Solution of non-linear boundary value problems in immobilized glucoamylase kinetics

Affiliation(s)

Department of Mathematics, Ganesar College of Arts and Science, Melaisivapuri, India.

Department of Mathematics, K. L. N. College of Engineering, Pottapalayam, India.

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

Department of Mathematics, Ganesar College of Arts and Science, Melaisivapuri, India.

Department of Mathematics, K. L. N. College of Engineering, Pottapalayam, India.

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

ABSTRACT

A mathematical model to describe the enzyme reaction, mass transfer and heat effects in the calorimetric system is discussed. The model is based on non-stationary diffusion Equation containing a nonlinear term related to immobilize liver esterase by flow calorimetry. This paper presents the complex numerical methods (Adomian decomposition method, Homotopy analysis and perturbation method) to solve the non-linear differential Equations that depict the diffusion coupled with a non-linear reaction terms. Approximate analytical expressions for substrate concentration have been derived for all values of parameters α, β and γE. These analytical results are compared with the available numerical results and are found to be in good agreement.

Cite this paper

Sevukaperumal, S. , Eswari, A. and Rajendran, L. (2013) Solution of non-linear boundary value problems in immobilized glucoamylase kinetics.*Natural Science*, **5**, 478-494. doi: 10.4236/ns.2013.54061.

Sevukaperumal, S. , Eswari, A. and Rajendran, L. (2013) Solution of non-linear boundary value problems in immobilized glucoamylase kinetics.

References

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[2] Burch, J. (1954) The purification and properties of horse liver esterase. Biochemistry, 58, 415-426.

[3] Stoops, J.K., Horgan, D.J., Runnegar, M.T.C., Jersey, J., dewebb, E.C. and Zerner, B. (1969) Carboxylesterases (EC 3.1.1). Kinetic Studies on Carboxylesterases Biochemistry, 8, 2026-33.

[4] Adler, A.J. and Kistiakowasky, G.B. (1962) Kinetics of pig liver esterase catalysis. Journal of the American Chemical Society, 84, 695-703. doi:10.1021/ja00864a001

[5] Stefuca, V., Gemenier, P. and Scheper, V., Eds. (1999) Advances in biochemical engineering biotechnology. Springer, Berlin, 71.

[6] Stefuca, V., Vikartovska-Welwardova, V. and Gemenier, P. (1999) Flow microcalorimeter auto-calibration for the ana lysis of immobilized enzyme kinetics. Analytica Chimica Acta, 355, 63. doi:10.1016/S0003-2670(97)81612-1

[7] Stefuca, V., Cipakova, I. and Gemeine, P. (2001) Investigation of immobilized glucoamy lase kinetics by flow calorimetry. Thermochimica Acta, 378, 79-85. doi:10.1016/S0040-6031(01)00589-5

[8] Malik, F., Stefuca, V. and Bales, V. (2004) Investigation of kinetics of immobilized liver esterase by flow calorimetry. Journal of Molecular Catalysis B: Enzymatic, 29, 81-87. doi:10.1016/S0040-6031(01)00589-5

[9] Jaradat, O.K. (2008) Adomian decomposition method for solving abelian differential equations. Journal of Applied Sciences, 8, 1962-1966. doi:10.1016/S0040-6031(01)00589-5

[10] Majid Wazwaz, A. and Gorguis, A. (2004) The decomposition method applied to systems of partial differential equations. Applied Mathematics and Computation, 149, 807-814.

[11] Makinde, O.D. (2007) Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Applied Mathematics and Computation, 184, 842-848. doi:10.1016/j.amc.2006.06.074

[12] Siddiqui, A.M., Hameed, M., Siddiqui, B.M. and Ghori, Q.K. (2010) Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid. Communications in Nonlinear Science and Numerical Simulation, 15, 2388-2399. doi:10.1016/j.cnsns.2009.05.073

[13] Mohamed, M.A. (2010) Comparison differential trans formation technique with Adomian decomposition method for dispersive long-wave equations in (2+1)-dimensions. Applied Mathematics, 5, 148-166.

