Two reliable computational methods pertaining to steady state substrate concentration of an immobilized enzyme system

Alexandria Engineering Journal (2017) xxx, xxx–xxx

H O S T E D BY

Alexandria University

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ORIGINAL ARTICLE

Two reliable computational methods pertaining to steady state substrate concentration of an immobilized enzyme system M. Salai Mathi Selvi a, G. Hariharan a,*, K. Kannan a, M.H. Heydari b a b

Department of Mathematics, School of Humanities and Sciences, SASTRA University, Thanjavur 613 401, Tamil Nadu, India Department of Mathematics, Faculty of Mathematics, Yazd University, Yazd, Islamic Republic of Iran

Received 22 October 2016; revised 23 August 2017; accepted 23 September 2017

KEYWORDS Mathematical modeling; Nonlinear differential equations; Legendre wavelet method

Abstract A mathematical model of an immobilized enzyme system with Michaelis-Menten mechanism for an irreversible reaction is discussed. The model is developed on the basis of diffusion equations containing a nonlinear term related to Michaelis-Menten (M-M) kinetics. In this paper, Legendre wavelets operational matrix of derivatives are used to solve the nonlinear reactiondiffusion equations. The concentration profile of substrate is computed for various parameter values. Also the dimensionless concentration of substrate is established for the slab, cylinder and spherical pallets in different cases. An approximate/analytical expression for substrate concentration is obtained as a function of the Thiele modulus and the Michaelis constant. The proposed wavelet solutions are compared with Adomian decomposition method (ADM) and numerical solutions. Satisfactory agreement with ADM and numerical is observed for all Thiele modulus and M-M constants. Power of the proposed methods is confirmed. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction In recent years, a great variety of phenomena in physics, chemistry and biology can be described by nonlinear reactiondiffusion equations (RDEs) and particularly by enzymekinetics related problems. Enzyme are often used in an immobilized form in industry because immobilized enzyme processes may be performed continuously and offer the possibility of utilizing the enzymes. Therefore a description of biosensors * Corresponding author. E-mail address: [email protected] (G. Hariharan). Peer review under responsibility of Faculty of Engineering, Alexandria University.

action is divided into the simplest cases for which analytical solutions still exist. Farhad et al. [8] used the intrinsic kinetic parameters for both reversible and irreversible unireactant immobilized enzyme systems that follow the MichaelisMenten mechanism. Kirthiga and Rajendran [10] established the homotopy analysis method (HAM) for solving nonlinear reaction-diffusion equations in enzyme kinetics. Many scientific problems in chemistry and experimental biology involve linear and nonlinear reaction-diffusion equations [13,14]. Recently, Rasi et al. [4] used the homotopy perturbation method (HPM) for investigating the current potential of redomax enzymatic homogenious system. Miyakawaa et al. [15] introduced the orthogonal collocation method for immobilized enzyme reactions. Chuan Zhang et al. [11]

https://doi.org/10.1016/j.aej.2017.09.012 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M. Salai Mathi Selvi et al., Two reliable computational methods pertaining to steady state substrate concentration of an immobilized enzyme system, Alexandria Eng. J. (2017), https://doi.org/10.1016/j.aej.2017.09.012

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M. Salai Mathi Selvi et al.

Nomenclature a Bi C De Ef gS K1 Km Kmf v0 Vm Vmf R S

volume of the fluid phase in the reactor (none) Biot number (none) dimensionless substrate concentration for irreversible reaction in the pellet (none) effective diffusivity of the substrate in the pellet (cm2 =min) effective factor (none) Pellet shape factor (none) external mass-transfer co-efficient (cm/s) irreversible reaction Michaelis constant (kg=m3 ) Michaelis constant of the forward reaction (M) initial reaction rates (none) irreversible maximum reaction rate (kg=s=m3 cat) maximum velocity of the forward reaction (mol/ min/l cat) half-thickness of the pellet (m) irreversible substrate concentration inside the pellet (lmol=cm3 )

