Solving 2nd Order Nonlinear Differential Equations Using Piecewise Analytic Method (Pendulum Equations)
Keywords:Nonlinear Differential Equation, Pade Approximants, Piecewise Analytic Method, Runge-Kutta Method, Pendulum Equations.
In this paper, piecewise analytic method (PAM) is used for solving highly nonlinear 2nd Â order differential equation (pendulum equations) which is a big problem for engineers and scientists. PAM is used for showing the nonlinear dynamics of the solution with and without linearizion. The error and accuracy of the solution are controlled easily according to our needs.
- S. H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley, 1994.
- D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition, Oxford University Press, 2007.
- Greenspan, D., Numerical Solution of Ordinary Differential Equations for Classical, Relativistic and Nano Systems. WILEY-VCH Verlag GmbH Co. KGaA, Weinheim, 2008.
- T. A. Abassy, Piecewise analytic method, International Journal of Applied Mathematical Research 1 (2012), no. 1, 77-107, DOI: 10.14419/ijamr.v1i1.22.
- T. A. Abassy, Introduction to piecewise analytic method, Journal of Fractional Calculus and Applications 3(S) (2012), 1-19.
- T. A. Abassy, Piecewise Analytic Method (Solving Any Nonlinear Ordinary Differential Equation of 1st Order with Any Initial Condition), International Journal of Applied Mathematical Research 2 (2013), no.1, 16-39, DOI: 10.14419/ijamr.v2i1.427.
- T. A. Abassy, Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study), International Journal of Applied Mathematical Research, 9 (2) (2020) 41-49, DOI: 10.14419/ijamr.v9i2.31118.
- T. A. Abassy, Piecewise Analytic Method (PAM) is a New Step in the Evolution of Solving Nonlinear Differential Equations, International Journal of Applied Mathematical Research, 8 (1) (2019) 12-19, DOI: 10.14419/ijamr.v8i1. 24984.
- C.ODE.E, ODE ARCHITECT Companion.JOHN WILEY and SONS, INC, New York (1966).
- A. A. Klimenko, Y. V. Mikhlin , J. Awrejcewicz, Nonlinear normal modes in pendulum systems, Nonlinear Dynamics, October 2012, Volume 70, Issue 1, pp 797-813.
- I. Malkin, Certain Problems of the Theory of Nonlinear Vibrations. Geotechteorizdat, Moscow (1956) (in Russian).
- A.Blaquiere, Nonlinear System Analysis. Academic Press, New York (1966).
- A.H. Nayfeh, Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979).
- J. Mawhin, Global results for the forced pendulum equation. Handbook of differential equations, 533-589, Elsevier/North-Holland, Amsterdam,2004.
- L.P. Pook, Understanding Pendulums: A Brief Introduction,Springer,2011.