Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study)

  • Authors

    • Tamer Abassy Benha University, Egypt
  • Nonlinear Differential Equation, Padé Approximants, Piecewise Analytic Method, Runge-Kutta Method.
  • Even though Runge-Kutta (RK) method is the most used by scientists and engineers, it is not the most powerful method. In this paper, a comparative study between Piecewise Analytic Method (PAM) and RK methods is achieved. The result of comparative study shows that PAM is more powerful and gives results better than RK Methods. PAM can be considered as a new step in the evolution of solving nonlinear differential equations.

  • References

    1. [1] D. Greenspan, Numerical Solution of Ordinary Differential Equations: for Classical, Relativistic and Nano Systems, WILEY-VCH Verlag GmbH and Co. KGaA ISBN: 978-3-527-40610-4, 2006.

      [2] B.P. Sommeijer, An Explicit Runge-Kutta Method of Order Twenty-five, CWI Quarterly, 11(1998) p.75–82.

      [3] C. Runge, Uber die numerische Auflosung von Differentialgleichungen. Math. Ann. 46(1895) p. 167-178.

      [4] W. Kutta, Beitrag zur naherungweisen Integration totaler Differentialgleichungen Z. Math. Phys. 46(1901) p. 435-453. ¨

      [5] T. A. Abassy, Piecewise analytic method, International Journal of Applied Mathematical Research 1(2012) p. 77-107.

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

      [7] 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.

      [8] T. A. Abassy, Solving nonlinear 2nd order differential equations using piecewise analytic method (Pendulum Equations), Proceeding of the 13th International Conference on Computational Structures Technology, 2018 Sep 4-6; Barcelona, Spain. Elsevier.

      [9] 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.

      [10] K. Heun, Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhangigen Ver ¨ anderlichen. ZAMP, 45(1900) p. 23. ¨

      [11] E.J. Nystrom, Uber die numerische Integration von Differentialgleichungen. Acta Soc. Sci. Fenn., 1925. 50(13): p. 1-55. ¨

      [12] H. A. Luther, H. P. Konen, Some Fifth-Order Classical Runge-Kutta Formulas. Journal of the Society for Industrial and Applied Mathematics, 4(1965), p. 551-558

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

      [14] D. Sarafyan, Continuous approximate solution of ordinary differential equations and their systems. Comp. Math. Applic. 10(1984) p. 139.

      [15] D. Sarafyan, New algorithms for the continuous approximate solution of ordinary differential equations and the upgrading of the order of the processes. Comp. Math. Applic. 20(1990) p. 77.

      [16] J.C. Butcher, On Runge-Kutta processes of high order, J. Aust. Math. Soc. 4(1964) p. 179-194.

      [17] D. Sarafyan, 7th-order 10-stage Runge–Kutta formulas. TR 38, Math. Dept., LSU in New Orleans, 1970.

      [18] A.R. Curtis, High-order explicit Runge–Kutta formulae, their uses, and limitations. JIMA 16(1975) p. 35.

      [19] E.B. Shanks, Solution of differential equations by evaluations of functions. Math. Comp. 20(1966) p. 21.

      [20] E. Hairer, A Runge–Kutta method of order 10. J. IMA 21(1978) p. 47

      [21] E. Fehlberg, New high order Runge–Kutta formulas with an arbitrarily small truncation error. ZAMM 46(1966) p. 1.

      [22] E. Fehlberg, Classical fifth-, sixth- seventh and eighth-order Runge–Kutta formulas with step size control. NASA, TR-R-287, Marshall Space Flight Center, Huntsville, Ala., 1968.

      [23] H.A. Luther, An explicit sixth-order Runge–Kutta formula. Math. Comp. 22(1968) p. 344.

      [24] A. Huta, Une amelioration de la m ´ ethode de Runge-Kutta-Nystr ´ om pour la r ¨ esolution num ´ erique des ´ equations diff ´ erentielles du premier ordre. Acta ´ Fac. Nat. Univ. Comenian. Math. 1(1956) p. 201-224.

      [25] A. Huta, Contribution a la formule de sixi ` eme ordre dans la m ` ethode de Runge- Kutta-Nystr ´ om. Acta Fac. Nat. Univ. Comenian. Math. 2(1957) p. 21-24. ¨

      [26] J.C. Butcher, Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and Linear Methods. Wiley, New York, 1987.

      [27] J.C. Butcher, Numerical methods for ordinary differential equations in the 20th century. Journal of Computational and Applied Mathematics, 125(2000) p. 1-29.

      [28] J.C. Butcher, Numerical Methods for Ordinary Differential Equations. New York: John Wiley and Sons, 2003.

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  • How to Cite

    Abassy, T. (2020). Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study). International Journal of Applied Mathematical Research, 9(2), 41-49. https://doi.org/10.14419/ijamr.v9i2.31118