS SS SS

SS SS

X YY X Y Y

X Y

-. ,! "$ $ #' (

(B2)

Solving the above Equations (B1) and (B2), we get

* 0SSY # (B3)

and

* 1SSX # . (B4)

Thus we can obtain *

SSX and as in the text

(Equation (10) and Equation (13)).

*

SSY

Appendix C: Non-Steady State Solution of the Equations using the HAM

The given differential equations for the non-steady state

condition are of the form as:

) ** *

* * *

* *

d1

d

TR TR TRTR TR TR

TR TR

*

*

X X YX X Y

t X

-! "# $ $' ( Y (C1)

) ** *

* * *

* *

d1

d

TR TR TRTR TR TR TR

TR TR

Y XY X Y

t X

- **

*

YY

Y. ,! "# $ $' (

(C2)

For the transient part, the initial conditions are rede-


S. USHA ET AL. 1095

fined as

) * ) ** * *

00 0TR SSX X X A# $ # 1$

1$

(C3)

) * ) ** * *

00 0TR SSY Y Y B# $ # (C4)

In order to solve the Equations (C1) and (C2) by

means of the HAM, we first construct the Zeroth-order

deformation equation by taking ,) * 1H t #

) * ) *

) *) *

) *

**

*

2**

*

* *

* * *2

* *

* * *

d1 1

d

d1

d

d

d

TRTR

TRTRTR

TR

TR TR TRTR TR

TR TR

Xp X

t

XXph X

t Y

X X XX Y

Y t Y

-

-

! "$ $% &

' (

!%# $ $%'

"& &(

(C5)

) * ) *

) *

) * ) *) *

**

*

* **

* *

2* *

2* *

* *

d1

d

d

d

TRTR

TR TR TRTR

TR

TR TRTR TR

TR TR

Yp Y

t

Y Yph Y

t X

Y YX Y

X X

, . -

, . -

, . .

! "$ $ $% &

' (

!%# $ $ %'

"$& &(

*

*d

dY

t (C6)

We have the solution series as

) * ) * ) ** * * * * *

,0 ,1

m

TR TR TR mm

X t X t X t p0

#

# 7 (C7)

and

) * ) * ) ** * * * * *

,0 ,1

m

TR TR TR mm

Y t Y t Y t p0

#

# 7 (C8)

where

) * ) *

) * ) *

* *

* *

, *

0

* *

* *

*

0

;1,

!

;1

!

m

TR

TR m m

p

m

TR

TR m

p

X t pX t

m t

Y t pY t

m t

#

#

8#

8

8#

8

(C9)

Substituting Equations (C7) and (C8) into Equations

(C5) and (C6) and comparing the coefficient of like

powers of p, we get

) **

,00

,0*

d: 1

d

TR

TR

Xp

t- $ #* 0X (C10)

) **

,0 *

,0*

d0

d

TR

TR

YY

t, . - $ $ # (C11)

) * ) *

) * ) *) *

) *

) *

* *

,1 ,01 *

,1* *

2*

,0*

,0 *

,0

3** *

,0,0 ,0

* * *

,0 ,0

2* * *

,0 ,0 ,0

d d: 1 1

d d

1 1

d

d

2

TR TR

TR

TR

TR

TR

TRTR TR

TR TR

TR TR TR

X Xp X h

t t

Xh X h

Y

h XX Xh

Y t Y

h X Y h X

-

-

$ #

$ $

(C12)

) * ) *

) * ) * ) *) *

) *

) *

* *

,1 ,01 *

,1* *

2*

,0*

,0 *

,0

3** *

,0,0 ,0

* * *

,0 ,0

2* * *

,0 ,0 ,0

d d: 1

d d

1 (

d

d

2

TR TR

TR

TR

TR

TR

TRTR TR

TR TR

TR TR TR

Y Yp Y h

t t

Yh Y h

X

h YY Yh

X t X

hX Y h Y

, . -

, . - , .

.

. .

$ $ #

$ $ $

(C13)

and so on.

