Modified Laplace Adomian decomposition method: A fast yet efficient numerical approach for partial differential equations

The research presents an extended version of the Laplace Adomian decomposition method (LADM) for application. The method originates from the combination of the series-based initial guess with the LADM. This paper examines the convergence properties of the algorithm while analyzing its computational performance and presents the approach to solve a general nonlinear partial differential equation through application. The Fornberg-Whitham equation, the Korteweg-de Vries model, and the Black-Scholes model are developed for an approximate analytical solution using the derived method. The results converged to the exact solution because of the proven convergence of the approximate solution to the exact solution. The absolute errors remained small even after a few iterations, and the series-based initial guess offered an efficient representation for handling nonlinear problems. Therefore, it is concluded that the approach shows general consistency and efficiency in solving this class of partial differential equations. The research findings reveal improvements over existing literature techniques, as the results show enhanced effectiveness and quicker convergence rates.