Classical Perturbation Method for the Solution of a Model of Diffusion and Reaction
Keywords:
Perturbation Method, Nonlinear Differential Equations, Porous Catalysts, Diffusion and Reaction.Abstract
In this paper, we employ perturbation method (PM) to solve nonlinear problems. As case study PM is employed to obtain approximate solutions for the nonlinear differential equation that models the diffusion and reaction in porous catalysts. We find that the square residual error (S.R.E) of our solutions is in the range and this requires only the third order approximation of PM, which shows the effectiveness of the method.
References
[1] Chow, T.L., “Classical Mechanics”, John Wiley and Sons Inc.,USA. (1995).
[2] Vasile Marinca and Nicolae Herisanu, “Nonlinear Dynamical Systems in Engineering”, first edition. Springer-Verlag Berlin Heidelberg, (2011).
[3] He, J.H., “A coupling method of a homotopy technique and a perturbation technique for nonlinear problems”, Int. J. Non-Linear Mech., 351: 37-43, (1998). DOI: 10.1016/S0020-7462(98)00085-7
[4] He, J.H., “Homotopy perturbation technique”, Comput. Methods Applied Mech. Eng., 178: 257-262, (1999). DOI: 10.1016/S0045-7825(99)00018-3
[5] Assas, L.M.B., “Approximate solutions for the generalized K-dV- Burgers’ equation by He’s variational iteration method”, Phys. Scr., 76: 161-164, (2007). DOI: 10.1088/0031-8949/76/2/008.
[6] He, J.H., “Variational approach for nonlinear oscillators”, Chaos, Solitons and Fractals, 34: 1430-1439, (2007). DOI: 10.1016/j.chaos.2006.10.026
[7] Kazemnia, M., S.A. Zahedi, M. Vaezi and N. Tolou, “Assessment of modified variational iteration method in BVPs high-order differential equations”, Journal of Applied Sciences, 8: 4192-4197, (2008). DOI:10.3923/jas.2008.4192.4197
[8] Evans, D.J. and K.R. Raslan, “The Tanh function method for solving some important nonlinear partial differential”, Int. J. Computat. Math., 82: 897-905, (2005). DOI: 10.1080/00207160412331336026
[9] Xu, F., A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method. Zeitschrift Naturforschung - Section A Journal of Physical Sciences, 62(12): 685-688, (2007).
[10] Mahmoudi, J., N. Tolou, I. Khatami, A. Barari and D.D. Ganji, “Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method”, Journal of Applied Sciences, 8: 358-363, (2008). DOI:10.3923/jas.2008.358.363
[11] Adomian, G., “A review of decomposition method in applied mathematics”, Mathematical Analysis and Applications. 135: 501-544, (1988).
[12] Babolian, E. and J. Biazar, “On the order of convergence of Adomian method”, Applied Mathematics and Computation, 130(2): 383-387, (2002). DOI: 10.1016/S0096-3003(01)00103-5.
[13] Zhang, L.-N. and L. Xu, “Determination of the limit cycle by He’s parameter expansion for oscillators in a potential”, Zeitschrift für Naturforschung - Section A Journal of Physical Sciences, 62(7-8): 396-398, (2007).
[14] Fereidon, A., Y. Rostamiyan, M. Akbarzade and D.D. Ganji, “Application of He’s homotopy perturbation method to nonlinear shock damper dynamics”, Archive of Applied Mechanics, 80(6): 641-649. DOI: 10.1007/s00419-009-0334-x, (2010).
[15] Hector Vazquez-Leal, Arturo Sarmiento-Reyes, Yasir Khan, Uriel Filobello-Nino, and Alejandro Diaz-Sanchez, “Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version”, Journal of Applied Mathematics, vol. (2012), Article ID 923975, 21 pages, (2012). doi:10.1155/2012/9239.
[16] Patel, T., M.N. Mehta and V.H. Pradhan, “The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method”, Asian Journal of Applied Sciences, 5: 60-66, (2012). DOI:10.3923/ajaps.2012.60.66
[17] Vazquez-Leal H., U. Filobello-Niño, R. Castañeda-Sheissa, L. Hernandez Martinez and A. Sarmiento-Reyes, “Modified HPMs inspired by homotopy continuation methods”, Mathematical Problems in Engineering, Vol. 2012, Article ID 309123, (2012). DOI: 10.155/2012/309123, 20 pages.
[18] Vazquez-Leal H., R. Castañeda-Sheissa, U. Filobello-Niño, A. Sarmiento-Reyes, and J. Sánchez-Orea, “High accurate simple approximation of normal distribution related integrals”, Mathematical Problems in Engineering, Vol. 2012, Article ID 124029, (2012). DOI: 10.1155/2012/124029, 22 pages.
[19] Filobello-Niño U., H. Vazquez-Leal, R. Castañeda-Sheissa, A. Yildirim, et al, “An approximate solution of Blasius equation by using HPM method”, Asian Journal of Mathematics and Statistics, Vol. 2012, 10 pages, (2012). DOI: 10.3923 /ajms.2012, ISSN 1994-5418.
