Solution of Continuous Nonlinear PDEs through Order CompletionThis work inaugurates a new and general solution method for arbitrary continuous nonlinear PDEs. The solution method is based on Dedekind order completion of usual spaces of smooth functions defined on domains in Euclidean spaces. However, the nonlinear PDEs dealt with need not satisfy any kind of monotonicity properties. Moreover, the solution method is completely type independent. In other words, it does not assume anything about the nonlinear PDEs, except for the continuity of their left hand term, which includes the unkown function. Furthermore the right hand term of such nonlinear PDEs can in fact be given any discontinuous and measurable function. |
Contents
APPLICATIONS TO SPECIFIC CLASSES OF LINEAR AND NONLINEAR PDEs | 159 |
GROUP INVARIANCE OF GLOBAL GENERALIZED SOLUTIONS OF NONLINEAR PDEs | 295 |
Appendix | 389 |
| 421 | |
| 429 | |
Other editions - View all
Common terms and phrases
algebra bijective C*-smooth classical solutions cº(n cº(ſ Colombeau commutative diagram continuous nonlinear PDEs Dedekind order completion denote dense set equivalence class equivalence relation existence and uniqueness existence results extension fact finite follows g E G give given global generalized solutions group invariance group transformations groups of transformations implies inclusion injective Lemma let us take Lie group linear manifolds mapping measurable functions nonlinear partial differential nonvoid Oberguggenberger obtain obviously open set open subset order completion method order isomorphical embedding partial differential equations partial differential operator partial order particular PDEs in 2.1 poset preserves infima projectable groups proof Proposition pull-back quotient space respective result in Theorem Riemann solvers Rosinger ſ ſ satisfies Schwartz distributions solutions of nonlinear surjective symmetry groups Theorem 5.1 uniform space uniformly continuous x e ſ