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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 problems: 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 Problems: 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 generalizations: the Cauchy problem
for inclusion and the Cauchy problem for second order equations.
Accessible to nonspecialists and beginning graduate students, this
volume brings together many different ideas to serve as a reference
on modern methods for abstract linear evolution equations.
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 problems: 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 Problems: 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 generalizations: the Cauchy problem for inclusion and the Cauchy problem for second order equations.
Accessible to nonspecialists and beginning graduate students, this volume brings together many different ideas to serve as a reference on modern methods for abstract linear evolution equations.
Stochastic Cauchy Problems in Infinite Dimensions: Generalized and
Regularized Solutions presents stochastic differential equations
for random processes with values in Hilbert spaces. Accessible to
non-specialists, the book explores how modern semi-group and
distribution methods relate to the methods of infinite-dimensional
stochastic analysis. It also shows how the idea of regularization
in a broad sense pervades all these methods and is useful for
numerical realization and applications of the theory. The book
presents generalized solutions to the Cauchy problem in its initial
form with white noise processes in spaces of distributions. It also
covers the "classical" approach to stochastic problems involving
the solution of corresponding integral equations. The first part of
the text gives a self-contained introduction to modern semi-group
and abstract distribution methods for solving the homogeneous
(deterministic) Cauchy problem. In the second part, the author
solves stochastic problems using semi-group and distribution
methods as well as the methods of infinite-dimensional stochastic
analysis.
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