The Adomian Decomposition Method for Nonlinear Partial Differential Equations

The goal of this report is to study the method introduced by Adomian known as the Adomian Decomposition Method (ADM), which is used to find an approximate solution to nonlinear partial differential equations (PDEs) as a series expansion involving the recursive solution of linear PDEs. We first describe the method, giving two specific examples with different nonlinearities and show exactly how the method works for these problems. Some analytical convergence results are then given, along with numerical solutions to the examples demonstrating these convergence results. A discussion of parameters inside of these nonlinearities follows, both for polynomial nonlinearities and for the more complicated hyperbolic sine nonlinearity problem. Finally, we compare the ADM with the Picard method, pointing out some important differences and demonstrating them by solving the given examples with both methods and comparing the results.