A New Approximate Analytical Method for ODEs

Aminikhah, Hossein

doi:10.2478/jamsi-2014-0002

Abstract

In this paper, we propose a new algorithm for solving ordinary differential equations. We show the superiority of this algorithm by applying the new method for some famous ODEs. Theoretical considerations are discussed. The first He's polynomials have used to reach the exact solution of these problems. This method which has good stability and accuracy properties is useful in deal with linear and nonlinear system of ordinary differential equations.

References

NAYFEH, A. H. 2011. Introduction to perturbation techniques, Wiley-VCH.
Search in Google Scholar
RAND, R. H., ARMBRUSTER, D. AND RAND, R. 1987. Perturbation methods, bifurcation theory and computer algebra, Springer-Verlag, New York.10.1007/978-1-4612-1060-3
Search in Google Scholar
HE, J. H. 1999. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 178, 257-262.
Search in Google Scholar
GANJI, D. D. 2006. The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters A 355, 337-341.
Search in Google Scholar
ABBASBANDY, S. 2006. Numerical solutions of the integral equations: Homotopy perturbation method and Adomian's decomposition method. Applied Mathematics and Computation 173, 493-500.
Search in Google Scholar
YILDIRIM, A. AND KOCAK, H. 2009. Homotopy perturbation method for solving the space-time fractional advection-dispersion equation. Advances in Water Resources 32, 1711-1716.
Search in Google Scholar
SHAKERI, F. AND DEHGHAN, M. 2008. Solution of delay differential equations via a homotopy perturbation method. Mathematical and Computer Modelling 48, 486-498.
Search in Google Scholar
AMINIKHAH, H. 2010. An analytical approximation for solving nonlinear Blasius equation by NHPM. Numerical Methods for Partial Differential Equations 26, 1291-1299.
Search in Google Scholar
AMINIKHAH, H. AND BIAZAR, J. 2009. A new HPM for ordinary differential equations. Numerical Methods for Partial Differential Equations 26, 480-489.
Search in Google Scholar
BILDIK, N., KONURALP, A., ORAKCL, B. AND KUCUKARSLAN, S. 2006. Solution of different type of the partial differential equation by differential transform method and Adomian's decomposition method. Applied Mathematics and Computation 172, 551-567.
Search in Google Scholar
BIAZAR, J., BABOLIAN, E. AND ISLAM, R. 2004. Solution of the system of ordinary differential equations by Adomian decomposition method. Applied Mathematics and Computation 147, 713-719.
Search in Google Scholar
BABOLIAN, E., Biazar, J. AND Vahidi, A.R. 2004. Solution of a system of nonlinear equations by Adomian decomposition method. Applied Mathematics and Computation 150, 847-854.
Search in Google Scholar
PAMUK, S. 2005. Solution of the porous media equation by Adomian's decomposition method. Physics Letters A 344, 184-188.
Search in Google Scholar
LIAO, S. 2004. On the homotopy analysis method for nonlinear problems. Applied Mathematics andComputation 147, 499-513.
Search in Google Scholar
ABBASBANDY, S. 2006. The application of homotopy analysis method to nonlinear equations arising in heat transfer. Physics Letters A 360, 109-113.
Search in Google Scholar
JAFARI, H. AND SEIFI, S. 2009. Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation 14, 2006-2012.
Search in Google Scholar
ODIBAT, Z., MOMANI, S. AND ERTURK, S. 2008. Generalized differential transform method: Application to differential equations of fractional order. Applied Mathematics and Computation 197, 467-477.
Search in Google Scholar
ERTURK, S. AND MOMANI, S. 2008. Solving systems of fractional differential equations using differential transform method. Journal of Computational and Applied Mathematics 215, 142-151.
Search in Google Scholar
AYAZ, F. 2004. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation 147, 547-567.
Search in Google Scholar
GHORBANI, A. 2009. Beyond Adomian polynomials: He polynomials. Chaos, Solitons & Fractals 39, 14861492.
Search in Google Scholar
SHAWAGFEH, N.T. 1993. Nonperturbative approximate solution for Lane-Emden equation. J. Math. Phys. 34, 4364-4369.
Search in Google Scholar

A New Approximate Analytical Method for ODEs

Abstract

Paradigm

My account