Adomian's method of decomposition: critical review and examples of failure

Adomian's method of decomposition is considered in application to initial-boundary value problems for the one space-dimensional spatially homogeneous heat conduction equation. It is shown that the fundamental equation of the method is well-defined only for certain restricted types of boundary conditions. Within the class of such boundary conditions, examples are given such that the fundamental equation fails to have a unique solution, and such that the sequence produced by iteration of this equation is divergent. The latter is a counterexample to a published assertion of convergence.