Piecewise Analytic Method (PAM) is a New Step in the Evolution of Solving Nonlinear Differential Equation

  • Authors

    • Tamer Abassy Benha University, Egypt
    2019-05-06
    https://doi.org/10.14419/ijamr.v8i1.24984
  • Nonlinear Differential equation, Piecewise Analytic Method, Runge-Kutta Method, Padé approximants.
  • In this paper, a new method is introduced for engineers and scientists which can be used for solving highly nonlinear differential equations. The method is called Piecewise Analytic Method (PAM). PAM is used to solve problems which other methods can't solve. The paper also shows how the accuracy and error can be controlled according to the needs.

  • References

    1. [1] D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition, Oxford University Press, (2007).

      [2] A. H. Nayfeh and B. Balachandran, Applied non-linear dynamics, Wiley, New York (1995).

      [3] D. Greenspan, Numerical Solution of Ordinary Differential Equations for Classical, Relativistic and Nano Systems. WILEY-VCH Verlag GmbH Co. KGaA, Weinheim, (2008).

      [4] M.K. Jain, Numerical Solution of Differential Equations. 2nd Ed., Wiley Eastern Ltd. New Delhi, (1984).

      [5] J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons New York, (2003).

      [6] G. Adomian, Solving Frontier Problem of Physics: the Decomposition Method, MA: Kluwer Academic Publishers, Boston, (1994).

      [7] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton: Chapman and Hall/ CRC Press, ISBN 1-58488-407-X.

      [8] T. A. Abassy, Piecewise analytic method, International Journal of Applied Mathematical Research, VOL.1, NO. 1, (2012), pp. 77-107.

      [9] T. A. Abassy, Introduction to piecewise analytic method, Journal of Fractional Calculus and Applications, 3(S), (2012), pp. 1-19.

      [10] 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, VOL.2, NO.1, (2013), pp.16-39.

      [11] T. A. Abassy, Solving nonlinear 2nd order differential equations using piecewise analytic method (Pendulum Equations), DOI: 10.13140/RG.2.2.19063.88489/1.

      [12] G. A. J. Baker, The theory and application of the Padé approximant method." In advances in theoretical physics, VOL. 1 (ed. K. A. Brueckner). New York: Academic Press (1965), pp. 1-58.

      [13] G. A. J. Baker and P. Graves-Morris, Padé approximants, Cambridge University Press, New York, (1996).

      [14] G. A. J. Baker, Essentials of Padé approximants in theoretical physics, Academic Press}, New York, (1975).

      [15] Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York, (1972), pp. 880.

      [16] T. A. Abassy, Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study), DOI: 10.13140/RG.2.2.15932.90241/1.

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  • How to Cite

    Abassy, T. (2019). Piecewise Analytic Method (PAM) is a New Step in the Evolution of Solving Nonlinear Differential Equation. International Journal of Applied Mathematical Research, 8(1), 12-19. https://doi.org/10.14419/ijamr.v8i1.24984