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This volume contains proceedings from the AMS conference on Applied
Analysis held at LSU (Baton Rouge) in April 1996. Topics include
partial differential equations, spectral theory, functional
analysis and operator theory, complex analysis, numerical analysis
and related mathematics. Applications include quantum theory, fluid
dynamics, control theory and abstract issues, such as
well-posedness, asymptotics, and more. The book presents the scope
and depth of the conference and its lectures. The state-of-the-art
surveys by Jerry Bona and Fritz Gesztesy contain topics of wide
interest. There have been a number of good conferences on related
topics, yet this volume offers readers a unique, varied viewpoint.
The scope of the material in the book will benefit readers
approaching the work from diverse perspectives. It will serve those
seeking motivational scientific problems, those interested in
techniques and subspecialties and those looking for current results
in the field.
This book investigates several classes of partial differential
equations of real time variable and complex spatial variables,
including the heat, Laplace, wave, telegraph, Burgers,
Black-Merton-Scholes, Schroedinger and Korteweg-de Vries
equations.The complexification of the spatial variable is done by
two different methods. The first method is that of complexifying
the spatial variable in the corresponding semigroups of operators.
In this case, the solutions are studied within the context of the
theory of semigroups of linear operators. It is also interesting to
observe that these solutions preserve some geometric properties of
the boundary function, like the univalence, starlikeness, convexity
and spirallikeness. The second method is that of complexifying the
spatial variable directly in the corresponding evolution equation
from the real case. More precisely, the real spatial variable is
replaced by a complex spatial variable in the corresponding
evolution equation and then analytic and non-analytic solutions are
sought.For the first time in the book literature, we aim to give a
comprehensive study of the most important evolution equations of
real time variable and complex spatial variables. In some cases,
potential physical interpretations are presented. The generality of
the methods used allows the study of evolution equations of spatial
variables in general domains of the complex plane.
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