Abstract
In this chapter we show how Maple can be used to derive explicit Runge-Kutta formulas which are used in numerical analysis to solve systems of differential equations of the first order. We show how the nonlinear system of equations for the coefficients of the Runge-Kutta formulas are constructed and how such a system can be solved. We close the chapter with an overall procedure to construct Runge-Kutta formulas for a given size and order. We will see up to which size such a general purpose program is capable of solving the equations obtained.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B. Buchberger,Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, in Progress, directions and open problems in multidimensional systems theory, ed. N.K. Bose, D. Reidel Publishing Co, p. 189–232, 1985
J.C. Butcher, The non-existence often Stage eight Order Explicit Runge-Kutta Methods, BIT Vol. 25, pp. 521–540, 1985.
S. Czapor and K. Geddes,On Implementing Buchbergers’s Algorithm for Gröbner Bases, ISSAC86, p. 233–238, 1986.
G.E. Collins, The Calculation of Multivariate Polynomial Resultants, Journal of the ACM, Vol. 18, No. 4, pp. 512–532, 1971.
K.O. Geddes, S.R. Czapor, and G. Labahn, Algorithms for Computer Algebra, Kluwer, 1992.
G.H. Gönnet and M.B. Monagan, Solving systems of Algebraic Equations, or the Interface between Software and Mathematics, Computers in mathematics, Conference at Stanford University, 1986.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Springer-Verlag Berlin Heidelberg, 1987.
R.J. Jenks, Problem #11: Generation of Runge-Kutta Equations, SIGSAM Bulletin, Vol. 10, No. 1, p. 6, 1976.
W. Kutta, Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Zeitschrift für Math. u. Phys., Vol. 46, p. 435–453, 1901.
M. Monagan and J.S. Devitt, The D Operator and Algorithmic Differentiation, Maple Technical Newsletter, No. 7 1992.
J. Moses, Solution of Systems of Polynomial Equations by Elimination, Comm. of the ACM, Vol. 9, No. 8, pp. 634–637, 1966.
B.L. van der Waerden, Algebra I, Springer-Verlag, Berlin, 1971.
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gruntz, D. (1993). Symbolic Computation of Explicit Runge-Kutta Formulas. In: Solving Problems in Scientific Computing Using Maple and Matlab ® . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-97533-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-97533-2_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57329-6
Online ISBN: 978-3-642-97533-2
eBook Packages: Springer Book Archive