Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A NEW FIFTH-ORDER WEIGHTED RUNGE-KUTTA ALGORITHM BASED ON HERONIAN MEAN FOR INITIAL VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS

  • CHANDRU, M. (Department of Mathematics, National Institute of Technology) ;
  • PONALAGUSAMY, R. (Department of Mathematics, National Institute of Technology) ;
  • ALPHONSE, P.J.A. (Department of Computer Applications, National Institute of Technology)
  • Received : 2016.03.19
  • Accepted : 2016.09.26
  • Published : 2017.01.30
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Abstract

A new fifth-order weighted Runge-Kutta algorithm based on heronian mean for solving initial value problem in ordinary differential equations is considered in this paper. Comparisons in terms of numerical accuracy and size of the stability region between new proposed Runge-Kutta(5,5) algorithm, Runge-Kutta (5,5) based on Harmonic Mean, Runge-Kutta(5,5) based on Contra Harmonic Mean and Runge-Kutta(5,5) based on Geometric Mean are carried out as well. The problems, methods and comparison criteria are specified very carefully. Numerical experiments show that the new algorithm performs better than other three methods in solving variety of initial value problems. The error analysis is discussed and stability polynomials and regions have also been presented.

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References

  1. J. C. Butcher, On Runge Processes of Higher Order, Journal of the Australian Mathematical Society, 4 (1964), No. 2, 179-189. https://doi.org/10.1017/S1446788700023387
  2. T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing Numerical Methods for Ordinary Differential Equations, SIAM, Journal of Numerical Analysis, Vol. 9 (1972), No. 4, 603-637. https://doi.org/10.1137/0709052
  3. L.F. Shampine and M.K. Gordon, Computer Solutions of Ordinary Differential Equations: the initial value problem, W.H. Freeman, San Francisco, CA, 85, 1975.
  4. L.F. Shampine and H.A. Watts, The art of a Runge-Kutta code. Part I, Mathematical Software, 3 (1977), 257-275. https://doi.org/10.1145/355744.355750
  5. L.F. Shampine and H.A. Watts, The Art of writing a Runge-Kutta Code. Part II, Applied Mathematical Computations, 5 (1979), 93-121. https://doi.org/10.1016/0096-3003(79)90001-8
  6. L.F. Shampine and H.A. Watts, Some Practical Runge-Kutta Formulas, Mathematics of Computations, 46 (1985), 135-150.
  7. J.C. Butcher, The numerical analysis of ordinary differential equations: RungeKutta and general linear methods, John Wiley & Sons, Chichester, New York, 1987.
  8. Z. Taha, Approach to variable structure control of Industrial Robots in Robot Control-Theory and Applications, Peter Peregrinus Ltd, North-Holland, (1988), 53-59.
  9. R.K. Alexander and J.J. Coyle, Runge-Kutta methods for differential-algebraic systems, SIAM Journal of Numerical Analysis, 27 (1990), No. 3, 736-752. https://doi.org/10.1137/0727043
  10. D.J. Evans, A new 4th order Runge-Kutta method for initial value problems with error control, International Journal of Computer Mathematics, 39 (1991), 217-227. https://doi.org/10.1080/00207169108803994
  11. D.J. Evans and A.R. Yaakub, A New Fourth-Order Runge-Kutta Formula Based on the Contraharmonic Mean, International Journal of Computer Mathematics, 57 (1995), 249-256. https://doi.org/10.1080/00207169508804428
  12. D.J. Evans and A.R. Yaakub, A New Fifth-Order Weighted Runge-Kutta Formula, International Journal of Computer Mathematics, 59 (1996), 227-243. https://doi.org/10.1080/00207169608804467
  13. A.R. Yaakub and D.J. Evans, A New Fifth Order Contraharmonic Mean Method Weighted Runge-Kutta Formula For Initial Value Problems, Loughborough University of Technology, Department of Computer Studies, Report No. 993, 1996.
  14. N. Yaacob and B. Sanugi, A New Fourth-order Embedded Method Based on the Harmonic Mean, Mathematika, 14 (1998), 1-6.
  15. G. Hung, Dissipitivity of Runge-Kutta methods for dynamical systems with delays, IMA Journal of Numerical Analysis, 20 (2000), 153-166. https://doi.org/10.1093/imanum/20.1.153
  16. R. Ponalagusamy, K. Murugesan, D. Paul Dhayabaran and E.C. Henry Amirtharaj, Numerical Solution of Heat-flow Problem by a Combined method of Rayleigh Ritz with STWS and RKHM, Advances in Modeling and Analysis Journal(France)-A, 38 (2001), 29-48.
  17. D.J. Evans and A.R. Yaakub, A Fifth-Order Runge-Kutta RK(5,5) Method with Error Control, International Journal of Computer Mathematics, 79 (2002), 1179-1185. https://doi.org/10.1080/00207160213937
  18. R. Ponalagusamy and S. Senthilkumar, Investigation on Multilayer Raster Cellular Neural Network by Arithmetic and Heronian Mean RKAHeM(4,4), Lecture Notes in Engineering and Compute Science[ISBN: 978-988-98671-5-7], WCE, IAENG, U.K., (2007), 713-718.
  19. D.J. Evans and N. Yaacob, A fourth-order Runge-Kutta method based on the heronian mean formula, International Journal of Computer Mathematics, 58 (2007), 103-115.
  20. B.B. Sanugi, N.B. Yaacob, A new fifth-order five-stage runge kutta method for initial value type problems in ODEs, International Journal of Computer Mathematics, 59 (2007), 187-207.
  21. R. Ponalagusamy, A Novel and Efficient Computational Algorithm of STWS for Generalized Linear Non-Singular/Singular Time Varying Systems, Journal of Software Engineering, 2 (2008), 1-9.
  22. N. Razali, R.R. Ahmad, M. Darus and A.S. Rambely, Fifth-Order Mean Runge-Kutta Methods Applied to the Lorenz System, Proc. Of the 13th WSEAS International Conference on Applied Mathematics, (2008), 333-338.
  23. R.Ponalagusamy and S.Senthilkumar, System of second order Robot Arm Problem by an Efficient Numerical Integration Algorithm, Archives of Computational Materials Science and Surface Engineering, 1 (2010), 38-44.
  24. R. Ponalagusamy and S. Senthilkumar, Investigation on Time-Multiplexing Cellular Neural Network Simulation by RKAHeM(4,4) Technique, International Journal of Advanced Intelligence Paradigms, 3 (2011), 43-66. https://doi.org/10.1504/IJAIP.2011.038115
  25. R. Ponalagusamy, P.J.A. Alphonse and M. Chandru, Development of New Fifth-Order Fifth-Stage Runge Kutta Method based on Heronian Mean, International Journal of Engineering Science, Advanced Computing and Bio-Technology, 2 (2011), 162-197.
  26. S. SenthilKumar, Malrey Lee and Tae-Kyu Kwon, Runge-Kutta(5,5) Methods for Robot Arm & Initial Value Problem, International Journal of Control and Automation, 6 (2013), 329-336.