described the effects of mass transfer and light intensity on performance of substrate degradation within an annular fiberillumination bioreactor using cell-immobilized PSB. Mireshghi et al. [16] developed a simple optimization algorithm to estimate the mass transfer parameters using initial rate data for irreversible unireactant immobilized enzymes. Chen and Chang [9] have analyzed the operational stability of immobilized D-glucose isomerase in tank reactor. Javidi [1] applied the spectral collocation method for the numerical solution of the Burgers-Huxley equations. Lepik [3,6] used the Haar wavelet method (HWM) for the solution of higher order differential equations. Yuanlu [5] applied the Chebyshev wavelet method (CWM) for solving the nonlinear fractional differential equations. Hariharan [2] established the Haar wavelet method (HWM) for a class of fractional order Klein-Gordon (KG) equations. Recently, Salai Mathiselvi and Hariharan [12] applied the wavelet based approximation algorithms for solving steadystate concentration of packed-bed reactor model. Doha et al. [7] derived the second kind Chebyshev operational matrix of derivatives for solving the Lane-Emden type singular boundary value problems(SBVPs). Recently, Gupta and Saha Ray [14] have utilized the Chebyshev wavelet method together with operational matrices for the time-fractional fifth-order Sawada-Kotera (SK) equations. In this paper, two reliable computational methods are established to find the substrate concentration in Michaelis–Menten enzyme kinetics model. Mohammadi and Hosseini [17] derived the Legendre operational matrices of derivatives for solving nonlinear differential equations. Recently, learned researchers Syed Tauseef Mohyud-Din and his coworkers [18–24] have introduced a few numerical techniques for solving MHD flow of nanofluid model problems. This paper is organized as follows: In Section 2, mathematical formulation of the problem is discussed. In Section 3, some properties of Legendre wavelet method (LWM) are presented. In Section 4, irreversible reaction models and method of

Seq Sb Sb0 t X x

equilibrium substrate concentration (lmol=cm3 ) irreversible substrate concentration in the bulk fluid phase (lmol=cm3 ) irreversible initial substrate concentration in the bulk fluid phase (lmol=cm3 ) time (none) dimensionless distance (none) distance to the center (none)

Greek symbols bb dimensionless parameter in irreversible reaction for bulk fluid phase (none) bb0 dimensionless parameter for initial fluid phase (none) / intrinsic modified Thiele modulus in irreversible reaction (none)

solution by the Legendre wavelet method (LWM) are presented. Results and discussions are given in Section 5. Concluding remarks are given in Section 6. 2. Mathematical model for irreversible reaction A differential mass balance equation for the substrate for irreversible reaction in dimensionless form can be represented as follows [15]: d2 C g  1 dC C ¼ /2 þ 2 X dX 1 þ bb C dX

ð1Þ

The boundary conditions are given by X ¼ 0; X ¼ 1;

dC ¼0 dX C¼1

ð2Þ ð3Þ

(without external mass transfer resistance) X ¼ 1;

dC ¼ Bi ð1  CÞ dX

ð4Þ

(with external mass transfer resistance) where C represents the dimensionless substrate concentration, X represents the dimensionless distance to the center or the surface of symmetry of the pellet and g is the pellet shape factor for slab, cylindrical and spherical respectively. The dimensionless parameters are defined as follows: qffiffiffiffiffiffiffiffiffi S x Sb C ¼ ; X ¼ ; bb ¼ ; / ¼ RKVmmDe ðThiele modulusÞ; Km Sb R K1 ðBiot numberÞ ð5Þ Bi ¼ R De In the above expressions, the parameters x; R; K1 ; De ; S and Sb represent the distance to the center, the half-thickness of the pellet, the external mass transfer coefficient, the effective diffusivity of the substrate in the pellet, the substrate concentration inside the pellet and substrate concentration in the bulk fluid

