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Yayın On a class of functional equations of the Wiener-Hopf type and their applications in n-part scattering problems(Oxford Univ Press, 2003-12) İdemen, Mehmet Mithat; Alkumru, AliAn asymptotic theory for the functional equation K-phi=f, where K : X-->Y stands for a matrix-valued linear operator of the form K=K1P1+K2P2+...+KnPn, is developed. Here X and Y refer to certain Hilbert spaces, {P-alpha} denotes a partition of the unit operator in X while K-alpha are certain operators from X to Y. One assumes that the partition {P-alpha} as well as the operators K-alpha depend on a complex parameter nu such that all K-alpha are multi-valued around certain branch points at nu=k(+) and nu=k(-) while the inverse operators K-alpha(-1) exist and are bounded in the appropriately cut nu-plane except for certain poles. Then, for a class of {P-alpha} having certain analytical properties, an asymptotic solution valid for \k(+/-)\-->infinity is given. The basic idea is the decomposition of phi into a sum of projections on n mutually orthogonal subspaces of X. The results can be extended in a straightforward manner to the cases of no or more branch points. If there is no branch point or n=2, then the results are all exact. The theory may have effective applications in solving some direct and inverse multi-part boundary-value problems connected with high-frequency waves. An illustrative example shows the applicability as well as the effectiveness of the method.Yayın Diffraction of two-dimensional high-frequency electromagnetic waves by a locally perturbed two-part impedance plane(2004) İdemen, Mehmet Mithat; Alkumru, AliAmong the wave propagation problems, those connected with half-spaces bounded by sectionally homogeneous boundaries take important place because they are motivated by microwave applications. If the boundary are of three or more parts, then the problem results, very frequently, in functional equations involving unknown functions, say Ψ+ (v), Ψ- (v) and P(v), which are regular in the upper half, lower half and whole of the complex v-plane, respectively, except at the point of infinity. A local (non-homogeneous) perturbation on a two-part boundary, which is of extreme importance from engineering point of view, gives also rise to a problem of this type. The aim of the present paper is to establish a method which is based on the elimination of the unknown functions Ψ+ (v) and Ψ- (v) to obtain an integral equation of the Fredholm type for the entire function P(v), which can be solved rather easily by numerical methods. The functions Ψ+ (v) and Ψ- (v) are then determined by the classical Wiener-Hopf technique.