The initial conditions are redefined as

) * ) ** * *

00 0TR SSX X X A0 1# $ # $ (C14)

) * ) ** * *

00 0TR SSY Y Y B0 1# $ # $ (C15)

and

) * ) ** *0 0, 0 0i iX Y# # for (C16) 2,3,i #

Solving the Equations (C10) and (C10) by using the

boundary conditions given in Equations (C14) and (C15),

we get

) * ) * ) * *1* *

0 0 1t

X t A e-$# $ (C17)

and

) * ) * ** *

0 0

tY t B e

. - , $# (C18)

Substituting the Equations (C17) and (C18) in Equa-

tions (C12) and (C12) and by using the boundary condi-

tions given in Equation (C16), we get

) *) * ) * ) *

) *

) * ) * ) *

) *

) * ) * ) *

) *

) * ) * ) *

) *

* *

* *

* *

* *

2 2 1 1

0* *

1

1 1

0 0

3 3 4 1

0

0

2 1 2 3

0

0

2 1

1

1

1

2 3

1

1 2

t t

t t

t t

t t

h A e eX t

B h A e e

h A e e

B

A h e e

B

- -

. , -

. - , -

- - . ,

-

. - ,

- . ,

-

- . ,

$ $

$ $

$ $ $

$ $ $

! "$ $% &' (#$

! "$ $% &' ( $

! "$ $% &' ( $ $

! "$ $% &' ( $ $

(C19)


S. USHA ET AL.


1096

and plot(t, X(:,1),'blue')

) * ) *) * ) *

) *) * ) *

) *) * ) *

) *) *) * ) *

) *) *

* *

*

* *

* *

* *

1

0* *

1 0

22

0

2 3 1 22

0

0

3 1 4 33

0

0

1

2

1 2 1

1 2 3 1 2

t t

t

t t

t t

t t

hB e eY t B e

hB e e

B h e e

A

hB e e

A

. , . - ,

. - ,

. - , . - ,

- , . - ,

. - , . - ,

.

-

.

. - ,

-

. - ,

.

. - ,

$ $

$

$ $

$ $ $

$ $ $

! "$% &' (# $

! "$% &' ( $

! $%' $ $ $

! $% &' ( $ $ $

plot(t, X(:,2),'green')

return

function [dx_dt]= TestFunction(t,x)

a=16;

b=3;

r=10;

"&(

"

(C20) dx_dt(1)

=x(1)*(1-(x(1)+x(2)))-(b*x(1)*x(2))/(x(1)+x(2));

dx_dt(2)=r*x(2)*(1-(x(1)+x(2)))+(b*x(1)*x(2))/(x(1)+x(

2))-a*x(2);

dx_dt = dx_dt';

Appendix E: Nomenclature

Symbol MeaningAdding the Equations (C17) and (C19) and the Equa-

tions (C18) and (C20), we get the Equations (9) and (12)

as in the text.

X Size of the uninfected cell population

Y Size of the infected cell population

a Rate of infected cell killing by the virus

b Transmission coefficient Appendix D: MATLAB Program to find the Numerical Solution of Non-Linear Equations (6) and (7)

1r Maximum per capita growth rates of unin-

fected cells

2r Maximum per capita growth rates of infected

cellsfunction main1

options= odeset('RelTol',1e-6,'Stats','on'); K Carrying capacity Xo = [10; 2]; t Time tspan = [0,10]; *X Size of the dimensionless uninfected cell po-

pulation tic

[t,X] = ode45(@TestFunction,tspan,Xo,options); *Y Size of the dimensionless infected cell popula-

tiontoc

figure *t Dimensionless time hold on

Vol.4, No.12, 983-991 (2012) Natural Science doi:10.4236/ns.2012.412127

Approximate analytical solution of non-linear reaction diffusion equation in fluidized bed biofilm reactor

Seetharaman Usha, Shanmugarajan Anitha, Lakshmanan Rajendran*

Department of Mathematics, The Madura College, Madurai, India; *Corresponding Author: [email protected]

Received 13 October 2012; revised 12 November 2012; accepted 27 November 2012

ABSTRACT

A mathematical model for the fluidized bed bio-

film reactor (FBBR) is discussed. An approxi-

mate analytical solution of concentration of phe-

nol is obtained using modified Adomian decom-

position method (MADM). The main objective is

to propose an analytical method of solution, which

do not require small parameters and avoid lin-

earization and physically unrealistic assump-

tions. Theoretical results obtained can be used

to predict the biofilm density of a single biopar-

ticle. Satisfactory agreement is obtained in the

comparison of approximate analytical solution

and numerical simulation.