Please cite this article in press as: M. Salai Mathi Selvi et al., Two reliable computational methods pertaining to steady state substrate concentration of an immobilized enzyme system, Alexandria Eng. J. (2017), https://doi.org/10.1016/j.aej.2017.09.012

Two reliable computational methods

3

phase respectively. Km and Vm are the kinetic parameters. The effectiveness factor Ef is given by Z 1 C Xg1 dX ð6Þ Ef ¼ gð1 þ bb Þ 0 1 þ bb C Vm Sb0 Km þ Sb0

ð12Þ 1 1 where nanm ¼ n5 ð2m3Þ 4 ; bn ¼ n3 and

The initial substrate reaction rate v0 is given by v0 ¼ Ef

0 112 1 X 1 1 M1 1 2X 2 X 2 X X M 3M 3M M 1 2 2 1 rm^ < @ 2k þ anm þ anm bn A l 2 n¼1 m¼M 2 n¼lk þ1 m¼2 3 n¼lk þ1

ð7Þ

0 Z rm^ ¼ @

where Sb0 denotes the initial substrate concentration.

1

fðxÞ  0

lk M 1 X X

112

!2 cnm wnm ðxÞ

dxA

n¼1 m¼0

3. Some properties of Legendre wavelets: [17]

Proof. See Ref. [17] h

Wavelet consists of a family of functions formed from the dilation and translation of a single function called the mother wavelet.   1 tb wa;b ðtÞ ¼ jaj2 w ; a; b 2 R; a – 0 ð8Þ a

Theorem 3.3. Let wðtÞ be the Legendre wavelets vector defined in Eq. (10). The derivative of the vector wðtÞ can be expressed by

where a and b are dilation and translation parameters respectively. On restriction of the parameters a and b to discrete values as a ¼ ak0 ; b ¼ nb0 ak0 ; a0 > 1; b0 > 0 and n and k are positive integers, discrete wavelets are determined. One such discrete wavelet is Legendre wavelet where a0 ¼ 2 and b0 ¼ 1, then the Legendre wavelet wk;n ðtÞ ¼

where D is the 2k ðM þ 1Þ operational matrix of derivative defined as follows:

k

ja0 j2 wðak0 t  nbÞforms an orthonormal basis in L2 ðRÞ. Hence Legendre wavelet is defined as wn;m ðtÞ ¼ wðk; n~; m; tÞ [25] having four arguments; ( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kþ1   m þ 12 2 2 Lm 2kþ1 t  ð2n þ 1Þ ; for 2nk 6 t < nþ1 2k wnm ðtÞ ¼ 0; for otherwise ð9Þ where m ¼ 0; 1; . . . ; Mand n ¼ 1; 2; . . . ; 2k  1. The coefficient qffiffiffiffiffiffiffiffiffiffiffiffi m þ 12 is for orthonormality. Here Lm ðtÞ is a well known Legendre polynomials of order m. h wðtÞ ¼ w0;0 ðtÞ; w0;1 ðtÞ; . . . ; w0;M ðtÞ; . . . ; w2k 1;M ðtÞ; . . . ; iT w2k 1;1 ðtÞ; . . . ; w2k 1;M ðtÞ

ð10Þ

Theorem 3.2.1. Any function fðxÞ defined on ½0; 1Þ with bounded first and second derivatives jf0 ðxÞj 6 M1 and jf00 ðxÞj 6 M2 , can be expanded as an infinite sum of the extended Legendre wavelets, and the series converges uniformly to fðxÞ, that is: fðxÞ ¼

cnm wnm ðxÞ

2

F

6 60 6 D¼6 6 .. 6. 4 0

ð13Þ

0

...

F

...

..

..

.

.

