The Runge-Kutta Method in Geometric Multiplicative Calculus
This work provides a novel numerical method for solving differential equations in the framework of geometric multiplicative calculus, but its impact is limited to a specialized domain and the improvements over existing methods are demonstrated only on a single example.
The authors derive a Multiplicative Runge-Kutta Method within geometric multiplicative calculus, removing restrictions to positive-valued functions and addressing derivative issues at roots. They demonstrate its applicability and efficiency through error analysis and comparisons with the Ordinary Runge-Kutta Method, showing advantages in computation time vs. relative error for one example.
This paper illuminates the derivation, the applicability, and the efficiency of the Multiplicative Runge-Kutta Method, derived in the frame- work of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus to positive-valued functions of real variable and the fact that the multiplicative derivative does not exist at the roots of the function, is presented explicitly to ensure that the proposed method is uni- versally applicable. The error analysis is also carried out in the framework of geometric multiplicative calculus explicitly. The presented method is applied to various problems and the results are compared to the ones obtained from the Ordinary Runge-Kutta Method. Moreover, for one example, a comparison of the computation time vs. relative error, is worked out, to illustrate the general advantage of the proposed method.