On the Numerical Solutions of Linear and Nonlinear Differential Equations by the Modified Laplace–Adomian Polynomials Method

Abstract

This study employs the Laplace–Adomian Polynomial Method (LAPM) to obtain approximate solutions for both linear and nonlinear ordinary differential equations. LAPM integrates the Laplace transform with Adomian polynomials to manage nonlinear terms effectively, avoiding the need for linearization or perturbation techniques. To evaluate the method’s accuracy and computational efficiency, three representative examples were solved, with the results benchmarked against corresponding exact solutions. The numerical outcomes, presented through tables and graphical comparisons, demonstrate that LAPM provides highly accurate approximations with minimal error using only a few series terms. The findings affirm that the method is not only straightforward and computationally efficient but also broadly applicable to various nonlinear problems. Given its robustness and simplicity, LAPM holds promise for extension to more complex systems, including partial differential equations and multi-dimensional models in applied sciences.