A Laplace-Embedded Residual Power Series Framework

Abstract

This paper presents a hybrid analytical approach, combining the residual power series method with the Laplace transform, to solve systems of linear and nonlinear differential equations. The proposed technique converts governing differential systems into algebraic representations, naturally incorporating initial conditions without numerical differentiation or discretization. We validate the method's robustness through applications to a coupled linear system, the nonlinear SIR epidemic model, and the chaotic Genesio-Tesi system. Comparisons with the fourth-order Runge-Kutta numerical method demonstrate exceptional accuracy, rapid convergence, and structural transparency, establishing this technique as a highly efficient tool for analyzing complex dynamic systems.