Analytical Solutions of Nonlinear Fractional Differential Equations via the Elzaki–Adomian Decomposition Method

Abstract

This paper presents a comprehensive analytical framework for solving nonlinear fractional dif-
ferential equations (FDEs) and ordinary differential equations (ODEs) using the Elzaki transform coupled with the Adomian decomposition approach, collectively termed the Elzaki–Adomian Decomposition Method (EADM). The Elzaki transform, a relatively new integral transform that generalizes both Laplace and Sumudu transforms, is employed to transform differential equations into algebraic equations. The Adomian polynomials are then used to systematically decompose nonlinear terms, permitting the development of convergent series solutions. The method is applied to several important nonlinear FDEs including the fractional logistic equation, fractional Van der Pol oscillator, Bratu-type equations, and mixed-order fractional equations. The convergence of the series solutions to exact solutions is validated both analytically and graphically. The influence of the fractional order parameter αon solution behavior is investigated, offering insights into the memory effects inherent in fractional systems. The proposed EADM proves to be an efficient, systematic, and computationally feasible technique for solving a wide class of nonlinear differential equations encountered in scientific and engineering applications.