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

Lieven Matthys, Hendrik Lambert, Gilbert De Mey
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/av75skmu6neednu63f27tpk7qu" style="color: black;">Journal of Computational and Applied Mathematics</a> </i> &nbsp;
If a system of coupled linear partial differential equations (PDEs)L. u = fin a compact simply connected region D of an n-dimensional space (e.g., n = 3) has a nonhomogeneous partfthat 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 pkg", v of where is the unit vector in the v direction. The sequence of particular solutions p~t .... can be determined recursively in a simple and
more &raquo; ... cient 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.
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