Analysis of non-linear reaction-diffusion processes with Michaelis-Menten kinetics by a new Homotopy perturbation method
Affiliation(s)
Government Higher Secondary School, Madurai, India. Department of Mathematics, The Madura College, Madurai, India.
Government Higher Secondary School, Madurai, India. Department of Mathematics, The Madura College, Madurai, India.
ABSTRACT
This paper demonstrates the approximate analytical solution to a non-linear singular two-point boundary-value problem which describes oxygen diffusion in a planar cell. The model is based on diffusion equation containing a non-linear term related to Michaelis-Menten kinetics of enzymatic reaction. Approximate analytical expression of concentration of oxygen is derived using new Homotopy perturbation method for various boundary conditions. The validity of the obtained solutions is verified by the numerical results.
Cite this paper
Shanthi, D. , Ananthaswamy, V. and Rajendran, L. (2013) Analysis of non-linear reaction-diffusion processes with Michaelis-Menten kinetics by a new Homotopy perturbation method. Natural Science, 5, 1034-1046. doi: 10.4236/ns.2013.59128.
Shanthi, D. , Ananthaswamy, V. and Rajendran, L. (2013) Analysis of non-linear reaction-diffusion processes with Michaelis-Menten kinetics by a new Homotopy perturbation method. Natural Science, 5, 1034-1046. doi: 10.4236/ns.2013.59128.
References
[1] Bassom, A.P., Ilchmann, A. and Vob, H. (1997) Oxygen diffusion in tissue preparations with Michaelis-Menten kinetics. Journal of Theoretical Biology, 185, 119-127. doi:10.1006/jtbi.1996.0298 [2] Rashevsky, N. (1960) Mathematical biophysics. Dover, New York. [3] Lin, S.H. (1976) Oxygen diffusion in a spherical cell with non-linear oxygen uptake kinetics. Journal of Theoretical Biology, 60, 449-457. doi:10.1016/0022-5193(76)90071-0 [4] Michaelis, L. and Menten, M. (1913) Die kinetic der inver tinwirkung. Biochemische Zeitschrift, 49, 333-369. [5] McElwain, D.L.S. (1978) A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. Journal of Theoretical Biology, 71, 255-263.
doi:10.1016/0022-5193(78)90270-9 [6] Hiltmann, P. and Lory, P. (1983) On oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics. Bulletin of Mathematical Biology, 45, 661-664. [7] Anderson, N. and Arthurs, A.M. (1985) Analytical bound ing functions for diffusion problems with MichaelisMentenkinetics. Bulletin of Mathematical Biology, 47, 145-153. [8] Asaithambi, N.S. and Garner, J.B. (1989) Point wise solution bounds for a class of singular diffusion problems in physiology. Applied Mathematics and Computation, 30, 215-222. doi:10.1016/0096-3003(89)90053-2 [9] Asaithambi, N.S. and Garner, J.B. (1992) Taylor series solutions of a class of diffusion problems in physiology. Mathematics and Computers in Simulation, 34, 563-570.
doi:10.1016/0378-4754(92)90042-F [10] Ghori, Q.K., Ahmed, M. and Siddiqui, A.M. (2007) Application of Homotopy perturbation method to squeezing flow of a Newtonian fluid. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 179184. doi:10.1515/IJNSNS.2007.8.2.179 [11] Ozis, T. and Yildirim, A. (2007) A comparative study of He’s Homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 243-248.
doi:10.1515/IJNSNS.2007.8.2.243 [12] Li, S.J. and Liu, X.Y. (2006) An Improved approach to non-linear dynamical system identification using PID neural networks. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 177-182. doi:10.1515/IJNSNS.2006.7.2.177 [13] Mousa, M.M., Ragab, S.F. and Nturforsch, Z. (2008) Application of the Homotopy perturbation method to linear and non-linear Schrodinger equations. Zeitschrift für Naturforschung, 63, 140-144. [14] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. doi:10.1016/S0045-7825(99)00018-3 [15] He, J.H. (2003) Homotopy perturbation method: A new non-linear analytical technique. Applied Mathematics and Computation, 135, 73-79. doi:10.1016/S0096-3003(01)00312-5 [16] He, J.H. (2003) A simple perturbation approach to Blasius equation. Applied Mathematics and Computation, 140, 217-222. doi:10.1016/S0096-3003(02)00189-3 [17] Ariel, P.D. (2010) Alternative approaches to construction of Homotopy perturbation algorithms. Nonlinear Science Letters A, 1, 43-52. [18] Loghambal, S. and Rajendran, L. (2010) Mathematical modeling of diffusion and kinetics of amperometric immobilized enzyme electrodes. Electrochimica Acta, 55, 5230-5238.
