References
- K. Abdul Hadi A., The Behaviour of the Global Error in the Numerical Solution of Ordinary and Integro-Differential Equations. Ph.D. Thesis, University of Dundee, 1997.
- H. Arndt, Numerical solution of retarded initial value problems with local and global error and stepsize control, Numer. Math., 43(1984), 343-360. https://doi.org/10.1007/BF01390178
- U. M. Ascher, R. M. M. Mattheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, NJ, 1988).
- A. Bellen, A Runge-Kutta-Nystrom method for Delay Differential Equations, Progress in Scientific Computing, Vol.5, Numerical Boundary Value ODEs, 1985, 271-283.
- A. Bellen, One step collocation for delay differential equations, J. Comp. Appl. Math., 10(1984), 275-283. https://doi.org/10.1016/0377-0427(84)90039-6
- A. Bellen and M. Zennaro,Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method, Numer. Math., 47(1985), 301-316. https://doi.org/10.1007/BF01389713
- J. R. Cash, A block 6(4) Runge-Kutta formula for nonstiff initial value problems, ACM Trans. Math. Software, 15(1989), 15-28. https://doi.org/10.1145/62038.62042
- P. Chocholaty and L. Slahor, A method to boundary value problems for delay equations, Numer. Math., 33(1979), 69-75. https://doi.org/10.1007/BF01396496
- J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Math., 6(1980), 19-26. https://doi.org/10.1016/0771-050X(80)90013-3
- J. R. Dormand and P. J. Prince, Runge-Kutta- Nystrom triples, Comput. Math. Appl., 12-13(1987), 937-949.
- J. R. Dormand, M. E. A. El-Mikkawy, and P. J. Prince, Families of embedded Runge-Kutta-Nystrom formulae, IMA J. Numer. Anal., 7(1987), 235-250. https://doi.org/10.1093/imanum/7.2.235
- W. H. Enrigh, K. R. Jackson, S. P. Norsett and P. G. Thomsen, Interpolants for Runge-Kutta formulas, ACM Trans. Math. Software, 12(1986), 193-218.
- W. H. Enrigh, K. R. Jackson, S. P. Norsett and P. G. Thomsen, Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants, Appl. Math. Comput., 27(1988), 313-335. https://doi.org/10.1016/0096-3003(88)90030-6
- W. H. Enrigh and J. D. Pryce, Tow FORTRAN packages for assessing initial value method, ACM Trans. Math. Software, 13(1987), 1-27. https://doi.org/10.1145/23002.27645
- J. M. Fine, A Low Order Runge-Kutta-NystrÄom Methods with Interpolations, Tech. Report 183/85, University of Toronto, Canada, 1985.
- C. W. Gear and O. Osterby, Solving ordinary differential equations with discontinuities, ACM Trans. Math. Software, 10(1984), 23-44. https://doi.org/10.1145/356068.356071
- E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equations I (Springer, Berlin, 1987).
- M. K. Horn, Developments in High Order Runge-Kutta-Nystrom Formulas, Dissertation, Texas University, Austin, Texas, 1977.
- A. V. Kim and V. G. Pimenov, Numerical methods for time-delay systems on the basis of i-smooth analysis, Proceedings of the 15th World Congress on Scientific Computation, Modelling and Applied Mathematics, Vol.1 : Computational Mathematics, pp. 193-196, 1997.
- A. V. Kim and V. G. Pimenov, Numerical methods for delay differential equations. Application of i-smooth calculus. (Lecture Notes in Mathematics, Vol. 44). Research Institute of Mathematics- Global Analysis Research Center. Seoul National University, 1999.
- A. Marthinsen. Continuous Extensions to Nystrom methods for the explicit solution of second order initial value problems. Technical report, Norwegian institute of Technology, Division of Mathematical Sciences, 1994.
- K. W. Neves, Automatic integration of functional differential equations: an approach, ACM Trans. Math. Software, 7(1981), 421-444. https://doi.org/10.1145/355972.355974
- K. W. Neves and A. Feldstein, Characterization of jump discontinuities for state dependent delay differential equations, J. Math. Anal. Appl., 56(1976), 689-707. https://doi.org/10.1016/0022-247X(76)90033-0
- K. W. Neves, Control of interpolatory error in retarded differential equations, ACM Trans. Math. Software, 1(1975), 357-368. https://doi.org/10.1145/355656.355661
- H. J. Oberle and H. J. Pesch,Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math., 37(1981), 235-255. https://doi.org/10.1007/BF01398255
- J. Oppelstrup, The RKFHB4 method for delay differential equations in: R. Burlisch, R.D. Grigorieff and J. Schroder, eds., Numerical Treatment of Differential Equations: Proceedings Oberwolfach, 1976, Lecture Notes in Mathematics 631 (Springer, Berlin, 1978), 133-146.
- G. Papageorgiou and Ch. Tsitouras, Practical Runge-Kutta-Nystrom methods for the equation y" = f(t; y) with interpolation properties, Dept. of Mathematics Tech. Report., 1/87, National Technical University of Athens, Greece, 1987.
- G. Papageorgiou and Ch. Tsitouras, Scaled Runge-Kutta-NystrÄom methods for the second order differential equation y" = f(t; y), Intern. J. of Computer Mathematics, 28(1989), 139-150. https://doi.org/10.1080/00207168908803734
- M. G. Roth, Difference methods for stiff delay differential equations, Ph.D. Thesis, Tech. Report UIUCDCS-R-80 ¡ 1012, Department of Computer Science, University of Illinois at Urbana- Champagne, IL (1980).
- L. F. Shampine, S. Thompson and J. Kierzenka, Solving Delay Differential Equations with dde23, Mathematics Department, Southern Methodist University, Dallas,TX 75275, May 2; (2002).
- L. F. Shampine, Numerical Solution of ODEs, Southern Methodist University, Chapman & Hall, London.
- P.W. Sharp and J.M. Fine, ERNY-an explicit Runge-Kutta-Nystrom integrator for second order initial value problems, TR 199/87, Department of Computer Science, University of Toronto, Toronto, 1987.
- H. J. Stetter, Considerations concerning a theory for ODE-solvers, in: R. Burlisch, R.D. Grigorieff and J. Schroder, eds., Numerical Treatment of Differential Equations: Proceedings Oberwolfach, 1976, Lecture Notes in Mathematics 631 (Springer, Berlin, 1978), 188-200.
- H. J. Stetter, Interpolation and error estimates in Adams PC-codes, SIAM J. Numer. Anal., 16(1979), 311-323. https://doi.org/10.1137/0716023
- H. J. Stetter, Tolerance proportionality in ODE-codes, in: R. Marz, ed., Proceedings Second Conference on Numerical Treatment of Differential Equations, Seminarberichte No.32, Humboldt University, Berlin (1980), 109-123; also in : R.D. Skeel, ed., Working Papers for the 1979 SIGNUM Meeting on Numerical Ordinary Differential Equations, Department of Computer Science, University of Illinois at Urbana-Champagne.
- D. R. Wille and C.T.H Baker, The propagation of derivative discontinuities in systems of delay differential equations, Numerical Analysis Report 160,University of Manchester, Manchester, England (1988).
- D. R. Wille and C.T.H Baker, The tracking of derivative discontinuities in systems of delay differential equations, Numerical Analysis Report 185,University of Manchester, Manchester,England (1990).
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