Matrix Wiener-Hopf Factorisation

Rawlins, A. D.; Williams, Bill

doi:10.1093/qjmam/34.1.1

Cited by 46 publications

(27 citation statements)

References 5 publications

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“…Matrix Wiener-Hopf kernels are fundamentally distinct from their scalar counterparts in that there is no algorithmic approach to determining the factorization (4) of the transformed kernel [16]. Exact factorization can be achieved for matrices with certain special features: those that are upper (or lower) triangular in form; those that are of Khrapkov-Daniele, i.e., commutative, form (see [17][18][19][20]); those whose elements comprise meromorphic functions [21,22]; kernels with special singularity structure that allows the Wiener-Hopf equation to be recast into uncoupled Riemann-Hilbert problems [23][24][25]; and N × N matrices with special algebraic or group structure [26][27][28]. For more details on exact matrix kernel factorization the interested reader is referred to the last mentioned article and to references cited in [29].…”

Section: Extensions Variations and Applications Of The Techniquementioning

confidence: 99%

A brief historical perspective of the Wiener–Hopf technique

2007

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It is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener-Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener-Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems.

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Section: Extensions Variations and Applications Of The Techniquementioning

confidence: 99%

A brief historical perspective of the Wiener–Hopf technique

2007

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“…There is, as yet, no general procedure of factorization of such matrices, although the factorization for a restricted class of matrices has been achieved. For example the Wiener-Hopf-Hilbert method, introduced by Hurd [25], Rawlins [26] and Rawlins and Williams [27], is a powerful tool in the case when the kernel matrix contains branch-point singularities, while the Daniele-Kharapkov method, proposed independently by Daniele [28] and Kharapkov [29], is effective for the class of matrices having only pole-singularties and branch-point singularities. Another detailed survey for the matrix factorization methods with reference to applications of these methods to different diffraction problems may also be found in a paper by Büyükaksoy et al [30].…”

Section: Introductionmentioning

confidence: 99%

Diffraction of Plane Waves by a Slit in an Infinite Soft-Hard Plane

Ayub¹,

Mann²,

Ramzan³

et al. 2009

PIER B

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Abstract-We have studied the problem of diffraction of plane waves by a finite slit in an infinitely long soft-hard plane. Analysis is based on the Fourier transform, the Wiener-Hopf technique and the method of steepest descent. The boundary value problem is reduced to a matrix Wiener-Hopf equation which is solved by using the factorization of the kernel matrix. The diffracted field, calculated in the farfield approximation, is shown to be the sum of the fields (separated and interaction fields) produced by the two edges of the slit. Some graphs showing the effects of various parameters on the diffracted field produced by two edges of the slit are also plotted.

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“…It can be seen that this approach includes also the case stated by Rawlins and Williams [6]. A more general procedure which is applicable to the cases for which the kernel matrix has only poles or poles and branch-point singularities is given by Khrapkov [7], Daniele [8], Rawlins [9] and Jones [10].…”

Section: Introductionmentioning

confidence: 99%

Multiple diffraction of plane waves by a soft/hard strip

Büyükaksoy

Alkumru

1995

J Eng Math

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A uniform asymptotic high-frequency solution is developed for the problem of diffraction of plane waves by a strip which is soft at one side and hard on the other. The related three-part boundary value problem is formulated into a "modified matrix Wiener-Hopf equation". By using the known factorization of the kernel matrix through the Daniele-Khrapkov method, the modified matrix Wiener-Hopf equation is first reduced to a pair of coupled Fredholm integral equations of the second kind and then solved by iterations. An interesting feature of the present solution is that the classical Wiener-Hopf arguments yield unknown constants which can be determined by means of the edge conditions.

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Matrix Wiener-Hopf Factorisation

Cited by 46 publications

References 5 publications

A brief historical perspective of the Wiener–Hopf technique

A brief historical perspective of the Wiener–Hopf technique

Diffraction of Plane Waves by a Slit in an Infinite Soft-Hard Plane

Multiple diffraction of plane waves by a soft/hard strip

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