[20] Filobello-Niño U., H. Vazquez-Leal, D. Pereyra Díaz, et al, “HPM Applied to Solve Nonlinear Circuits: A Study Case”, Applied Mathematics Sciences, Vol. 6, 2012, no. 85-88, 4331-4344, (2012).
[21] Filobello-Niño U, H. Vazquez-Leal, Y. Khan, et al, “Using perturbation methods and Laplace–Padé approximation to solve nonlinear problems”, Miskolc Mathematical Notes, 14 (1), 89-101, (2013),
[22] Uriel Filobello-Nino, Hector Vazquez-Leal, Juan Cervantes-Perez, et al, “A handy approximate solution for a squeezing flow between two infinite plates by using of Laplace transform homotopy perturbation method”, Springer Plus 3: 421, (2014)
[23] Filobello-Nino U., H. Vazquez-Leal, Y. Khan, et al, “Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals”, Computational and Applied Mathematics, ISSN: 0101-8205, (2013). DOI= 10.1007/s40314-013-0073-z.
[24] Yan-Ping Sun, Shi-Bin Liu, Scott Keith, “Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by the decomposition method”. Chemical Engineering Journal 102 (2004) 1-10.
[25] S. Abbasbandy. “Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method”, Chemical Engineering Journal 136 (2008) 144-150.
[26] U. Filobello-Nino, H. Vazquez-Leal, K. Boubaker, et al, “Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation”, Asian Journal of Mathematics & Statistics, 6: 76-82, (2013)
[27] Filobello-Nino U, Vazquez-Leal H, Benhammouda B, et al, “A handy approximation for a mediated bioelectrocatalysis process, related to Michaelis-Mentem equation”. Springer Plus, 3: 162. (2014)
[28] Holmes, M.H., Introduction to Perturbation Methods.Springer-Verlag, New York, (1995).
[29] U. Filobello-Nino, H. Vazquez-Leal, A. K. Shukla, A. Sarmiento-Reyes, et al. A novel study for the nonlinear model of diffusion and reaction in porous catalysts by using Laplace transform-homotopy perturbation method. Currently submitted to Revista Mexicana de Ingeniería Química.
[2] Vasile Marinca and Nicolae Herisanu, “Nonlinear Dynamical Systems in Engineering”, first edition. Springer-Verlag Berlin Heidelberg, (2011).
[3] He, J.H., “A coupling method of a homotopy technique and a perturbation technique for nonlinear problems”, Int. J. Non-Linear Mech., 351: 37-43, (1998). DOI: 10.1016/S0020-7462(98)00085-7
[4] He, J.H., “Homotopy perturbation technique”, Comput. Methods Applied Mech. Eng., 178: 257-262, (1999). DOI: 10.1016/S0045-7825(99)00018-3
[5] Assas, L.M.B., “Approximate solutions for the generalized K-dV- Burgers’ equation by He’s variational iteration method”, Phys. Scr., 76: 161-164, (2007). DOI: 10.1088/0031-8949/76/2/008.
[6] He, J.H., “Variational approach for nonlinear oscillators”, Chaos, Solitons and Fractals, 34: 1430-1439, (2007). DOI: 10.1016/j.chaos.2006.10.026
[7] Kazemnia, M., S.A. Zahedi, M. Vaezi and N. Tolou, “Assessment of modified variational iteration method in BVPs high-order differential equations”, Journal of Applied Sciences, 8: 4192-4197, (2008). DOI:10.3923/jas.2008.4192.4197
[8] Evans, D.J. and K.R. Raslan, “The Tanh function method for solving some important nonlinear partial differential”, Int. J. Computat. Math., 82: 897-905, (2005). DOI: 10.1080/00207160412331336026
[9] Xu, F., A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method. Zeitschrift Naturforschung - Section A Journal of Physical Sciences, 62(12): 685-688, (2007).
[10] Mahmoudi, J., N. Tolou, I. Khatami, A. Barari and D.D. Ganji, “Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method”, Journal of Applied Sciences, 8: 358-363, (2008). DOI:10.3923/jas.2008.358.363
[11] Adomian, G., “A review of decomposition method in applied mathematics”, Mathematical Analysis and Applications. 135: 501-544, (1988).
[12] Babolian, E. and J. Biazar, “On the order of convergence of Adomian method”, Applied Mathematics and Computation, 130(2): 383-387, (2002). DOI: 10.1016/S0096-3003(01)00103-5.
[13] Zhang, L.-N. and L. Xu, “Determination of the limit cycle by He’s parameter expansion for oscillators in a potential”, Zeitschrift für Naturforschung - Section A Journal of Physical Sciences, 62(7-8): 396-398, (2007).
