Bigeometric Calculus and Runge Kutta Method

TL;DR

This paper introduces the Bigeometric calculus, develops a Bigeometric Runge-Kutta method, and demonstrates its superior accuracy and efficiency over the traditional method in specific initial value problems.

Contribution

It presents the Bigeometric derivative, derives the Bigeometric Taylor theorem, and develops a novel Bigeometric Runge-Kutta method for solving differential equations.

Findings

Bigeometric Runge-Kutta method outperforms ordinary in accuracy.

The method reduces computation time for certain problems.

Application to biological models shows practical effectiveness.

Abstract

The properties of the Bigeometric or proportional derivative are presented and discussed explicitly. Based on this derivative, the Bigeometric Taylor theorem is worked out. As an application of this calculus, the Bigeometric Runge-Kutta method is derived and is applied to academic examples, with known closed form solutions, and a sample problem from mathematical modelling in biology. The comparison of the results of the Bigeometric Runge-Kutta method with the ordinary Runge-Kutta method shows that the Bigeometric Runge-Kutta method is at least for a particular set of initial value problems superior with respect to accuracy and computation time to the ordinary Runge-Kutta method.