A Comparison of the Adomian Decomposition Method and Variational Iteration Method for a Two-Dimensional Nonlinear Wave Equation

Sanaullah Dehraj, Asgher Ali Maitlo, Wajid Ahmed Siyal, Muhammad Memon, Lubna Naz Arain, Laraib Arain, Kamran Umrani


This paper studies a two-dimensional wave equation in the presence of power and derivative nonlinearity subject to suitable initial conditions. It aims to construct the exact-analytical solution of a two-dimensional nonlinear wave equation using two different semi-analytical methods and to compare the obtained results. First, a well-known Adomian decomposition method (ADM) based on operators is employed. Many researchers use the ADM to investigate several scientific applications, and this method straightforwardly attacks the problem without using linearization, discretization, perturbation, or any other restrictive assumption that may change the physical behavior of the problem. Second, the variational iteration method (VIM) also provides rapid convergent successive approximations of the closed-form solutions if it exists; otherwise, it provides an approximation of a high degree accuracy level even in case of few obtained iterations. The obtained results are drawn graphically and presented in the tables. Both methods provide almost the same solutions in each nonlinearity case, and VIM has computational advantages over ADM in computation size. Further, ADM and VIM provide a series of solutions that converge in a very small time domain, which limits these methods.


Keywords: Adomian decomposition method, variational iteration method, approximations, partial differential equations, two-dimensional wave equation.



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