[14] Liao, S.J. (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. The sis, Shanghai Jiao Tong University, Shanghai.

[15] Liao, S.J. (2003) Beyond perturbation: Introduction to the homotopy analysis method. CRC Press, Boca Raton. doi:10.1201/9780203491164

[16] Awaadeh, F., Jaradat, H.M. and Alsyyed, O. (2009) Analytical solution for nonlinear gas dynamic equation by homotopy analysis method. Chaos, Solitons & Fractals, 42, 1422-1427.

[17] Jafari, H., Chun, C. and Saeidy, S. (2009) Analytical solution for nonlinear gas dynamic equation by homotopy analysis method. Applied Mathematics, 4, 149-154.

[18] Sohouli, A.R., Famouri, M., Kimiaeifar, A. and Domairry, G. (2010) Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux. Communications in Nonlinear Science and Numerical Simulation, 15, 1691-1999. doi:10.1016/j.cnsns.2009.07.015

[19] Domarriy, G. and Fazeli, M. (2009) Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity. Communications in Nonlinear Science and Numerical Simulation, 14, 489-499. doi:10.1016/j.cnsns.2007.09.007

[20] Liao, S.J. (2004) On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147, 499-513. doi:10.1016/S0096-3003(02)00790-7

[21] Domairry, G. and Barring, H. (2008) An approximation of the analytic solution of some nonlinear heat transfer equations: A survey by using homotopy analysis method. Ad vanced Studies in Theoretical Physics, 2, 507-518.

[22] He, J.H. (2000) A coupling method of homotopy tech nique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35, 37-43. doi:10.1016/S0020-7462(98)00085-7

[23] Ganji, D.D., Amini, M. and Kolahdooz, A. (2008) Ana lytical investigation of hyperbolic equations via He’s methods. American Journal of Engineering and Applied Sciences, 1, 399-407. doi:10.3844/ajeassp.2008.399.407

[24] Ariel, P.D. (2010) Homotopy perumbation method and natural convention flow of a third grade fluid through a circular tubenon. Nonlinear Science Letters A, 1, 43-52.

[25] Fereidoon, A., Rostamiyan, Y., Davoudabadi, M.R., Farahani, S.D. and Ganji, D.D. (2010) Analytic approach to investigation of distributions of stresses and radial displacement at the thick-wall cylinder under the internal and external pressures. Middle-East Journal of Scientific Research, 5, 321-328.

[1] Monk, P. and Wadso, I. (1968) A flow micro reaction. Acta Chemica Scandinavica, 22, 1842-1852. doi:10.3891/acta.chem.scand.22-1842

[2] Burch, J. (1954) The purification and properties of horse liver esterase. Biochemistry, 58, 415-426.

[3] Stoops, J.K., Horgan, D.J., Runnegar, M.T.C., Jersey, J., dewebb, E.C. and Zerner, B. (1969) Carboxylesterases (EC 3.1.1). Kinetic Studies on Carboxylesterases Biochemistry, 8, 2026-33.

[4] Adler, A.J. and Kistiakowasky, G.B. (1962) Kinetics of pig liver esterase catalysis. Journal of the American Chemical Society, 84, 695-703. doi:10.1021/ja00864a001

[5] Stefuca, V., Gemenier, P. and Scheper, V., Eds. (1999) Advances in biochemical engineering biotechnology. Springer, Berlin, 71.