Keywords: Fluidized Bed Biofilm Reactor;

Non-Linear Reaction Diffusion Equation; Phenol;

Effectiveness Factor; Modified Adomian

Decomposition Method

1. INTRODUCTION

There has been much interest in the development of

biofilms. Biofilms play significant roles in many natural

and engineered systems. The importance of biofilms has

steadily emerged since their first scientific description in

1936 [1]. Mechanistically based modeling of biofilms

began in the 1970s. The early efforts focused mainly on

substrate flux from the bulk liquid into the biofilm. Bio-

films have been used to treat wastewater since the end of

the 19th century. Biofilm reactors with larger specific

surface areas were developed starting in the 1980s [2-4].

Biofilm modeling was advanced by Rittmann and Mc-

Carty [5,6], who based their models on diffusion (Fick’s

law) and biological reaction (Monod kinetics) within the

biofilm and liquid-layer mass transfer from the bulk liq-

uid. The mathematical model to describe the oxygen uti-

lization for a TFBBR in wastewater treatment was devel-

oped by Choi [7], which was proposed to describe the

oxygen concentration distribution. This model consisted

of the biofilm model that described the oxygen uptake

rate and the hydraulic model that presented characteris-

tics of liquid and gas phase [8].

The fluidized bed reactor (FBB) is the reactor which

carries on the mass transfer or heat transfer operation

using the fluidization concept. At first it was mainly used

in the chemical synthesis and the petrochemistry industry.

Because this kind of reactor displayed in many aspects

its unique superiority, its application scope was enlarged

gradually to metal smelting, air purification and many

other fields. Since 1970’s, people have successfully ap-

plied the fluidization technology to the wastewater bio-

chemical process field. An FBB is capable of achieving

treatment in low retention time because of the high bio-

mass concentrations that can be achieved. A bioreactor

has been successfully applied to an aerobic biological

treatment of industrial and domestic wastewaters. An

FBB offers distinct mechanical advantages, which allow

small and high surface area media to be used for biomass

growth [9-12].

Fluidization overcomes operating problems such as

bed clogging and the high-pressure drop, which would

occur if small and high surface area media were em-

ployed in packed-bed operation. Rather than clog with

new biomass growth, the fluidized bed simply expands.

Thus for a comparable treatment efficiency, the required

bioreactor volume is greatly reduced. A further advan-

tage is the possible elimination of the secondary clarifier,

although this must be weighed against the mediumbio-

mass separator [10-12].

Abdurrahman Tanyolac and Haluk Beyenal proposed

to evaluate average biofilm density of a spherical bio-

particle in a differential fluidized bed system [13]. To our

knowledge, no general analytical expressions for the

concentration of phenol and effectiveness factor have

been reported for all values of the parameters , ! and .

However, in general, analytical solutions of non-linear

differential equations are more interesting and useful

Copyright © 2012 SciRes. Openly accessible at http://www.scirp.org/journal/ns/

S. Usha et al. / Natural Science 4 (2012) 983-991 984

their numerical solutions, as they are used to various

kinds data analysis. Therefore, herein, we employ ana-

lytical method to evaluate the phenol concentration and

effectiveness factor for all possible values of parameters.

2. MATHEMATICAL FORMULATION OF THE PROBLEM

The details of the model adopted have been fully des-

cribed in Haulk Beyenal and Abdurrahman Tanyolac [14].

Figure 1 represents a general kinetic scheme of differen-

tial fluidized bed biofilm reactor (DFBBR).

