FRACTIONAL EULER METHOD FOR NUMERICAL SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH ORDER 0< α≤1

Abstract

This paper aims to explore numerical techniques for solving linear and non-linear fractional differential equations (FDEs) with initial conditions. The effectiveness of the Fractional Forward Euler Method (FFEM) and the Fractional Backward Euler Method (FBEM) for obtaining numerical solutions to linear and non-linear Fractional Differential Equations (FDEs) of order 0 < α ≤ 1 under the initial conditions is examined in this research study. The proposed approaches in this research study are based on fractional calculus principles, which include the Caputo fractional derivative definition and the Mittag - Leffler function characteristics. The performance of both the methods is tested using a number of instructive instances, comparing their numerical approximations to known exact answers. The results demonstrate that the accuracy of both Fractional Forward Euler Method (FFEM) and Fractional Backward Euler Method (FBEM) is highly sensitive to the step size h, with significantly improved precision observed as h decreases. While observing the results FFEM provides a straightforward approach whereas FBEM is shown to exhibit potential instability near the initial point before converging to an accurate long-term solution. This research study confirms that these Fractional Euler methods serve as effective and feasible computationally tools for approximating solutions to FDEs, provided careful attention is paid to step size selection.