Modified nonlinearities distribution Homotopy Perturbation method as a tool to find power series solutions to ordinary differential equations
DOI:
https://doi.org/10.21640/ns.v6i12.22Keywords:
differential equations, power series solutions, Homotopy perturbation method, Homotopy perturbation method with no linearities distributions, approximate methodsAbstract
In this article, modified non-linearities distribution homotopy perturbation method (MNDHPM) is used in order to find power series solutions to ordinary differential equations with initial conditions, both linear and nonlinear. We will see that the method is particularly relevant in some cases of equations with non-polynomial coefficients and inhomogeneous non-polynomial termsDownloads
References
Agida M, Kumar AS. (2010). A Boubaker Polynomials Expansion Scheme solution to random Love equation in the case of a rational kernel. El. J. Theor. Phys, 7 (2010) 319-326.
Aminikha, Hossein. (2011) Analytical Approximation to the Solution of Nonlinear Blasius Viscous Flow Equation by LTNHPM. International Scholarly Research Network ISRN Mathematical Analysis, Volume 2012, Article ID 957473, 10 pages.
Arfken, George y Hans Weber (1995). Mathematical Methods for Physicists, Fourth Edition. Academic Press, Inc.
Biazar, J.y H. Aminikha (2009a). Study of convergence of homotopy perturbation method for systems of partial differential equations. Computers and Mathematics with Applications, Vol. 58, No. 11-12, (2221-2230).
Biazar, J. y H. Ghazvini (2009b). Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Analysis: Real World Applications, Vol. 10, No 5, (2633-2640).
Chowdhury, SH. (2011). A comparison between the modified homotopy perturbation method and a decomposition method for solving nonlinear heat transfer equations, Journal of Applied Sciences, 11: 1416-1420.
Evans DJ, Raslan. KR (2005). The Tanh function method for solving some important nonlinear partial differential. Int. J. Computat. Math., 82: 897-905.
Filobello-Nino, U, H. Vazquez-Leal, R. Castaneda-Sheissa, A. Yildirim, L. Hernandez-Martinez et al (2012a). An approximate solution of Blasius equation by using HPM method. Asian J. Math. Stat., 5: 50-59.
Filobello-Nino, U, H. Vazquez-Leal, Y. Khan, A. Yildirim, D. Pereyra-Diaz et al (2012b). HPM applied to solve nonlinear circuits: A study case. Applied Math. Sci., 4331 - 4344.
Filobello-Nino, U, H. Vazquez-Leal, Y. Khan, A. Yildirim, D, VM. Jimenez-Fernandez et al (2012c). Perturbation method and Laplace-Pade approximation to solve nonlinear problems. Miskolc Mathematical Notes. (In Press).
Ganji, DD, H. Babazadeh, F. MM. Noori, Pirouz, M. Janipour (2009). An application of homotopy perturbation method for non linear blasius equation to boundary layer flow over a flat plate. Int. J. Nonlinear Sci. Vol.7, No.4: 309-404.
Ganji, DD, H. Mirgolbabaei, M. Miansari (2008). Application of homotopy perturbation method to solve linear and non-linear systems of ordinary differential equations and differential equation of order three. Journal of Applied Sciences, 8: 1256-1261.
Ghanouchi, J, H. Labiadh, K. Boubaker (2008). An attempt to solve the heat transfert equation in a model of pyrolysis spray using 4q-order m-Boubaker polynomials Intl. J. of Heat and Technology, 26: 49-53.
He, JH. (1998). A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int. J. Non-Linear Mech., 351: 37-43.
He, JH. (1999). Homotopy perturbation technique. Comput. Methods Applied Mech. Eng., 178: 257-262.
He, JH. (2000). A coupling method of a homotopy and a perturbation technique for nonlinear problems. International Journal of Nonlinear Mechanics, 35(1): 37-43.
He, JH (2006). Homotopy perturbation method for solving boundary value problems. Physics Letters A, 350(1-2): 87-88.
He, JH. (2008). Recent Development of the Homotopy Perturbation Method. Topological Methods in Nonlinear Analysis, 31.2: 205-209.
Kazemnia M, SA. Zahedi, M. Vaezi, N. Tolou (2008). Assessment of modified variational iteration method in BVPs high-order differential equations. Journal of Applied Sciences, 8: 4192-4197.
Khan, Y y Q. Wu (2011). Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Computers and Mathematics with Applications, Vol. 61, No. 8: 1963-1967.
Kooch, A y M. Abadyan (2011). Evaluating the ability of modified Adomian decomposition method to simulate the instability of freestanding carbon nanotube: comparison with conventional decomposition method. Journal of Applied Sciences, 11: 3421-3428.
Kooch, A y M. Abadyan (2012). Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method. Trends in Applied Sciences Research, 7: 57-67.
Mahmoudi, JN. Tolou, I. Khatami, A. Barari, DD. Ganji (2008). Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method. Journal of Applied Sciences, 8: 358-363.
Mirgolbabaei, H y DD Ganji (2009). Application of homotopy perturbation method to solve combined Korteweg de Vries-Modified Korteweg de Vries equation. Journal of Applied Sciences, 9: 3587-3592.
Noorzad R, AT Poor, M. Omidvar (2008). Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics. Journal of Applied Sciences, 8: 369-373.
Patel, T y MN. Mehta, VH. Pradhan (2012). 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.
Saravi, M, M. Hermann, D. Kaiser (2013). Solution of Bratu’s equation by He’s Variational Iteration method. American Journal of computational and applied mathematics, 3(1):46-48.
Sharma, PR y G Methi (2011). Applications of homotopy perturbation method to partial differential equations. Asian Journal of Mathematics & Statistics, 4: 140-150.
Tolou, N J, M. Mahmoudi, I. Ghasemi, A. Khatami, Barari, DD. Ganji (2008). On the non-linear deformation of elastic beams in an analytic solution. Asian Journal of Scientific Research, 1: 437-443.
Vanani, SK, S. Heidari, M. Avaji (2011). A low-cost numerical algorithm for the solution of nonlinear delay boundary equations. Journal of Applied Sciences, 11: 3504-3509.
Vazquez-Leal, H, U. Filobello-Nino, R. Castaneda-Sheissa, L. Hernandez-Martinez, A. Sarmiento-Reyes (2012a). Modified HPMs inspired by homotopy continuation methods. Mathematical Problems in Engineering. Vol. 2012.
Vazquez-Leal, H, R. Castaneda-Sheissa, U. Filobello-Nino, A. Sarmiento-Reyes, J. Sánchez-Orea (2012b). High accurate simple approximation of normal distribution related integrals. Mathematical Problems in Engineering, Vol. 2012.
Zill Dennis (2012). A First Course in Differential Equations with Modeling Applications, 0th Edition. Brooks /Cole Cengage Learning.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Nova Scientia
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Conditions for the freedom of publication: the journal, due to its scientific nature, must not have political or institutional undertones to groups that are foreign to the original objective of the same, or its mission, so that there is no censorship derived from the rigorous ruling process.
Due to this, the contents of the articles will be the responsibility of the authors, and once published, the considerations made to the same will be sent to the authors so that they resolve any possible controversies regarding their work.
The complete or partial reproduction of the work is authorized as long as the source is cited.
