Risc Colloquium

Fritz Schwarz, Fraunhofer Gesellschaft, Institut SCAI, Sankt Augustin Research during the last two decades on Loewy~Rs theory of linear ordinary differential equations is reviewed. It is shown how it may be applied for solving such equations in closed form. In order to extend it to linear partial differential equations it has to be eformulated in terms of differential algebra. In this way the relevant concepts may be suitably generalized and a similar theory is obtained for the partial case. As a result new classes of linear linear partial differential equations may be solved. These applications are supported by computer algebra software that implements the solution procedures following from this proceeding. Finally a few preliminary results are presented indicating how Loewy~Rs results may be extended to special classes of nonlinear equations.