Approaches Developed to Solve Fractional Differential Equations

Abstract

Fractional calculus is not a new topic, in reality it has almost the same history as that of classical calculus. In this paper, new approaches are developed to solve a class of nonlinear fractional differential initial value problems with fractional derivative defined in Caputo sense. Two numerical methods have been presented, Fractional Adams-Bashforth-Moulton Method (FABMM) and Fractional Differential Transformation Method (FDTM) compared with Adomian Decomposition Method (ADM), used for very specific type of problems. The methods are used on two different nonlinear fractional differential equations of the form, , with and without exact solution for the same initial condition.  We present new results that deal with the Adomian Decomposition Method (ADM), suitable to handle fractional calculus applications. The results are obtained with comparisons made between FDTM, FABMM and the exact solutions at each integration point, given, both graphically and tabularly, for different fractional orders , constant step-size and small time interval. Our work, with using symbolic software packages as Wolfram Mathematica 12.1, can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.