1. Base storage; 2. Heating strip; 3. Trap; 4 & 5.

Sponge plug; 6. Dissolved oxygen electrode; 7. Com-

bined gas feed; 8. Temperature control unit; 9. Thermo-

couple; 10. Thermometer; 11. Pulse dampener; 12. Steel

screen; 13. Non-fluidized medium (river sand); 14. Air

bubbles; 15. Support particle (active carbon; 16. Biofilm;

17. Acid storage; 18. Air sparger; 19. Oxygenerator; 20.

Differential fluidized bed biofilm reactor (DFBBR); 21.

pH electrode; AT—air trap; FCF—fresh culture feed (only

for start up); GCV—gas flow control valve; GFM—gas

flow meter; P1—pump for fresh feed; P2—fluidized bed

combined feed pump; P3—pump for base; P4—pump for

acid; R-rotameter; DOM—dissolved oxygen meter; pHM

—pH meter; pHC—pH controller; TC—temperature

controller; TM—temperature measurement; VC—volt-

age control.

The biological reaction is described by the Monod re-

lationship, which is a nonlinear expression. The differen-

tial equation for diffusion with Monod reaction within

6

9

TM

DOM

DOM

TM

TC

VC

Medium solrage

tank

13

12

15

16

14

18

21

17 P4

P3

8

4

10 6

Effluent tank

5

1

pHMpHC

R

1920

2

3

R

P1

P2

11R

AT

FCF

GFM

GCV

GFM

GFM

GFM

7

Oxygen

Nitrogen

Air

Fresh feed

Figure 1. A schematic diagram of differential fluidized bed biofilm reactor (DFBBR) system [14].




the biofilm is [13]

! "max2

2

d d

d d

f f

X P o

D SSr

r r Y K Sr

#$ % &' ( )* +

X (1)

where S is the phenol concentration, max# is the maxi-

mum specific growth rate of substrate, fD is the aver-

age effective diffusion coefficient of the limiting sub

strate, fX is the average biofilm density, X PY is the

yield coefficient for phenol, and oK is the half rate

kinetic constant for phenol. The equation can be solved

subject to the following boundary conditions [13]:

d0 at

dp

Sr r

r& & (2)

atbS S r r& & b (3)

where b denotes biofilm surface substrate concentra-

tion, r is the radial distance, rp is the radius of clean par-

ticle, and rb is the radius of biofilm covered bioparticle.

The effectiveness factor for a spherical bioparticle is

S

! "! " ! ! ""

2

3

max

4 d d

4 3

bb f X P r r

s

b f b o

r D Y S r

r X S K S,

#&&) b

(4)

Normalized Form

The above differential equation (Eq.1) for the model

can be simplified by defining the following normalized

variables,

max; ; ; ;

p fbb

b b b o f X P

r XSS ru r

S r r K D Y

#- . / & & & & &

oK

(5)

where , and u - . represent normalized concentra-

tion, distance and radius parameters, respectively. ! de-

notes a saturation parameter and is the Thiele modulus.

Furthermore, the saturation parameter ! describes the

ratio of the phenol concentration within the biofilm

to the rate kinetic constant for phenol ! "bS oK . Then

Eq.1 reduces to the following normalized form

2 2

2

d 2 d

d 1d

u u

u

u

- - /-) &

) (6)

The boundary conditions reduce to

d0 when

d

u- .

-& & (7)

1 when 1u -& & (8)

The effectiveness factor in normalized form is as fol-

lows:

! "2

1

3 1 d

d

u

-

/,

- &

) $ %& ' (

* + (9)

3. ANALYTICAL EXPRESSION OF CONCENTRATION OF PHENOL USING MODIFIED ADOMIAN DECOMPOSITION METHOD (MADM)

MADM [15-17] is a powerful analytic technique for

solving the strongly nonlinear problems. This MADM

yields, without linearization, perturbation, transformation

or discretisation, an analytical solution in terms of a rap-

idly convergent infinite power series with easily com-

putable terms. The decomposition method is simple and

easy to use and produces reliable results with few itera-

tion used. The results show that the rate of convergence

of Modified Adomian decomposition method is higher

than standard Adomian decomposition method [18-22].