0

3

7 07 7 7 .. 7 .7 5

... F

0

where F is ðM þ 1ÞðM þ 1Þ matrix and its ðr; sÞth element is defined as follows: 8 kþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ð2r  1Þð2s  1Þ; <2 Fr ;s ¼ r ¼ 2; . . . ; ðM þ 1Þ; s ¼ 1; . . . ; r  1 for ðr þ sÞodd; > : 0; for otherwise ð14Þ 4. Irreversible reaction model

3.1. Convergence and error analysis

1 X 1 X

dwðtÞ ¼ DwðtÞ dt

ð11Þ

n¼1 m¼0

4.1. Without external mass transfer resistance In recent years, much attention is devoted to the application of the Legendre wavelet method for various non-linear reaction– diffusion models. This method is used to find the approximate/analytical solutions in terms of series with easily computable terms. The basic concepts of the Adomian decomposition method (ADM) are given in Appendix B. 4.1.1. Case-I [Legendre wavelet Method (LWM)] We apply the Legendre wavelet method (LWM) for solving Eqs. (1)–(3) to obtain the substrate concentration with various boundary conditions (See Appendix A). g1 T C C DwðxÞ ¼ /2 x 1 þ bb C

Proof. See Ref. [17] h

CT D2 wðxÞ þ

Theorem 3.2.2. Suppose fðxÞ be a continuous function defined on ½0; 1Þ, with bounded first and second derivatives M1 and M2 Pk P respectively, and ln¼0 M1 m¼0 cnm wnm ðxÞ be the approximate solution using the extended Legendre wavelets. Then for the error bound we have:

The boundary conditions are given by

ð15Þ

X ¼ 0;

CT DwðxÞ ¼ 0

ð16Þ

X ¼ 1;

C¼1

ð17Þ

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M. Salai Mathi Selvi et al.

4.1.2. Case-II [Adomian decomposition method (ADM)]

4.2. With external mass transfer resistance

The substrate concentration without external mass transfer resistance for the initial and boundary conditions Eqs. (1)– (3) is obtained by " # 1 /2 /4 2 4 2 CðXÞ ¼ 1 þ ðX  1Þ þ ð3X  10X þ 7Þ 2g 1 þ bb 60ð1 þ bb Þ3 ð18Þ Effective factor Ef for slab,cylindrical and spherical is Ef ¼ 1 

/2

4.2.1. Case-III [Legendre wavelet Method (LWM)] We apply the LWM for the equations Eqs. (1)–(4) to obtain the substrate concentration. CT D2 wðxÞ þ

g1 T C C DwðxÞ ¼ /2 x 1 þ bb C

ð21Þ

The boundary conditions are given by ð19Þ

X ¼ 0;

CT DwðxÞ ¼ 0

ð22Þ

Using Eqs. (6) and (7) the initial substrate reaction rate v0 can be obtained as follows. ! /2 Vm Sb0 v0 ¼ 1  ð20Þ 15ð1 þ bb Þ2 Km þ Sb0

X ¼ 1;

CT DwðxÞ ¼ Bi ð1  CÞ

ð23Þ

Dimensionless substrate concentration C

1.2 LWM ADM NM

1.1

1 β =50 b

βb=10

0.9

βb=5 0.8 βb=3 0.7 β =2 b 0.6

0.5

1.4

Dimensionless substrate concentration C

15ð1 þ bb Þ2

βb=1

1.2

1 β =300 b 0.8 βb=100 0.6

βb=50

0.4

βb=35

0.2

βb=25

0

0

0.2

0.4

0.6

0.8

1

Dimensionless distance X 0

0.2

0.4

0.6

0.8

1

Dimensionless distance X

Fig. 1 Dimensionless concentration of substrate C vs dimensionless distance X in the slab pellet for various values of bb and fixed value of Theile Modulus / ¼ 1:5.

Fig. 3 Dimensionless concentration of substrate C vs dimensionless distance X in the cylindrical pellet for various values of bb and fixed value of Theile Modulus / ¼ 10.

