On the Adomian Decomposition Method for Solving PDEs (Communications in Mathematical Research, 2021)
In this paper, we explore some issues related to adopting the Adomiandecomposition method (ADM) to solve partial differential equations (PDEs), particularly linear diffusion equations. Through a proposition, we show that extendingthe ADM from ODEs to PDEs poses some strong requirements on the initial andboundary 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 alsopresent an argument that would address a long-term standing "pitfall" of the originalADM 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 solveany linear diffusion equation (homogeneous or non-homogeneous), with reasonablesmoothness of the initial and boundary data.
Keywords: Adomian decomposition method; non-smooth initial condition; linear PDEs.