On the Adomian Decomp osition Metho d for Solving PDEs

In this paper, we explore some issues related to adopting the Adomian decomposition method (ADM) to solve partial differential equations (PDEs), par-ticularly linear diffusion equations. Through a proposition, we show that extending the ADM from ODEs to PDEs poses some strong requirements on the initial and boundary conditions, which quite often are violated for problems encountered in en-gineering, physics and applied mathematics. We then propose a modified approach, based on combining the ADM with the Fourier series decomposition, to provide solu-tions for those problems when these conditions are not met. In passing, we shall also present an argument that would address a long-term standing“pitfall”of the original ADM and make this powerful approach much more rigorous in its setup. Numeri-cal examples are provided to show that our modified approach can be used to solve any linear diffusion equation (homogeneous or non-homogeneous), with reasonable smoothness of the initial and boundary data.