1.1

1.2 LWM ADM NM

φ=0.1

1

Dimensionless substrate concentration C

Dimensionles substrate concentration C

LWM ADM NM

φ=1.2

0.9 0.8

φ=2

0.7 0.6

φ=2.5 0.5 0.4

φ=3 0

0.2

0.4

0.6

0.8

1

Dimensionless distance X

Fig. 2 Dimensionless concentration of substrate C vs dimensionless distance X in the slab pellet for various values of / and fixed value of bb ¼ 5.

LWM ADM NM

1.1 φ=0.1

1

φ=1

0.9 0.8

φ=1.5

0.7 φ=2

0.6 0.5

φ=2.5

0.4 φ=3

0.3 0.2

0

0.2

0.4

0.6

0.8

1

Dimensionless distance X

Fig. 4 Dimensionless concentration of substrate C vs dimensionless distance X in the cylindrical pellet for various values of / and fixed value of bb ¼ 1.

Please cite this article in press as: M. Salai Mathi Selvi et al., Two reliable computational methods pertaining to steady state substrate concentration of an immobilized enzyme system, Alexandria Eng. J. (2017), https://doi.org/10.1016/j.aej.2017.09.012

Two reliable computational methods

5 1 LWM ADM NM

1 β =100 b

Dimensionless substrate concentration C

Dimensionless substrate concentration C

1.1

0.9 0.8

βb=15

0.7

βb=10

0.6 β =7 b

0.5

β =5 b

0.4 0.3

0.2 β =2 b 0.1

0

0.2

0.4

0.6

0.8

0.9

0.8 B =15 i

B =5 i

0.7

Bi=2

0.6

Bi=1

0.5

0.4

1

LWM ADM NM

0

0.2

Dimensionless distance X

Fig. 5 Dimensionless concentration of substrate C vs dimensionless distance X in the spherical pellet for various values of bb and fixed value of Theile Modulus / ¼ 5.

0.6

0.8

1

Fig. 7 Dimensionless concentration of substrate C vs dimensionless distance X in the slab pellet for various values of Bi and value of / ¼ 1; bb ¼ 1.

1

1.2 LWM ADM NM

1.1 φ=0.1

1 0.9

Dimensionless substrate concentration C

Dimensionless substrate concentration C

0.4

Dimensionless distance X

φ=1.5

0.8 φ=2

0.7

0.6 φ=2.5 0.5 φ=3 0.4

LWM ADM NM

0.95 β =10 b

0.9 β =5 b

0.85 βb=2.5 0.8 β =1.50 b

0.75

0.3 φ=4 0.2

0.7

0

0.2

0.4

0.6

0.8

1

0

0.2

Dimensionless distance X

Fig. 6 Dimensionless concentration of substrate C vs dimensionless distance X in the spherical pellet for various values of / and fixed value of bb ¼ 1.

The substrate concentration with external mass transfer resistance for the initial and boundary conditions Eqs. (1)–(4) is obtained by    1 /2 2 X2   1 2g ð1 þ bb Þ Bi     /4 1 1 1 1 2 4 2 ðX X  1Þ  þ   1 þ 3 Bi 2 Bi ð1 þ bb Þ3 20  1 ð24Þ  5Bi

CðXÞ ¼ 1 þ

Using Eq. (6), effective factor for slab, cylindrical and spherical is

0.6

0.8

1

Fig. 8 Dimensionless concentration of substrate C vs dimensionless distance X in the slab pellet for various values of bb and value of / ¼ 1; Bi ¼ 5.

