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Generalized Runge-Kutta method with respect to the non-Newtonian calculus. (English) Zbl 1347.65122

Summary: Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
26A99 Functions of one variable
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