This 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  Noether theorems, for the case of a finite-part singular integral equation.

Beyond the above, Chapter 3 is devoted with the application of the finite-part singular integral equations in elasticity and fracture mechanics, by considering several crack problems which are reduced to the solution of such a type (or systems) of integral equations.Chapter 4 provides the application of singular integral equations in aerodynamics, by studying planar airfoils in  two - dimensional  aerodynamics.  In  Chapter 5  the  multidimensional  singular  integral  equations  are  being investigated in the Lebesque spaces Lp. In the  same  Chapter  the  Singular  Integral  Operators  Method  (S.I.O.M.) is introduced for the numerical evaluation of the above type of integral equations. This approximation 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.

Chapter 6  provides the application of the multidimensional  singular  integral  equations  in  elasticity,  viscoelasticity  and fracture mechanics of isotropic solids, by considering several elastic stress analysis methods and crack problems. Special attention should be given by the reader of this book for Chapter 7. In this Chapter is being investigated  the  new  field  of applied mechanics, under the name Relativistic Elasticity, which results as a combination of the classical theory of elasticity and general relativity. According to this theory, the relative stress tensor for moving structures has been formulated and its connection with the absolute stress tensor of the stationary frame has been given. Although at present this method looks to be more theoretical, with the rapid growth of technology in 20 or 30 years it will be well applied.

Furthermore, Chapter 8 is devoted with the application of the multidimensional singular integral equations in elasticity and fracture mechanics of anisotropic solids, by considering two- and three-dimensional elastic stress analysis.  On  the  other hand, Chapter 9 provides the application of the multidimensional singular integral equations in plasticity of isotropic solids by studying several applications of two- and three-dimensional plasticity and thermoelastoplasticity.

Chapter 10 is devoted with the investigation of the non-linear singular integral equations, by proving some existence and uniqueness theorems for the above type of non-linear integral equations defined on Banach spaces. Also, same Chapter provides a uniqueness theory for  the  non - linear  finite-part  singular  integral  equations  and  the  non - linear  singular integrodifferential equations defined on Banach spaces. Beyond the above, Chapter 11 is devoted with the investigation of several approximation methods (like the collocation and the quadrature methods) for the numerical evaluation  of  the non-linear singular integral equations, as closed form solutions of such type of  non - linear  integral  equations  are  very difficult to be determined.

Chapter 12  provides  further the  application  of  the  non - linear  singular  integral  equations  in  fluid  mechanics  and aerodynamics, for the solution of generalized problems of turbomachines and aircrafts.Furthermore in the same Chapter the non-linear singular integral equations are applied to the determination of the velocity  and  pressure  coefficient  field around a NACA airfoil in 2-D inviscid and unsteady flow.

Finally, Chapter 13 is devoted with the application of the non-linear integro-differential equations in structural analysis, by considering an orthotropic shallow  spherical  shell  analysis  and  a  sandwich  plates  stress  analysis. Also , Chapter 14 provides the application of the non-linear singular integral equations in the theory of elastodynamics, for the solution of the seismic wave equation.