Improvement on a Central Theory of PDEs

One of the two basic theorems in [5] on the existence of solutions of PDEs is improved with the use of a group analysis type argument. Recently, two rather different and general, that is, type independent solution methods have been developed for very large classes of linear and nonlinear systems of PDEs with possibly associated initial and/or boundary conditions. One of them, [5], is using standard Functional Analytic methods, while the other, [6,1,2,7-11], is based on a new idea in the realms of PDEs, namely, the Dedekind order completion of spaces of smooth functions. Contrary to widespread perceptions, it thus proves to be possible to implement no less than two powerful solution methods for a very large variety of linear and nonlinear PDEs. These two methods are type independent in the sense that they are no longer dependent on specifics of one or another of the countless particular types of PDEs. In fact, the essence of both methods is that, each in its own way is able to solve far more general equations than PDEs. And it is precisely in this lifting to a higher level of generality, one beyond PDEs, that the two methods attain their respective type independent power.