<p>In this study, the Laplace Adomian decomposition method is employed to solve the initial value problem of standard wave equation with sine source term known as sine–Gordon equation. The Laplace transform is coupled with the Adomian decomposition method to attain analytical approximation of the aforesaid equation. The numerical simulation is carried out to obtain the analytical approximations. The method's solution will be derived as an infinite series that converges rapidly to its analytical solution. The applicability of the present technique is justified with four numerical examples and compared the approximate result and absolute errors with the existing results in literature. Several three-dimensional and two-dimensional figures are depicted to demonstrate the method's reliability and simplicity. The solution behaviour of the numerical examples are illustrated through tables. Furthermore, it is demonstrated that the suggested approach is straightforward, efficient, and applicable to a wide range of other nonlinear equations in mathematical physics.</p>