Factorization of a class of matrices generated by Sommerfeld diffraction problems with oblique derivatives

Lebre, Amarino B.; Santos, Ana Moura; Speck, F.-O.

doi:10.1002/(sici)1099-1476(19970925)20:14<1185::aid-mma909>3.0.co;2-t

Cited by 9 publications

(5 citation statements)

References 13 publications

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“…However, generalised inverses can be constructed as before according to the FIS interpretation. One can find plenty of further boundary value and transmission problems in higher dimensions (n ≥ 3), where the associated operators are not anymore Fredholm but generalised invertible, just adding one or more variables, see [28,64,81,113], for instance.…”

Section: +mentioning

confidence: 99%

From Sommerfeld Diffraction Problems to Operator Factorisation

Speck¹

2019

Constructive Mathematical Analysis

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This article presents a brief survey devoted to the interplay between diffraction and operator theory, particularly between the so-called canonical diffraction problems (exemplified by Sommerfeld half-plane problems) on one hand and operator factorisation theory on the other hand. It is shown how operator factorisation concepts appear in a natural way from applications and how they can help to find solutions rigorously in case of well-posed problems or how to normalise problems by an adequate change of function spaces.

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Section: +mentioning

confidence: 99%

From Sommerfeld Diffraction Problems to Operator Factorisation

Speck¹

2019

Constructive Mathematical Analysis

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“…In particular, the operator P associated to the problem (for the case = 1 and s = (2n+1)=4) can also be transformed into a Fredholm operator by taking a corresponding image restriction. For this purpose, the identities (25), (16), (13) and (12) are of fundamental importance.…”

Section: Normalization In Critical Casesmentioning

confidence: 99%

“…Although the di raction problem with oblique derivative conditions for the half-plane can be solved rigorously [13], the analysis of the boundary value problem associated with a strip is more di cult. The main di culty is due to the emergence of semi-almost periodic elements in the entries of the Fourier matrix symbol of the corresponding Wiener-Hopf operators.…”

Section: Introductionmentioning

confidence: 99%

An operator approach for an oblique derivative boundary‐transmission problem

Castro

Santos

2004

Math Methods in App Sciences

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We consider a boundary-transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave di raction theory. The boundary under consideration consists of a strip, and certain conditions are assumed on it in the form of oblique derivatives. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well-posedness of the problem is shown for a range of non-critical regularity orders of the Bessel potential spaces, which include the ÿnite energy norm space. In addition, an operator normalization method is applied to the critical orders case.

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“…This is a subclass of C(Q 1 ; Q 2 ) that includes an interesting example from di¬raction theory [2,11]. It consists of the set of all matrix functions of the form…”

Section: Generalized Daniele{khrapkov Classmentioning

confidence: 99%

“…The present paper deals with a general method for explicit Wiener{Hopf factorization of non-rational matrix-valued functions that appear in several areas of mathematics and its applications, such as linear operator theory, di¬raction theory [8,11,13] and integrable systems [17].…”

Section: Introductionmentioning

confidence: 99%

Explicit Wiener–Hopf factorization and nonlinear Riemann–Hilbert problems

Câmara¹,

Santos²,

Carpentier³

2002

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A method for explicit Wiener–Hopf factorization of 2 × 2 matrix-valued functions is presented together with an abstract definition of a class of functions, C(Q1, Q2), to which it applies. The method involves the reduction of the original factorization problem to certain nonlinear scalar Riemann–Hilbert problems, which are easier to solve. The class C(Q1, Q2) contains a wide range of classes dealt with in the literature, including the well-known Daniele–Khrapkov class. The structure of the factors in the factorization of any element of the class C(Q1, Q2) is studied and a relation between the two columns of the factors, which gives one of the columns in terms of the other through a linear transformation, is established. This greatly simplifies the complete determination of the factors and gives relevant information on the nature of the factorization. Two examples suggested by applications are completely worked out.

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Factorization of a class of matrices generated by Sommerfeld diffraction problems with oblique derivatives

Cited by 9 publications

References 13 publications

From Sommerfeld Diffraction Problems to Operator Factorisation

From Sommerfeld Diffraction Problems to Operator Factorisation

An operator approach for an oblique derivative boundary‐transmission problem

Explicit Wiener–Hopf factorization and nonlinear Riemann–Hilbert problems

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