ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Numeric-Analytic Solutions of Dynamical Systems Using a New Iterative Method

Journal of Applied Nonlinear Dynamics 1(2) (2012) 141--158 | DOI:10.5890/JAND.2012.05.003

Sachin Bhalekar$^{1}$; Varsha Daftardar-Gejji$^{2}$

$^{1}$ Department of Mathematics, Shivaji University, Kolhapur - 416004, India

$^{2}$ Department of Mathematics, University of Pune, Pune - 411007, India

Abstract

In this article, we couple the New Iterative Method proposed by Daftardar-Gejji and Jafari [J. Math. Anal. Appl. 2006;316:753– 763] with classical discretization technique to integrate some linear and nonlinear systems. The numerical examples are presented to explain the method. Examples include some nonlinear dynamical systems such as Financial system, Duffing oscillator, Van der Pol oscillator and a non-autonomous system.

Acknowledgments

V. Daftardar-Gejji acknowledges the Department of Science and Technology, N. Delhi, India for the Research Grants [Project No SR/S2/HEP-024/2009].

References

1.  [1] Daftardar-Gejji, V. and Jafari, H. (2006), An iterative method for solving non linear functional equations, J. Math. Anal. Appl., 316, 753-763.
2.  [2] Bhalekar, S. and Daftardar-Gejji, V. (2008), New Iterative Method: Application to Partial Differential Equations, Appl. Math. Comput., 203, 778-783.
3.  [3] Daftardar-Gejji, V. and Bhalekar, S. (2008), Solving fractional diffusion-wave equations using the New Iterative Method, Frac. Calc. Appl. Anal., 11(2),193-202.
4.  [4] Daftardar-Gejji, V. and Bhalekar, S. (2010), Solving fractional boundary value problems with Dirichlet boundary conditions, Comput. Math. Appl., 59, 1801-1809.
5.  [5] Daftardar-Gejji, V. and Bhalekar, S. (2008), An Iterative method for solving fractional differential equations, Proc. Appl. Math. Mech., 7(1), 2050017-2050018.
6.  [6] Bhalekar, S., Daftardar-Gejji, V. (2010), Solving evolution equations using a new iterative method, Numer. Methods Partial Differential Eq., 26(4), 906-916. DOI 10.1002/num.20463.
7.  [7] Guellal, S., Grimalt, P. and Cherruault, Y. (1997), Numerical study of Lorenz's equation by the Adomian method, Computers Math. Applic., 33(3), 25-29.
8.  [8] Hashim, I. Noorani, M.S.M., Ahmad, R., Bakar, S.A., Ismail, E.S., Zakaria, A.M. (2006), Accuracy of Adomian decomposition method applied to the Lorenz system, Chaos, Solitons and Fractals, 28, 1149- 1158.
9.  [9] Ghosh, S., Roy, A. and Roy, D. (2007), An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators, Comput. Methods Appl. Mech. Engrg., 196, 1133-1153.
10.  [10] Alomari, A.K., Noorani, M.S.M. and Nazar, R. (2009), Adaptation of homotopy analysis method for numeric-analytic solution of Chen system, Commun. Nonlinear Sci. Numer. Simulat., 14(5), 2336-2346.
11.  [11] Hashim, I. and Chowdhury, M.S.H. (2008), Adaptation of homotopy-perturbation method for numericanalytic solution of system of ODEs, Physics Letters A, 372, 470-481.
12.  [12] Bhalekar, S., Daftardar-Gejji, V. (2011), Convergence of the new iterative method, Int. J. Differential Equations, 2011, Article ID 989065
13.  [13] Ma, J.H. and Chen, Y.S.(2001), Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I), Appl Math Mech (English Ed), 22, 1240-1251.
14.  [14] Ma, J.H., Chen, Y.S. (2001), Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (II), Appl Math Mech (English Ed), 22, 1375-1382.