ABSTRACT
This chapter aims to present a number of typical applications of the nonstandard theory of generalized functions to partial differential equations. The exposition covers part of a general program which might be called Nonlinear Theory of Generalized Functions. This program has been developed to some extent directly in the framework of Nonstandard Analysis, to a larger extent by standard methods, but implicitly using a substantial amount of ideas from Nonstandard Analysis, like internal functions, extended number systems, and transfer. Some remarks on the history of the subject, more details concerning the relation between standard and nonstandard models, and relevant literature can be found in the Appendix. Applications of Nonstandard Analysis to ordinary/partial differential equations as well as linear/nonlinear operations on distributions appear to be more scattered in the nonstandard literature. A wealth of material can be found in S. Albeverio, J. E. Fenstad, R. Hoegh-Krohn, T. Lindstrom and J. E. Rubio as well as the extensive bibliographies there.
