A numerical method for the solution of an autonomous initial value problem


Using a known interpolation formula we introduce a class of numerical methods for approximating the solutions of scalar initial value problems for first order differential equations, which can be identified as explicit Runge-Kutta methods. We determine bounds for the local truncation error and we also compare the convergence order and the stability region with those for explicit Runge-Kutta methods, which have convergence order equal with number of stages (i.e. with 2, 3 and 4 stages). The convergence order is only two, but our methods have a larger absolute stability region than the above mentioned methods. In the last section a numerical example is provided, and the obtained numerical approximation is compared with the corresponding exact solution.


Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)


initial value problem; stability region; convergence order; local truncation error

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F. Pătrulescu, A numerical method for the solution of an autonomous initial value problem, Carpathian J. Math. vol. 28 (2012), pp. 289-296


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Carpathian Journal of Mathematics

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North University of Baia Mare, Department of Mathematics and Computer Science, Baia Mare

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[1] Crouzeix, A.L. Mignot, Analyse numerique des equations differentielles, Masson, Paris (1989).
[2] D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, Chichester (1990).
[3] Pavaloiu, On an approximation formula, Rev. Anal. Numer. Theor. Approx.,  26, no. 1-2 (1997), 179-183.
[4] Ralston, Runge-Kutta Methods with Minimum Error Bounds, Math. Comp.,16, no. 80 (1962), 431-437.
[5] F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1964).

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