[14] Fereidon, A., Y. Rostamiyan, M. Akbarzade and D.D. Ganji, “Application of He’s homotopy perturbation method to nonlinear shock damper dynamics”, Archive of Applied Mechanics, 80(6): 641-649. DOI: 10.1007/s00419-009-0334-x, (2010).
[15] Hector Vazquez-Leal, Arturo Sarmiento-Reyes, Yasir Khan, Uriel Filobello-Nino, and Alejandro Diaz-Sanchez, “Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version”, Journal of Applied Mathematics, vol. (2012), Article ID 923975, 21 pages, (2012). doi:10.1155/2012/9239.
[16] Patel, T., M.N. Mehta and V.H. Pradhan, “The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method”, Asian Journal of Applied Sciences, 5: 60-66, (2012). DOI:10.3923/ajaps.2012.60.66
[17] Vazquez-Leal H., U. Filobello-Niño, R. Castañeda-Sheissa, L. Hernandez Martinez and A. Sarmiento-Reyes, “Modified HPMs inspired by homotopy continuation methods”, Mathematical Problems in Engineering, Vol. 2012, Article ID 309123, (2012). DOI: 10.155/2012/309123, 20 pages.
[18] Vazquez-Leal H., R. Castañeda-Sheissa, U. Filobello-Niño, A. Sarmiento-Reyes, and J. Sánchez-Orea, “High accurate simple approximation of normal distribution related integrals”, Mathematical Problems in Engineering, Vol. 2012, Article ID 124029, (2012). DOI: 10.1155/2012/124029, 22 pages.
[19] Filobello-Niño U., H. Vazquez-Leal, R. Castañeda-Sheissa, A. Yildirim, et al, “An approximate solution of Blasius equation by using HPM method”, Asian Journal of Mathematics and Statistics, Vol. 2012, 10 pages, (2012). DOI: 10.3923 /ajms.2012, ISSN 1994-5418.
[20] Filobello-Niño U., H. Vazquez-Leal, D. Pereyra Díaz, et al, “HPM Applied to Solve Nonlinear Circuits: A Study Case”, Applied Mathematics Sciences, Vol. 6, 2012, no. 85-88, 4331-4344, (2012).
[21] Filobello-Niño U, H. Vazquez-Leal, Y. Khan, et al, “Using perturbation methods and Laplace–Padé approximation to solve nonlinear problems”, Miskolc Mathematical Notes, 14 (1), 89-101, (2013),
[22] Uriel Filobello-Nino, Hector Vazquez-Leal, Juan Cervantes-Perez, et al, “A handy approximate solution for a squeezing flow between two infinite plates by using of Laplace transform homotopy perturbation method”, Springer Plus 3: 421, (2014)
[23] Filobello-Nino U., H. Vazquez-Leal, Y. Khan, et al, “Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals”, Computational and Applied Mathematics, ISSN: 0101-8205, (2013). DOI= 10.1007/s40314-013-0073-z.
[24] Yan-Ping Sun, Shi-Bin Liu, Scott Keith, “Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by the decomposition method”. Chemical Engineering Journal 102 (2004) 1-10.
[25] S. Abbasbandy. “Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method”, Chemical Engineering Journal 136 (2008) 144-150.
[26] U. Filobello-Nino, H. Vazquez-Leal, K. Boubaker, et al, “Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation”, Asian Journal of Mathematics & Statistics, 6: 76-82, (2013)
[27] Filobello-Nino U, Vazquez-Leal H, Benhammouda B, et al, “A handy approximation for a mediated bioelectrocatalysis process, related to Michaelis-Mentem equation”. Springer Plus, 3: 162. (2014)
[28] Holmes, M.H., Introduction to Perturbation Methods.Springer-Verlag, New York, (1995).
[29] U. Filobello-Nino, H. Vazquez-Leal, A. K. Shukla, A. Sarmiento-Reyes, et al. A novel study for the nonlinear model of diffusion and reaction in porous catalysts by using Laplace transform-homotopy perturbation method. Currently submitted to Revista Mexicana de Ingeniería Química.
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Published
2017-01-18
How to Cite
Filobello-Nino, U., Vazquez-Leal, H., Perez-Sesma, J. A. A., Perez-Sesma, A., Sandoval-Hernandez, M., Sarmiento-Reyes, A., Huerta-Chua, J., Jimenez-Fernandez, V. M., Pereyra-Diaz, D., Hernandez-Machuca, S. F., Cuellar-Hernandez, L., Castro-Gonzalez, F., Gasca-Herrera, A. E., Pretelin Canela, J. E., Contreras-Hernandez, A. D., Alvarez-Gasca, O., Palma-Grayeb, B. E., Rocha-Fernandez, J. L., Sanchez-Orea, J., Sampieri-Gonzalez, C. E., & Gonzalez-Martinez, F. J. (2017). Classical Perturbation Method for the Solution of a Model of Diffusion and Reaction. American Scientific Research Journal for Engineering, Technology, and Sciences, 27(1), 151–160. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/1357
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