Examples of Solving PDEs by Order Completion

So far, the order completion method for solving PDEs, introduced in 1990, can solve by far the most general linear and nonlinear systems of PDEs, with possible initial and/or boundary data. Examples of solv-ing various PDEs with the order completion method are presented. Some of such PDEs do not have global solutions by any other known methods, or are even proved not to have such global solutions. The presentation next aims to be as summary, and in fact, sketchy as possible, even if by that it may create some difficulty. However, nowa-days, being subjected to an ever growing “information overload”, that approach may turn out to be not the worst among two bad alterna-tives. Details can be found in [1], while on the other hand, alternative longer ”short presentations ” are in [6-8]. 0. Bare bones presentation The most general systems of nonlinear PDEs and possibly associated initial and/or boundary problems have equations of the form (0.1) T (x,D)U(x) = F (x, U(x),..., DpxU(x),...) = f(x), x ∈ Ω where Ω ⊆ Rn are open, possibly unbounded, p ∈ Nn, |p | ≤ m, F jointly continuous in all arguments, f continuous, except for certain possible discontinuities on closed, nowhere dense subsets of Ω. 1 Clearly, we have the associated mapping