The decomposition method and distributions

AbstractThe decomposition method is a powerful, nonnumerical method that has been developed by G. Adomian in order to solve many equations such as Partial Differential Equations (PDE). The theoretical study of this method has been performed by Y. Cherruault and L. Gabet. Several difficulties remain when trying to solve PDE. The operator used to obtain a canonical form u = Gu depends on several integration constants and, therefore, its contractance, that is needed for the convergence of the iterative scheme, is not easy to prove. Moreover, boundary conditions can't always be taken into account. In this paper, we explain how distributions spaces can be used in order to apply rigourously the decomposition methods. When the PDE are linear and defined on a bounded or half-bounded domain with respect to several variables and unbounded with respect to the others, the canonical form can be obtained, the computation performed and the boundary conditions taken into account thanks to distributions. This approach also explains why the classical decomposition method does not generally lead to a solution verifying all boundary conditions when only one differential operator is inverted