Eur. Phys. J. Plus (2018) 133: 37
https://doi.org/10.1140/epjp/i2018-11895-1

Regular Article

A new numerical approximation of the fractal ordinary differential equation

¹ Institute for Groundwater Studies, Faculty for Natural and Agricultural Sciences, University of the Free State, 9300, Bloemfontein, South Africa
² Department of Mathematics, Faculty of Science & Technology, The ICFAI University, 302031, Jaipur, India

^* e-mail: sonaljainmaths@gmail.com

Received: 26 October 2017
Accepted: 9 January 2018
Published online: 6 February 2018

Abstract

The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.