Ef ¼ 1  4.2.2. Case-IV [Adomian decomposition method (ADM)]

0.4

Dimensionless distance X

2/2 5ð1 þ bb Þ2

ð25Þ

Using Eq. (7), the initial substrate reaction rate v0 can be obtained as follows: ! 2/2 Vm Sb0 ð26Þ v0 ¼ 1  5ð1 þ bb Þ2 Km þ Sb0 5. Result and discussion Substrate concentration C vs the dimensionless radial distance X for the irreversible reactions with various parameter values is presented in Figs. 1–6. The proposed LWM results have been compared with ADM and NM results. For small values of k and M, the LWM results closer to ADM results

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M. Salai Mathi Selvi et al. 1 LWM ADM NM

0.95

Dimensionless substrate concentration C

Dimensionless substrate concentration C

1

0.9 Bi=5 0.85 Bi=2 0.8 B =1 i

0.75 0.7 Bi=0.5

0.65

LWM ADM NM

0.95 Bi=20 0.9 B =5 i

0.85

Bi=2

0.8 B =1 i

0.75

0.7

B =0.5 i

0

0.2

0.4

0.6

0.8

0.65

1

0

0.2

Dimensionless distance X

Fig. 9 Dimensionless concentration of substrate C vs dimensionless distance X in the cylindrical pellet for various values of Bi and the value of / ¼ 1; bb ¼ 2.

1

0.8

β =100

0.9 β =10 b

0.85

β =6 b

0.8 βb=4 0.75 βb=3 0

β =10 b

0.95 β =5 b

0.9

0.85 βb=2

0.8

0.75 0.2

0.4

0.6

1

LWM ADM NM

b

LWM ADM NM

Dimensionless substrate concentration C

Dimensionless substrate concentration C

0.6

Fig. 11 Dimensionless concentration of substrate C vs dimensionless distance X in the spherical pellet for various values of Bi and the value of / ¼ 1; bb ¼ 1.

0.95

0.7

0.4

Dimensionless distance X

0.8

βb=1

0

0.2

0.4

0.6

0.8

1

Dimensionless distance X

1

Dimensionless distance X

Fig. 10 Dimensionless concentration of substrate C vs dimensionless distance X in the cylindrical pellet for various values of bb and the value of / ¼ 1; Bi ¼ 0:5.

Fig. 12 Dimensionless concentration of substrate C vs dimensionless distance X in the spherical pellet for various values of bb and the value of / ¼ 1; Bi ¼ 1.

6. Conclusion (See Figs. 1–6). Fig. 7 shows that the substrate concentration C versus distance X for / and bb without external mass transfer resistance. Figs. 7–14 represent the substrate concentration C versus distance X for / and bb with external mass transfer resistance. Moreover, the Legendre operational matrices have large numbers of zero elements and they are sparse types of matrices.Thus, approximation with the Legendre wavelets has short CPU time. The substrate concentration C vs the dimensionless radial distance X for the both irreversible reactions is plotted in Figs. 1–6 for various values of the Thiele modulus / and bb for the three shapes. When dimensionless parameter bb increases the substrate concentration is also increases and the Thiele modulus for thickness of the pellet / increases, the substrate concentration inside catalyst will also decrease in all the cases.

In this research work, two reliable approximate/analytical methods have been successfully applied for investigating the substrate concentrations in an immobilized enzyme system. A mathematical model is discussed on the basis of diffusion equations containing a nonlinear term related to Michaelis-Menten (M-M) kinetics. The proposed approximate/analytical results have been validated with ADM and NM results. Satisfactory agreement with the existing results is observed. Moreover, the use of LWM and ADM is found to be simple, straight forward, reliable and they require less computation costs. Acknowledgment This work is supported by the Department of Science and Technology (DST-SERB), Government of India (Ref. No. SB/FTP/MS-012/2013). This project is partly funded under a

Please cite this article in press as: M. Salai Mathi Selvi et al., Two reliable computational methods pertaining to steady state substrate concentration of an immobilized enzyme system, Alexandria Eng. J. (2017), https://doi.org/10.1016/j.aej.2017.09.012

Two reliable computational methods

7 with the initial conditions

Dimensionless substrate concentration C

1.2 LWM ADM NM

1.1

ð28Þ

or boundary conditions y0 ð1Þ ¼ B

yð0Þ ¼ A;

