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As the title of the book indicates, this is primarily a book on
partial differential equations (PDEs) with two definite slants:
toward inverse problems and to the inclusion of fractional
derivatives. The standard paradigm, or direct problem, is to take a
PDE, including all coefficients and initial/boundary conditions,
and to determine the solution. The inverse problem reverses this
approach asking what information about coefficients of the model
can be obtained from partial information on the solution. Answering
this question requires knowledge of the underlying physical model,
including the exact dependence on material parameters. The last
feature of the approach taken by the authors is the inclusion of
fractional derivatives. This is driven by direct physical
applications: a fractional derivative model often allows greater
adherence to physical observations than the traditional integer
order case. The book also has an extensive historical section and
the material that can be called ""fractional calculus"" and
ordinary differential equations with fractional derivatives. This
part is accessible to advanced undergraduates with basic knowledge
on real and complex analysis. At the other end of the spectrum, lie
nonlinear fractional PDEs that require a standard graduate level
course on PDEs.
As the title of the book indicates, this is primarily a book on
partial differential equations (PDEs) with two definite slants:
toward inverse problems and to the inclusion of fractional
derivatives. The standard paradigm, or direct problem, is to take a
PDE, including all coefficients and initial/boundary conditions,
and to determine the solution. The inverse problem reverses this
approach asking what information about coefficients of the model
can be obtained from partial information on the solution. Answering
this question requires knowledge of the underlying physical model,
including the exact dependence on material parameters. The last
feature of the approach taken by the authors is the inclusion of
fractional derivatives. This is driven by direct physical
applications: a fractional derivative model often allows greater
adherence to physical observations than the traditional integer
order case. The book also has an extensive historical section and
the material that can be called ""fractional calculus"" and
ordinary differential equations with fractional derivatives. This
part is accessible to advanced undergraduates with basic knowledge
on real and complex analysis. At the other end of the spectrum, lie
nonlinear fractional PDEs that require a standard graduate level
course on PDEs.
Inverse problems are concerned with determining causes for observed
or desired effects. Problems of this type appear in many
application fields both in science and in engineering. The
mathematical modelling of inverse problems usually leads to
ill-posed problems, i.e., problems where solutions need not exist,
need not be unique or may depend discontinuously on the data. For
this reason, numerical methods for solving inverse problems are
especially difficult, special methods have to be developed which
are known under the term "regularization methods." This volume
contains twelve survey papers about solution methods for inverse
and ill-posed problems and about their application to specific
types of inverse problems, e.g., in scattering theory, in
tomography and medical applications, in geophysics and in image
processing. The papers have been written by leading experts in the
field and provide an up-to-date account of solution methods for
inverse problems.
14 contributions present mathematical models for different imaging
techniques in medicine and nondestructive testing. The underlying
mathematical models are presented in a way that also newcomers in
the field have a chance to understand the relation between the
special applications and the mathematics needed for successfully
treating these problems. The reader gets an insight into a modern
field of scientific computing with applications formerly not
presented in such form, leading from the basics to actual research
activities.
Here is a clearly written introduction to three central areas of
inverse problems: inverse problems in electromagnetic scattering
theory, inverse spectral theory, and inverse problems in quantum
scattering theory. Inverse problems, one of the most attractive
parts of applied mathematics, attempt to obtain information about
structures by nondestructive measurements. Based on a series of
lectures presented by three of the authors, all experts in the
field, the book provides a quick and easy way for readers to become
familiar with the area through a survey of recent developments in
inverse spectral and inverse scattering problems. In the opening
chapter, Paivarinta collects the mathematical tools needed in the
subsequent chapters and gives references for further study.
Colton's chapter focuses on electromagnetic scattering problems. As
an application he considers the problem of detecting and monitoring
leukemia. Rundell's chapter deals with inverse spectral problems.
He describes several exact and algorithmic methods for
reconstructing an unknown function from the spectral data. Chadan
provides an introduction to quantum mechanical inverse scattering
problems. As an application he explains the celebrated method of
Gardner, Greene, Kruskal, and Miura for solving nonlinear evolution
equations such as the Korteweg_de Vries equation. Each chapter
provides full references for further study.
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