Abstract Cauchy Problems: Three Approaches. Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 120.

by Alexei Filinkov, Irina V. Melnikova

Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches were extensively developed over the last decades by many researchers, nowhere could one find a comprehensive treatment of all three approaches.Abstract Cauchy Three Approaches provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, but also to important the Cauchy problem for inclusion and the Cauchy problem for second order equations.

259 pages, Kindle Edition

First published March 27, 2001