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Direct and Indirect Boundary Integral Equation Methods (Hardcover)
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Direct and Indirect Boundary Integral Equation Methods (Hardcover)
Series: Monographs and Surveys in Pure and Applied Mathematics
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The computational power currently available means that
practitioners can find extremely accurate approximations to the
solutions of more and more sophisticated mathematical
models-providing they know the right analytical techniques. In
relatively simple terms, this book describes a class of techniques
that fulfill this need by providing closed-form solutions to many
boundary value problems that arise in science and engineering.
Boundary integral equation methods (BIEM's) have certain advantages
over other procedures for solving such problems: BIEM's are
powerful, applicable to a wide variety of situations, elegant, and
ideal for numerical treatment. Certain fundamental constructs in
BIEM's are also essential ingredients in boundary element methods,
often used by scientists and engineers. However, BIEM's are also
sometimes more difficult to use in plane cases than in their
three-dimensional counterparts. Consequently, the full, detailed
BIEM treatment of two-dimensional problems has been largely
neglected in the literature-even when it is more than marginally
different from that applied to the corresponding three-dimensional
versions. This volume discusses three typical cases where such
differences are clear: the Laplace equation (one unknown function),
plane strain (two unknown functions), and the bending of plates
with transverse shear deformation (three unknown functions). The
author considers each of these with Dirichlet, Neumann, and Robin
boundary conditions. He subjects each to a thorough
investigation-with respect to the existence and uniqueness of
regular solutions-through several BIEM's. He proposes suitable
generalizations of the concept of logarithmic capacity for plane
strain and bending of plates, then uses these to identify contours
where non-uniqueness may occur. In the final section, the author
compares and contrasts the various solution representations, links
them by means of boundary operators, and evaluates them for their
suitability for numeric computation.
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