Using MADM method, we can obtain the concentration

of phenol (see Appendix A & B) as follows:

! "

! "! "

! "! "

! "! "

! "

! "! "! " ! "

! "

4 3 22

3

2 22

2

22 42 3

2 3

44

3

18 40 30 71 21

6 1 360 1

9 20 101

3 1 60 1

1 21

6 1 6 1 36 1

120 1

u -

. . . .

/ /

. . .-

/ /

. .- -

/ / /

-

/

0 ) 00& 0 0

) )

1 20 )3 40 0

) 3 4)5 6

1 20) 0 03 4

) ) )3 45 6

))

(10)

provided ! " ! "2 1 2 6 1 .0 7 )/ . Using Eq.9, we can

obtain the simple approximate expression of effective-

ness factor as follows:

! "! "

2 3 2

2

9 20 15 41

60 1

. . ., .

/

0 ) 0& 0 )

) (11)

From Eq.11, we see that the effectiveness factor is a

function of the Thiele modulus , the saturation parame-

ter ! and the radius parameter . This Eq.11 is valid only

when

! "! "

2 3 2

2

9 20 15 41

60 1

. . ..

/

0 ) 00 7

).

4. NUMERICAL SIMULATION

The non-linear equation [Eq.1] for the boundary con-

ditions [Eqs.7 and 8] are solved by numerically. The

function pdex4 in Scilab/Matlab software is used to solve

the initial-boundary value problems for parabolic-elliptic

partial differential equations numerically. The Scilab/

Matlab program is also given in Appendix C. Its nu-

merical solution is compared with the analytical results


obtained using MADM method.

5. RESULTS AND DISCUSSION

Eq.10 is the new, simple and approximate analytical

expression of the concentration of phenol. Concentration

of phenol depends upon the following three parameters

. , ! and . Figures 2(a)-(d) represent a series of nor-

malized phenol concentration for the different values of

the Thiele modulus. In this Figure 2, the concentration of

phenol decreases with the increasing values of the Thiele

modulus . Moreover, the phenol concentration tends to

one as the Thiele modulus ! 0.1. Upon careful evalua-

tion of these figures, it is evident that there is a simulta-

neous increase in the values of concentration of phenol u

when decreases. Furthermore, the phenol concentration

increases slowly and rises suddenly when the normalized

radial distance 0.3- 8 . Figure 3 represents the effect-

tiveness factor " versus normalized Thiele modulus for

different values of normalized saturation parameter !.

From this figure, it is inferred that, a constant value of

normalized saturation parameter !, the effectiveness fac-

tor decreases quite rapidly as the Thiele modulus in-

creases. Moreover, it is also well known that, a constant

value of normalized Thiele modulus , the effectiveness

factor increases with increasing values of !.

The normalized effectiveness factor " versus normal-

ized saturation parameter ! is plotted in Figure 4. The

effectiveness factor " is equal to one (steady state value)

(a) (b)

(c) (d)

Figure 2. Plot of normalized phenol concentration u as a function of - in fluidized bed biofilm reactor. The concen-

tration were computed for various values of the Thiele modulus and the radius parameter = 0.01 using Eq.10 when

the normalized saturation parameter (a) ! = 0.1; (b) ! = 1; (c) ! = 10; and (d) ! = 100.



! = 100

! = 20

! = 15

! = 10

! = 7

! = 5

0 2 4 6 8 10 12 14 16 18 20

Normalized Thiele modulus

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

No

rmal

ized

eff

ecti

ven

ess

fact

or "

Figure 3. Plot of the normalized effectiveness factor " versus the Thiele

modulus . The effectiveness factor " were computed using Eq.11 for various

values of the normalized saturation parameter ! when the normalized radius

parameter = 0.01.

0 1 2 3 4 5 6 7 8

Normalized saturation parameter #

1

0.95

0.9

0.85

0.8

0.75

0.7

No

rmal

ized

eff

ecti

ven

ess

fact

or "

= 2

= 1

= 0.01, 0.1

= 0.5

Figure 4. Plot of the normalized effectiveness factor " versus normalized

saturation parameter !. The effectiveness factor " were computed using Eq.11

for different values of the Thiele modulus when the normalized radius

parameter = 0.01.

when 58/ and all values of . Also the effectiveness

factor , is uniform when 0.5 7 and for all values of

!. From this figure, it is concluded that the effectiveness

factor decreases when increases at x = 0. A three di-

mensional effectiveness factor " computed using Eq.11

for 100/ & as shown in Figure 5. In this Figure 5, we

notice that the effectiveness factor tends to one as the

Thiele modulus decreases.