[6] Stefuca, V., Vikartovska-Welwardova, V. and Gemenier, P. (1999) Flow microcalorimeter auto-calibration for the ana lysis of immobilized enzyme kinetics. Analytica Chimica Acta, 355, 63. doi:10.1016/S0003-2670(97)81612-1

[7] Stefuca, V., Cipakova, I. and Gemeine, P. (2001) Investigation of immobilized glucoamy lase kinetics by flow calorimetry. Thermochimica Acta, 378, 79-85. doi:10.1016/S0040-6031(01)00589-5

[8] Malik, F., Stefuca, V. and Bales, V. (2004) Investigation of kinetics of immobilized liver esterase by flow calorimetry. Journal of Molecular Catalysis B: Enzymatic, 29, 81-87. doi:10.1016/S0040-6031(01)00589-5

[9] Jaradat, O.K. (2008) Adomian decomposition method for solving abelian differential equations. Journal of Applied Sciences, 8, 1962-1966. doi:10.1016/S0040-6031(01)00589-5

[10] Majid Wazwaz, A. and Gorguis, A. (2004) The decomposition method applied to systems of partial differential equations. Applied Mathematics and Computation, 149, 807-814.

[11] Makinde, O.D. (2007) Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Applied Mathematics and Computation, 184, 842-848. doi:10.1016/j.amc.2006.06.074

[12] Siddiqui, A.M., Hameed, M., Siddiqui, B.M. and Ghori, Q.K. (2010) Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid. Communications in Nonlinear Science and Numerical Simulation, 15, 2388-2399. doi:10.1016/j.cnsns.2009.05.073

[13] Mohamed, M.A. (2010) Comparison differential trans formation technique with Adomian decomposition method for dispersive long-wave equations in (2+1)-dimensions. Applied Mathematics, 5, 148-166.

[14] Liao, S.J. (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. The sis, Shanghai Jiao Tong University, Shanghai.

[15] Liao, S.J. (2003) Beyond perturbation: Introduction to the homotopy analysis method. CRC Press, Boca Raton. doi:10.1201/9780203491164

[16] Awaadeh, F., Jaradat, H.M. and Alsyyed, O. (2009) Analytical solution for nonlinear gas dynamic equation by homotopy analysis method. Chaos, Solitons & Fractals, 42, 1422-1427.

[17] Jafari, H., Chun, C. and Saeidy, S. (2009) Analytical solution for nonlinear gas dynamic equation by homotopy analysis method. Applied Mathematics, 4, 149-154.

[18] Sohouli, A.R., Famouri, M., Kimiaeifar, A. and Domairry, G. (2010) Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux. Communications in Nonlinear Science and Numerical Simulation, 15, 1691-1999. doi:10.1016/j.cnsns.2009.07.015

[19] Domarriy, G. and Fazeli, M. (2009) Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity. Communications in Nonlinear Science and Numerical Simulation, 14, 489-499. doi:10.1016/j.cnsns.2007.09.007

[20] Liao, S.J. (2004) On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147, 499-513. doi:10.1016/S0096-3003(02)00790-7

[21] Domairry, G. and Barring, H. (2008) An approximation of the analytic solution of some nonlinear heat transfer equations: A survey by using homotopy analysis method. Ad vanced Studies in Theoretical Physics, 2, 507-518.

[22] He, J.H. (2000) A coupling method of homotopy tech nique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35, 37-43. doi:10.1016/S0020-7462(98)00085-7

[23] Ganji, D.D., Amini, M. and Kolahdooz, A. (2008) Ana lytical investigation of hyperbolic equations via He’s methods. American Journal of Engineering and Applied Sciences, 1, 399-407. doi:10.3844/ajeassp.2008.399.407

[24] Ariel, P.D. (2010) Homotopy perumbation method and natural convention flow of a third grade fluid through a circular tubenon. Nonlinear Science Letters A, 1, 43-52.

[25] Fereidoon, A., Rostamiyan, Y., Davoudabadi, M.R., Farahani, S.D. and Ganji, D.D. (2010) Analytic approach to investigation of distributions of stresses and radial displacement at the thick-wall cylinder under the internal and external pressures. Middle-East Journal of Scientific Research, 5, 321-328.