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The present book deals with the finite-part singular integral
equations, the multidimensional singular integral equations and the
non-linear singular integral equations, which are currently used in
many fields of engineering mechanics with applied character, like
elasticity, plasticity, thermoelastoplasticity, viscoelasticity,
viscoplasticity, fracture mechanics, structural analysis, fluid
mechanics, aerodynamics and elastodynamics. These types of singular
integral equations form the latest high technology on the solution
of very important problems of solid and fluid mechanics and
therefore special attention should be given by the reader of the
present book, who is interested for the new technology of the
twentieth-one century. Chapter 1 is devoted with a historical
report and an extended outline of References, for the finite-part
singular integral equations, the multidimensional singular integral
equations and the non-linear singular integral equations. Chapter 2
provides a finite-part singular integral representation analysis in
Lp spaces and in general Hilbert spaces. In the same Chapter are
investigated all possible approximation methods for the numerical
evaluation of the finite-part singular integral equations, as
closed form solutions for the above type of integral equations are
available only in simple cases. Also, Chapter 2 provides further a
generalization of the well known Sokhotski-Plemelj formulae and the
Nother theorems, for the case of a finite-part singular integral
equation.
The book includes the latest high technology on solving very important theoretical and practical problems on solid mechanics, fracture mechanics, structural analysis, elastodynamics, fluid mechanis and aerodynamics, by using linear and non-linear singular integral equation methods. Analytical theories and numerical evaluation methods are investigated and introduced for the finite-part singular integral equations, the multidimensional singular integral equations and the non-linear singular integral equations with a very important use in a wide field of engineering mechanics. The proposed Singular Interal Operator Method in many cases offers important advantages over "domain" type solutions, like finite elements and finite difference, as well as analytical methods, such as complex variable methods.
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