Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations (original) (raw)

Abstract

In this paper three pairs of embedded Runge-Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y 000 ¼ f ðx; yÞ denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge-Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs.

Key takeaways

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  1. The paper introduces three pairs of embedded Runge-Kutta methods for third-order ordinary differential equations.
  2. RKD4(3), RKD5(4), and RKD6(5) methods outperform existing embedded Runge-Kutta pairs in efficiency.
  3. The RKD4(3) method maintains the first same as last (FSAL) property for improved accuracy.
  4. Numerical results demonstrate RKD pairs' superiority over traditional first-order reduction methods.
  5. The work emphasizes the computational efficiency of direct solutions for third-order ODEs without requiring initial value conversions.

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References (21)

  1. B.R. Duffy, S.K. Wilson, A third-order differential equation arising in thin-film flows and relevant to Tanner's law, Appl. Math. Lett. 10 (1997) 63-68.
  2. T.G. Myers, Thin films with high surface tension, SIAM Rev. 40 (1998) 441-462.
  3. E. Momoniat, Symmetries, first integrals and phase planes of a third-order ordinary differential equation from thin film flow, Math. Comput. Model. 49 (2009) 215-225.
  4. D.O. Awoyemi, O.M. Idowu, A class of hybrid collecations methods for third-order ordinary differential equations, Int. J. Comput. Math. 82 (2005) 1287-1293.
  5. N. Waeleh, Z.A. Majid, F. Ismail, A new algorithm for solving higher order IVPs of ODEs, Appl. Math. Sci. 5 (2011) 2795-2805.
  6. N. Zainuddin, 2-Point block backward differentiation formula for solving higher order ODEs (Ph.D. thesis), Universiti Putra Malaysia, 2011.
  7. S.N. Jator, Solving second order initial value problems by a hybrid multistep method without predictors, Appl. Math. Comput. 217 (2011) 4036-4046.
  8. M. Mechee, N. Senu, F. Ismail, B. Nikouravan, Z. Siri, A three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film flow problem, Math. Prob. Eng. 2013 (2013), http://dx.doi.org/10.1155/2013/795397\. 7p.
  9. X. You, Z. Chen, Direct integrators of Runge-Kutta type for special third-order ordinary differential equations, Appl. Numer. Math. 74 (2013) 128-150.
  10. W. Gader, D. Gruntz, Derivation of numerical methods using computer algebra, SIAM Rev. 41 (1999) 577-593.
  11. J.R. Dormand, Numerical Methods for Differential, A computational Approach, CRC Press, Inc., Florida, 1996.
  12. J.R. Dormand, P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19-26.
  13. J.R. Dormand, P.J. Prince, High order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7 (1981) 67-75.
  14. F. Ismail, B. Suleiman, Embedded singly diagonally implicit Runge-Kutta method (4, 5) in (5, 6) for the integration of stiff systems of ODEs, Int. J. Comput. Math. 2 (1998) 325-341.
  15. M. El-Mikkawy, R. El-Desouky, A new optimized non-FSAL embedded Runge-Kutta-Nyström algorithm of orders 6 and 4 in six stages, Appl. Math. Comput. 145 (2003) 33-43.
  16. J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of Runge-Kutta-Nyström formulae, IMA J. Numer. Anal. 7 (1987) 235-250.
  17. N. Senu, M. Suleiman, F. Ismail, An embedded explicit Runge-Kutta-Nyström method for solving oscillatory problems, Phys. Scr. 80 015005, http:// dx.doi.org/10.1088/0031-8949/80/01/015005.
  18. E. Fehlberg, Lower order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems, NASA TR R-315, 1969.
  19. J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley and Sons Ltd., England, 2008.
  20. E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, second ed., Springer, Berlin, 2010.
  21. E. Fehlberg, Classical fifth sixth, seventh and eighth order Runge-Kutta formulas with stepsize control, NASA TR R-287, 1968.