NUMERICAL SOLUTIONS OF BIGEOMETRIC INITIAL VALUE PROBLEMS USING THE BIGEOMETRIC RUNGE KUTTA METHOD

Abstract

In many studies based on Bigeometric Calculus, an approximation to the Bigeometric Taylor series is used without knowing the correct version. The reason for that can be seen easily in the proof of the Bigeometric Taylor Series presented in the current paper. Based on this Taylor series the Bigeometric Runge-Kutta method is derived explicitly. As the Runge-Kutta method is based on the first derivative, it is not surprising that the coefficients and parameters in the Bigeometric RungeKutta behave according to the Butcher Tableau. The convergence and stability tests are also applied to the Bigeometric Runge-Kutta. Application of the Bigeometric Runge-Kutta method to problems with known closed form solutions show the superiority of this method for a certain family of problems compared to the one in Newtonian calculus. Furthermore, the Bigeometric Runge-Kutta method is applied to mathematical modelling in biology and the Bigeometric R¨ossler attractor, showing the general applicability of the method

Keywords

• Multiplicative Calculus
• Runge-Kutta
• Numerical Approximation

References

1. [1] M. Agarwal and A. S. Bhadauria, Mathematical modeling and analysis of tumor therapy with oncolytic virus, Applied Mathematics, 2, 131–140, 2011