Multiplicative Runge-Kutta methods. (English) Zbl 1178.65082
Summary: Numerical multiplicative algorithm of Runge-Kutta type for solving multiplicative differential equations is presented. The multiplicative Rössler system has been briefly examined to test the method proposed.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
numerical examples; Runge-Kutta methods; Rössler system; multiplicative algorithm; multiplicative differential equationsReferences:
| [1] | Volterra, V., Hostinsky, B.: Operations Infinitesimales Lineares. Herman, Paris (1938). |
| [2] | Rybaczuk, M., Kedzia, A., Zielinski, W.: The concept of physical and fractal dimension II. The differential calculus in dimensional spaces. Chaos Solitons Fractals 12, 2537–2552 (2001). · Zbl 0994.28003 · doi:10.1016/S0960-0779(00)00231-9 |
| [3] | Kasprzak, W., Lysik, B., Rybaczuk, M.: Measurements, Dimensions, Invariants Models and Fractals. Ukrainian Society on Fracture Mechanics, SPOLOM, Wroclaw-Lviv, Poland (2004). |
| [4] | Prosnak, J.: Introduction to Numerical Fluid Mechanics. Approximate Methods of Solving Ordinary Differential Problems (in Polish). Zaklad Narodowy im. Ossolinskich, Wroclaw, Poland (1993). |
| [5] | Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester, England (2003). · Zbl 1040.65057 |
| [6] | Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C; The Art of Scientific Computing. Cambridge University Press, Cambridge, MA (1992). · Zbl 0845.65001 |
| [7] | Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An Introduction to Dynamical Systems. Springer-Verlag, New York (2000). · Zbl 0867.58043 |
| [8] | Aniszewska, D., Rybaczuk, M.: Analysis of the multiplicative Lorenz system. Chaos Solitons Fractals 25, 79–90 (2005). · Zbl 1079.37020 · doi:10.1016/j.chaos.2004.09.060 |
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