A recursive construction of particular solutions to a system of coupled linear partial differential equations with polynomial source term

If a system of coupled linear partial differential equations (PDEs) L·u = f in a compact simply connected region D of an n-dimensional space (e.g., n = 3) has a nonhomogeneous part f that is a vector of polynomials or can be approximated by a vector of polynomials, a particular solution can be written as a linear combination of particular solutions P klm,v of L·u = x 1 kx 2 lx 3 m e v , where e v is the unit vector in the v direction. The sequence of particular solutions P klm,v can be determined recursively in a simple and efficient way. This technique is an extension of an article of Janssen and Lambert (1992) who stated the theorem for a single PDE. Before extending it to systems of PDEs, their theorem is reviewed and slightly modified; an extra condition is added to ensure the explicit recursivity of the recursion formula. In the case of a system of coupled PDEs this condition can exclude the existence of a recursion formula. The technique is generally applicable to reduce a nonhomogeneous problem to a homogeneous one, for which several solution techniques can be used.