doi:10.1016/j.electacta.2010.04.050 [19] Meena, A. and Rajendran, L. (2010) Mathematical modeling of amperometric and potentiometric biosensors and system of non-linear equations—Homotopy perturbation approach. Journal of Electroanalytical Chemistry, 644, 50-59. doi:10.1016/j.jelechem.2010.03.027 [20] Anitha, S., Subbiah, A., Subramaniam, S. and Rajendran, L. (2011) Analytical solution of amperometric enzymatic reactions based on Homotopy perturbation method. Electrochimica Acta, 56, 3345-3352. doi:10.1016/j.electacta.2011.01.014 [21] Ananthaswamy, V. and Rajendran, L. (2012) Analytical solution of two-point non-linear boundary value problems in a porous catalyst particles. International Journal of Mathematical Archive, 3, 810-821. [22] Ananthaswamy, V. and Rajendran, L. (2012) Analytical solutions of some two-point non-linear elliptic boundary value problems. Applied Mathematics, 3, 2012, 10441058. doi:10.4236/am.2012.39154 [23] Ananthaswamy, V. and Rajendran, L. (2012) Analytical solution of non-isothermal diffusion-reaction processes and effectiveness factors. ISRN Physical Chemistry, 2012, 1-14. [24] Rajendran, L. and Anitha, S. (2013) Comments on analytical solution of amperometric enzymatic reactions based on Homotopy perturbation method” by Ji-Huan He, LuFeng Mo. Electrochimica Acta, 102, 474-476. doi:10.1016/j.electacta.2013.03.163
[1] Bassom, A.P., Ilchmann, A. and Vob, H. (1997) Oxygen diffusion in tissue preparations with Michaelis-Menten kinetics. Journal of Theoretical Biology, 185, 119-127. doi:10.1006/jtbi.1996.0298 [2] Rashevsky, N. (1960) Mathematical biophysics. Dover, New York. [3] Lin, S.H. (1976) Oxygen diffusion in a spherical cell with non-linear oxygen uptake kinetics. Journal of Theoretical Biology, 60, 449-457. doi:10.1016/0022-5193(76)90071-0 [4] Michaelis, L. and Menten, M. (1913) Die kinetic der inver tinwirkung. Biochemische Zeitschrift, 49, 333-369. [5] McElwain, D.L.S. (1978) A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. Journal of Theoretical Biology, 71, 255-263.
doi:10.1016/0022-5193(78)90270-9 [6] Hiltmann, P. and Lory, P. (1983) On oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics. Bulletin of Mathematical Biology, 45, 661-664. [7] Anderson, N. and Arthurs, A.M. (1985) Analytical bound ing functions for diffusion problems with MichaelisMentenkinetics. Bulletin of Mathematical Biology, 47, 145-153. [8] Asaithambi, N.S. and Garner, J.B. (1989) Point wise solution bounds for a class of singular diffusion problems in physiology. Applied Mathematics and Computation, 30, 215-222. doi:10.1016/0096-3003(89)90053-2 [9] Asaithambi, N.S. and Garner, J.B. (1992) Taylor series solutions of a class of diffusion problems in physiology. Mathematics and Computers in Simulation, 34, 563-570.
doi:10.1016/0378-4754(92)90042-F [10] Ghori, Q.K., Ahmed, M. and Siddiqui, A.M. (2007) Application of Homotopy perturbation method to squeezing flow of a Newtonian fluid. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 179184. doi:10.1515/IJNSNS.2007.8.2.179 [11] Ozis, T. and Yildirim, A. (2007) A comparative study of He’s Homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 243-248.
doi:10.1515/IJNSNS.2007.8.2.243 [12] Li, S.J. and Liu, X.Y. (2006) An Improved approach to non-linear dynamical system identification using PID neural networks. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 177-182. doi:10.1515/IJNSNS.2006.7.2.177 [13] Mousa, M.M., Ragab, S.F. and Nturforsch, Z. (2008) Application of the Homotopy perturbation method to linear and non-linear Schrodinger equations. Zeitschrift für Naturforschung, 63, 140-144. [14] He, J.H. (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. doi:10.1016/S0045-7825(99)00018-3 [15] He, J.H. (2003) Homotopy perturbation method: A new non-linear analytical technique. Applied Mathematics and Computation, 135, 73-79. doi:10.1016/S0096-3003(01)00312-5 [16] He, J.H. (2003) A simple perturbation approach to Blasius equation. Applied Mathematics and Computation, 140, 217-222. doi:10.1016/S0096-3003(02)00189-3 [17] Ariel, P.D. (2010) Alternative approaches to construction of Homotopy perturbation algorithms. Nonlinear Science Letters A, 1, 43-52. [18] Loghambal, S. and Rajendran, L. (2010) Mathematical modeling of diffusion and kinetics of amperometric immobilized enzyme electrodes. Electrochimica Acta, 55, 5230-5238.
doi:10.1016/j.electacta.2010.04.050 [19] Meena, A. and Rajendran, L. (2010) Mathematical modeling of amperometric and potentiometric biosensors and system of non-linear equations—Homotopy perturbation approach. Journal of Electroanalytical Chemistry, 644, 50-59. doi:10.1016/j.jelechem.2010.03.027 [20] Anitha, S., Subbiah, A., Subramaniam, S. and Rajendran, L. (2011) Analytical solution of amperometric enzymatic reactions based on Homotopy perturbation method. Electrochimica Acta, 56, 3345-3352. doi:10.1016/j.electacta.2011.01.014 [21] Ananthaswamy, V. and Rajendran, L. (2012) Analytical solution of two-point non-linear boundary value problems in a porous catalyst particles. International Journal of Mathematical Archive, 3, 810-821. [22] Ananthaswamy, V. and Rajendran, L. (2012) Analytical solutions of some two-point non-linear elliptic boundary value problems. Applied Mathematics, 3, 2012, 10441058. doi:10.4236/am.2012.39154 [23] Ananthaswamy, V. and Rajendran, L. (2012) Analytical solution of non-isothermal diffusion-reaction processes and effectiveness factors. ISRN Physical Chemistry, 2012, 1-14. [24] Rajendran, L. and Anitha, S. (2013) Comments on analytical solution of amperometric enzymatic reactions based on Homotopy perturbation method” by Ji-Huan He, LuFeng Mo. Electrochimica Acta, 102, 474-476. doi:10.1016/j.electacta.2013.03.163