ADOMIAN DECOMPOSITION METHOD FOR NUMERICAL SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS (0 < α ≤ 1)

Abstract

The Idea of Fractional calculus extends the classical concepts of differentiation and integration to derivatives of non-integer order not a whole number. These operators are particularly useful for modeling physical systems in which memory and hereditary characteristics play an important role. Because of this capability, fractional differential equations have been applied in many areas such as viscoelasticity, diffusion phenomena, control theory, biological systems, and signal processing.. Finding exact solutions for nonlinear fractional differential equations is often challenging to us. Therefore, several analytical and numerical techniques have been developed to obtain approximate solutions. Among them, the Adomian Decomposition Method (ADM) is a simple and effective technique that represents the solution as a convergent series while handling nonlinear terms through Adomian polynomials eliminating the need of discretization or linearization. In this study, the Adomian Decomposition Method(ADM) is applied to obtain approximate solutions of nonlinear fractional differential equations defined using the Caputo fractional derivative. A representative nonlinear fractional differential equation is solved and the resulting approximate solution is compared with the known exact solution. The absolute error between the exact and approximate solutions is also evaluated. The numerical results show that the ADM approach provides accurate approximations with only a few terms of the series expansion.