In this paper, we explore some issues related to adopting the Adomian
decomposition method (ADM) to solve partial differential equations (PDEs), particularly 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 engineering, physics and applied mathematics. We then propose a modified approach,
based on combining the ADM with the Fourier series decomposition, to provide solutions 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. Numerical 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.
