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The Runge–Kutta method in geometric multiplicative calculus

Part of: Numerical analysis: Ordinary differential equations Functional-differential and differential-difference equations

Published online by Cambridge University Press:  01 August 2015

Mustafa Riza  and
Hatice Aktöre
Mustafa Riza
Affiliation:
Department of Physics, Eastern Mediterranean University, Gazimağusa, North Cyprus, via Mersin 10, Turkey email [email protected]
Hatice Aktöre
Affiliation:
Department of Mathematics, Eastern Mediterranean University, Gazimağusa, North Cyprus, via Mersin 10, Turkey email [email protected]

Abstract

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This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge–Kutta method, derived in the framework of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus on positive-valued functions of real variables and the fact that the multiplicative derivative does not exist at the roots of the function are presented explicitly to ensure that the proposed method is universally applicable. The error and stability analyses are also carried out explicitly in the framework of geometric multiplicative calculus. The method presented is applied to various problems and the results are compared to those obtained from the ordinary Runge–Kutta method. Moreover, for one example, a comparison of the computation time against relative error is worked out to illustrate the general advantage of the proposed method.

MSC classification

Primary: 65L06: Multistep, Runge-Kutta and extrapolation methods
Secondary: 34K28: Numerical approximation of solutions
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 539 - 554
Copyright
© The Author(s) 2015 

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