1

0.9

0.8

ð29Þ

Spherical

To solve problem (27) we approximate yðxÞ; f1 ðxÞ; f2 ðxÞ and gðxÞ by the Legendre wavelets as

Cylindrical

yðxÞ ¼ CT wðxÞ f1 ðxÞ ¼ FT1 wðxÞ f2 ðxÞ ¼ FT2 wðxÞ

0.7

gðxÞ ¼ GT wðxÞ 0.6

y0 ðxÞ ¼ CT DwðxÞ

Slab 0.5

00

0

0.2

0.4

0.6

0.8

1

Dimensionless distance X

Fig. 13 Dimensionless concentration of substrate C vs dimensionless distance X the different shape factor which for slab, cylindrical, spherical pellets without mass-transfer resistance of / ¼ 5; bb ¼ 25. 0.8

Dimensionless substrate concentration C

y0 ð0Þ ¼ B

yð0Þ ¼ A;

LWM ADM NM

0.7

y ðxÞ ¼ C D wðxÞ T

2

ð30Þ ð31Þ

Employing Eqs. (30) and (31), the residual RðxÞ for Eq. (27) can be written as RðxÞ ¼ ½CT D2 wðxÞ þ FT1 wðxÞwT ðxÞDT C þ FT2 wðxÞwT ðxÞC  GT wðxÞ

ð32Þ

By using the product operation matrix of Legendre wavelets, we have T RðxÞ ¼ ½wT ðxÞðD2 Þ C þ wT ðxÞ Fb1 DT C þ wðxÞ Fb2 C

 wT ðxÞG

ð33Þ

Consider the non-linear equation

0.6 00

y ðxÞ ¼ F½x; y0 ðxÞ; yðxÞ

0.5

ð34Þ

with the initial conditions Spherical

0.4

yð0Þ ¼ A;

Cylindrical

ð35Þ

or boundary conditions

0.3

yð0Þ ¼ A; 0.2

0.1

y0 ð0Þ ¼ B

ð36Þ

In order to use shifted Legendre polynomials for this problem, we first approximate yðxÞ by the Legendre wavelets

Slab

0

y0 ð1Þ ¼ B

0.2

0.4

0.6

0.8

1

Dimensionless distance X

Fig. 14 Dimensionless concentration of substrate C vs dimensionless distance X the different shape factor which for slab, cylindrical, spherical pellets with mass-transfer resistance of / ¼ 5; bb ¼ 25.

SERB-DST Fast track young scientist to Dr.G.Hariharan. The author gratefully acknowledges the continued financial support from the Department of Science and Technology (DST), Government of India (SR/FST/MSI-107/2015). Our hearty thanks are due to Prof. R. Sethuraman, ViceChancellor, SASTRA University, Dr.S.Vaidhyasubramaniam, Dean/Planning and development and Dr.S.Swaminathan, Dean/Sponsored research for their kind encouragement and for providing good research environment. Appendix A. A.1. Applications of the operational matrix of derivative The numerical solution of non-linear singular boundary value problems arising in physiology, y00 ðxÞ þ f1 ðxÞy0 ðxÞ þ f2 ðxÞyðxÞ ¼ gðxÞ

ð27Þ

yðxÞ ¼ CT wðxÞ

ð37Þ

By using Eqs. (27) and (30) we have C D2 wðxÞ ¼ F½x; CT DwðxÞ; CT wðxÞ T

ð38Þ

Also, initial and boundary value conditions Eqs. (28) and (29) yield, yð0Þ ¼ CT wð0Þ ¼ A y0 ð0Þ ¼ CT Dwð0Þ ¼ B

ð39Þ

and yð0Þ ¼ CT wð0Þ ¼ A yð1Þ ¼ CT wð1Þ ¼ B

ð40Þ

To find the solution yðxÞ, we first calculate Eq. (38) at 2k ðM þ 1Þ  2 points. For a better result, we use the first 2k ðM þ 1Þ  2 roots of shifted Legendre P2kðMþ1Þ ðxÞ. These equations collectively with Eq. (39) or (40) generate 2k ðM þ 1Þ non-linear equations which can be solved using Newton’s iterative method. Consequently yðxÞ given in Eq. (37) can be calculated. The above-mentioned applications for linear and non-linear differential equations can be easily execute for higher order differential equations.

