| Περίληψη: | The main concern of this Dissertation is focused on the derivation of novel integral formulation for simple problems. This alternative integral representations display a rapid decay as the complex parameter involved tends to infinity and are therefore suitable for numerical computations and for the study of the asymptotic properties of those solutions. There is also another important advantage attached to the novel formulae presented. These integral representations are useful for solving changing-type boundary value problems (such as Dirichlet data on part of the boundary and Neumann data on the complementary of the boundary).
The following problems are analyzed: (a) The Laplacian operator in the interior of a Square, (b) the Laplacian operator in the interior and exterior of a Sphere and, (c) the Stokes' operator concerning the irrotational flow of an incompressible, viscous fluid.
Moreover, the behaviour of the Gegenbauer functions of the first and second kind of general complex degree and order on the cut (-1, +1) are examined.
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