A Novel Computational Framework for Nonlinear Differential Equations Employing the Modified Laplace Adomian Polynomial Method

Abstract

Nonlinear differential equations pose significant challenges for conventional analytical and numerical techniques, particularly in efficiently handling complex nonlinear terms while maintaining solution accuracy and stability. This paper presents a novel computational framework for solving such equations using the Modified Laplace–Adomian Polynomial Method (LAPM), which integrates the Laplace transform with an enhanced form of the Adomian Decomposition Method. In the proposed approach, nonlinear terms are systematically decomposed into rapidly convergent Adomian polynomials, simplifying the solution process and reducing computational complexity without compromising precision. The performance of LAPM is evaluated using several benchmark nonlinear and linear differential equations, where it exhibits superior convergence speed, accuracy, and stability when compared with traditional methods. These results demonstrate that the Modified Laplace–Adomian Polynomial Method is a reliable and efficient tool for addressing a wide class of nonlinear differential equations in applied mathematics, physics, and engineering, and contributes to the growing repertoire of semi-analytical techniques for nonlinear problem solving.