© 2003 by Institute of Mathematics and its Applications
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On a class of functional equations of the Wiener–Hopf type and their applications in n-part scattering problems
1 ISIK University, Büyükdere Cad., 80670 Maslak, Istanbul, Turkey 2 Gebze Institute of Technology, Gebze, Kocaeli, Turkey
An asymptotic theory for the functional equation K
= 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
} denotes a partition of the unit operator in X while K are certain operators from X to Y. One assumes that the partition {P} as well as the operators K depend on a complex parameter 
such that all K are multi-valued around certain branch points at = k+ and = k– while the inverse operators K–1 exist and are bounded in the appropriately cut -plane except for certain poles. Then, for a class of {P} having certain analytical properties, an asymptotic solution valid for |k±| 
is given. The basic idea is the decomposition of 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.
Keywords: mixed boundary-value problems; matrix Wiener–Hopf equation; diffraction of high-frequency waves.
Received 26 August 2001. Revised 29 July 2002.
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