6. CONCLUSIONS

We have developed a comprehensive analytical for-

malism to understand and predict the behavior of fluid-

ized bed biofilm reactor. We have presented analytical

expression corresponding to the concentration of phenol

in terms of , , , and- . / using the modified Adomian

decomposition method. The approximate solution is used



Figure 5. Plot of the three-dimensional effectiveness factor " against and , calculated using Eq.11

for ! = 100.

to estimate the effectiveness factor of this kind of sys-

tems. The analytical results will be useful for the deter-

mination of the biofilm density in this differential fluid-

ized bed biofilm reactor. The theoretical results obtained

can be used for the optimization of the performance of

the differential fluidized bed biofilm reactor. Also the

theoretical model described here can be used to obtain

the parameters required to improve the design of the dif-

ferential fluidized bed biofilm reactor.

7. ACKNOWLEDGEMENTS

This work was supported by the Council of Scientific and Industrial

Research (CSIR No.: 01(2442)/10/EMR-II), Government of India. The

authors also thank the Secretary, The Madura College Board, and the

Principal, The Madura College, Madurai, Tamilnadu, India for their

constant encouragement.

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APPENDIX A

Basic Concept of the Modified Adomian Decompo-

sition Method (MADM)

Consider the nonlinear differential equation in the

form

! " ! " ! "2

12, ;

n nny y y F x y g x n

x x

099 9) ) ) & 8 0 (A1)

with initial condition

! " ! "0 , 0y A y B9& &

"

(A2)

where ! ,F x y is a real function, ! "g x is the given

function and A and B are constants. We propose the new

differential operator, as below

2

2

d

d

n nL x x yx

0& (A3)

So, the problem (A1) can be written as,

! " ! " ! ",L y g x F x y& 0 (A4)

The inverse operator is therefore considered a

two-fold integral operator, as below.

1L0

! " ! "1

0 0

d dx x

n nL x x x0 0: & :; ; x (A5)

Applying of (A4) to the first three terms 1L0

! "2

12 n nny y y

x x

099 9) )

of Eq.A1, we find

! "

! "

! " ! "

1

2

2

0 0

1

0

12

12d d

d 0

x xn n

xn n n

n nnL y y y

x x

n nnx x y y y x x

x x

x x y nx y x y y

0

0

0 0

0$ %99 9) )' (

* +

0$ %99 9& ) )' (

* +

9& ) & 0

; ;

;

By operating 1L0 on (A4), we have

! " ! " !1 1 ,y x A L g x L F x y0 0& ) 0 " (A6)

The Adomian decomposition method introduce the

solution ! "y x and the nonlinear function ! ",F x y by

infinity series

! " ! "0

,nn

y x y x<

&

&= (A7)

and

! "0

, nn

F x y A<

&

&= (A8)

where the components ! "ny x of the solution ! "y x

will be determined recurrently and the Adomian poly-

nomials An of ! ",F x y are evaluated [23-25] using the

formula

! " ! " 00

1 d

! d

nn

n nnn

A x N yn

,,,

<

&&

$ %& ' (

* += (A9)

By substituting (A7) and (A8) into (A6),



! " ! "1 1

0n n

n

y x A L g x L A< <

0 0

&

& ) 0= =0n&

(A10)

Through using Adomian decomposition method, the

components ! "ny x can be determined as

! " ! "! " ! "

1

0

1

1 , 0n n

y x A L g x

y x L A n

0

0)

& )

& 0 8 (A11)

which gives

! " ! "! " ! "! " ! "! " ! "

1

0

1

1 0

1

2

1

3 2

...

y x A L g x

y x L A

y x L A

y x L A

0

0

0

0

& )

& 0

& 0

& 0

1 (A12)

From (A9) and (A10), we can determine the compo-

nents ! "ny x , and hence the series solution of ! "y x in

(A7) can be immediately obtained.

