On a class of functional equations of the Wiener-Hopf type and their applications in n-part scattering problems

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Tarih

2003-12

Dergi Başlığı

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Yayıncı

Oxford Univ Press

Erişim Hakkı

info:eu-repo/semantics/closedAccess

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Özet

An 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.

Açıklama

This work was partly supported by the Turkish Academy of Sciences (TUBA). The authors are indebted to a referee who informed them of some references and made valuable suggestions.

Anahtar Kelimeler

Mixed boundary-value problems, Matrix Wiener-Hopf equation, Diffraction of high-frequency waves, Diffraction, Factorization, Field, Integral equations, High-frequency waves, Functional equations, Poles and zeros, Matrix algebra, Mathematical operators, Frequencies, Boundary value problems, Approximation theory

Kaynak

IMA Journal of Applied Mathematics

WoS Q Değeri

Q3

Scopus Q Değeri

Q3

Cilt

68

Sayı

6

Künye

İdemen, M. M. & Alkumru, A. (2003). On a class of functional equations of the wiener-hopf type and their applications in n-part scattering problems. IMA Journal of Applied Mathematics, 68(6), 563-586. doi:10.1093/imamat/68.6.563