Implementation of the Laplace Variational Iteration Method (LT-VIM) for various types of nonlinear partial differential equations (PDEs)

Abstract

This study implements the Laplace Transform Variational Iteration Method (LT-VIM) to solve one-dimensional linear and nonlinear partial differential equations (PDEs) that resist exact solutions. LT-VIM merges the Laplace transform with the variational iteration principle, enabling the Lagrange multiplier to be determined algebraically in the transform domain. This bypasses the complex variational calculus of the classical VIM, yielding a straightforward and rapidly convergent iterative scheme.The method is validated on three canonical models: the linear Airy equation (dispersive waves), and the nonlinear Allen-Cahn and Sine-Gordon equations (phase separation and soliton dynamics). Results demonstrate that LT-VIM achieves high accuracy, often converging to exact or benchmark solutions in few iterations. The work confirms LT-VIM as an efficient, powerful semi-analytical tool for complex PDEs in mathematical physics and engineering.