APPENDIX B

Analytical Expression of Concentration of Phenol

Using the Modified Adomian Decomposition Method

In this appendix, we derive the general solution of

nonlinear Eq.7 by using Adomian decomposition

method. We write the Eq.7 in the operator form,

! "2

1

uL u

u

/

&)

(B.1)

where 2

1

2

d

dL - -

-0& .

Applying the inverse operator on both sides of

Eq.B.1 yields

1L0

! "2

1

1 2

1

uu c c L

u

- -

/0 $ %

& ) ) ')* +

( (B.2)

where A and B are the constants of integration. We let,

! " ! "0

nn

u u- -<

&

&= (B.3)

! "0

nn

N u A-<

&

&1 25 6 = (B.4)

where

! " ! "! "1

uN u

u

--

/ -

$ %&1 2 '5 6 ' )* +

((

A

(B.5)

Now Eq.B.2 becomes

! " 1

1 20 0

n nn n

u c c L- -< <

0

& &

& ) )= = (B.6)

We identify the zeroth component as

! "0 1u c- -& ) 2c (B.7)

and the remaining components as the recurrence relation

! " 1

1 0 n nu L A n- 0) & 8 (B.8)

We can find An as follows:

! " 00

1 d

! d

nn

n nnn

A N un

,,,

<

&&

$ %& ' (

* += (B.9)

The initial approximations (boundary conditions

Eqs.7 and 8 are as follows

! "0 1 1u - & & (B.10)

! "0d du - . - 0& & (B.11)

and

! "i 1 0; 1, 2,3,u i- & & & (B.12)

! "id d 0; 1,2,u i- . -& & & 3, (B.13)

Solving the Eq.B.7 and using the boundary conditions

Eqs.B.10 and B.11, we get

0 1u & (B.14)

Now substituting n = 0 in Eqs.B.8 and B.9, we can

obtain

! " 1

1u L- 0& 0A (B.15)

and ! "2

0 01

A N u /

& &)

(B.16)

By operating 1L0 on (B.16),

2 21 1 1

0 0

d d1 1

L

- - - -

/ /0 0$ % $ %

&' ( ' () )* + * +

; ; - - (B.17)

Now Eq.B.15 becomes

! "! "

22

16 1

u

- -/

a b-& ) ))

(B.18)

Solving the Eq.B.18 and using the boundary condi-

tions Eqs.B.12 and B.13, we get

! " ! "! " ! " ! "

2 2 22

1

2 1

6 1 3 1 6 1u

. . - -

/ / /

0& 0 )

) ) )- (B.19)

Similarly we can get

! "

! "! "

! "! "

! "! " ! " ! "

2

4 2 3 4

3 3

4 4 42 3

3 3 3

7 30 40 18 10 20 9

360 1 180 1

1 2

36 1 36 1 120 1

u -

. . . . . . 2

4

-/ /

. . - -

/ / /

0 ) 0 0 )& )

) )

00 0 )

) ) )-

(B.20)

S. Usha et al. / Natural Science 4 (2012) 983-991

Copyright © 2012 SciRes. http://www.scirp.org/journal/ns/Openly accessible at

991

Adding Eqs.B.14, B.19 and B.20, we get Eq.11 in the

text.

APPENDIX C

Scilab/Matlab program to find the numerical solu-

tion of Eq.8 is as follows

function pdex1

m = 2;

x = linspace(0.01,1);

t = linspace(0,1000);

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

u = sol(:,:,1);

figure

plot(x,u(end,:))

title(‘u(x,t)’)

%

--------------------------------------------------------------

function [c,f,s] = pdex4pde(x,t,u,DuDx)

c = 1;

f = DuDx;

phi=24.5;

alpha=100;

s =-(phi^2*u)/(1+alpha*u);

%

--------------------------------------------------------------

function u0 = pdex1ic(x)

u0 = 1;

% --------------------------------------------------------------

function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t)

pl = 0;

ql = 1;

pr = ur-1;

qr = 0;

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