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8

M. Salai Mathi Selvi et al.

Appendix B. B.1. Adomian decomposition method

Through using Adomian decomposition method, the components yn ðxÞ can be determined as

Algorithm Consider the singular initial value problem in the second order ordinary differential equation in the form

y0 ðxÞ ¼ A þ L1 gðxÞ

2 y00 þ y0 þ fðx; yÞ ¼ gðxÞ x

ð41Þ

ð49Þ

which gives y0 ðxÞ ¼ A þ L1 gðxÞ y1 ðxÞ ¼ L1 ðA0 Þ

yð0Þ0 ¼ B;

yð0Þ ¼ A;

ykþ1 ¼ L1 ðAk Þ; k P 0

where fðx; yÞ is a real function, gðxÞ is given function and A and B are constants. Here, we propose the new differential operator, as below d2 L ¼ x1 2 xy dx

y2 ðxÞ ¼ L1 ðA1 Þ y3 ðxÞ ¼ L1 ðA2 Þ

From Eqs. (49) and (50) we can determain the components yn ðxÞ and hence the series solution of yðxÞ in Eq. (48) can be immediately obtained. For numerical purposes, the n-term approximate

so, the problem Eq. (41) can be written as, Ly ¼ gðxÞ  fðx; yÞ

ð42Þ

The inverse operator L1 is therefore considered a twofold integral operator, as below, ZZ x L1 ð:Þ ¼ x1 xð:Þdx dx ð43Þ

ð50Þ

Wn ¼

n1 X yk n¼0

can be used to approximate the exact solution. References

0

Applying L1 of Eq. (43) to the first two terms y00 þ x2 y0 of Eq. (41) we find    ZZ x  2 2 x y00 þ y0 dxdx L1 y00 þ y0 ¼ x1 x x 0 ¼ x1

Z

x

ðxy0 þ y  yð0ÞÞdx ¼ y  yð0Þ

0

By operating L1 on Eq. (41), we have yðxÞ ¼ A þ L1 gðxÞ  L1 fðx; yÞ

ð44Þ

The Adomian decomposition method introduce the solution yðxÞ and the nonlinear function fðx; yÞ by infinity series yðxÞ ¼

1 X yn ðxÞ

ð45Þ

n¼0

and fðx; yÞ ¼

1 X An ðxÞ

ð46Þ

n¼0

where the components yn ðxÞ of the solution yðxÞ will be determined recurrently. Specific algorithms were seen to formulate Adomian polynomials. The following algorithm: A0 ¼ Fðu0 Þ; A1 ¼ u1 F0 ðu0 Þ A2 ¼ u2 F0 ðu0 Þ þ

u21 00 F ðu0 Þ; 2!

A3 ¼ u3 F0 ðu0 Þ þ u1 u2 F00 ðu0 Þ þ

u31 000 F ðu0 Þ; 3!

ð47Þ

can be used to construct Adomian polynomials, when FðuÞ is a nonlinear function. By substituting Eqs. (41)–(45), 1 1 X X yn ðxÞ ¼ A þ L1 gðxÞ  L1 An n¼0

n¼0

ð48Þ

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Please cite this article in press as: M. Salai Mathi Selvi et al., Two reliable computational methods pertaining to steady state substrate concentration of an immobilized enzyme system, Alexandria Eng. J. (2017), https://doi.org/10.1016/j.aej.2017.09.012