A Simplified Hybrid Analytical Method for Solving Integer and Fractional-Order Differential Equations without Adomian Polynomials or Lagrange Multipliers

In this study, we propose a novel hybrid analytical technique that combines the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve a class of linear and nonlinear first-order initial value problems (IVPs), including those of fractional order. The principal aim of this approach is to overcome the computational challenges typically encountered in each individual method—namely, the complexity of generating Adomian polynomials in ADM and the requirement for Lagrange multipliers in VIM. By synthesizing the strengths of both methods, the hybrid scheme constructs analytical series solutions without necessitating linearization, Adomian polynomials, or the explicit formulation of Lagrange multipliers. This significantly streamlines the solution process while preserving accuracy and generality. The validity and computational efficiency of the proposed method are substantiated through a series of illustrative examples, encompassing both integer-order and fractional differential equations. The results demonstrate that the hybrid approach not only simplifies implementation but also yields precise and rapidly converging solutions, making it a robust alternative for tackling a broad spectrum of initial value problems in mathematical modeling and applied sciences.