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# On The Solution Of Wiener-Hopf Problems Involving Noncommutative Matrix Kernel Decompositions

(Impact Factor: 1.43). 11/2000; 57(2). DOI: 10.1137/S0036139995287673
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ABSTRACT

Many problems in physics and engineering with semi-infinite boundaries or interfaces are exactly solvable by the Wiener-Hopf technique. It has been used successfully in a multitude of different disciplines when the Wiener-Hopf functional equation contains a single scalar kernel. For complex boundary value problems, however, the procedure often leads to coupled equations which therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel into a product of two functions, one analytic in an upper region of a complex (transform) plane and the other analytic in an overlapping lower half-plane. This is straightforward for scalar kernels but no method has yet been proposed for general matrices. In this article a new procedure is introduced whereby Pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel. As well as being simple to apply, the use of approximants allows the accuracy of the factorization to be increased almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves by a semi-infinite screen at the interface between two compressible media with different physical properties is examined in detail. Numerical evaluation of the approximate factorizations together with results on an upper bound of the absolute error reveal that convergence with increasing Pade number is extremely rapid. The procedure is computationally efficient and of simple analytic form and offers high accuracy (error < 10 -7 % typically). The method is applicable to a wide range of initial or boundary value problems.

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• "A number of other approaches have been proposed to factorize non-rational matrix-function s, especially those related to certain problems appearing in applications (see, e.g., survey paper Bart et al., 1986). In Abrahams (1997), the coupled Wiener–Hopf equations were considered with matrix coefficients of the form "
##### Article: Constructive methods for factorization of matrix-functions
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ABSTRACT: A survey of constructive methods for the factorization of $n \times n$ matrix-functions is presented. The importance of these methods for theoretical and practical applications is singled out. Several classes of matrices are considered which are factorized by the proper technique. The perspective of the constructive methods and procedures is discussed and open questions are formulated.
Full-text · Article · Dec 2015 · IMA Journal of Applied Mathematics
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• "To remove poles we can consider the factorisation K N (t) = (Q N − M)(M −1 Q N + ), (5.3) where M is a rational matrix, which is chosen such that the resulting factorisation has no poles in the required half-planes, see [2] for further details. We are turning to illustrations of this method. "
##### Article: Stability Analysis of Matrix Wiener--Hopf Factorisation of Daniele--Khrapkov Class and Reliable Approximate Factorisation
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ABSTRACT: This paper presents new stability results for matrix Wiener--Hopf factorisation. The first part of the paper examines conditions for stability of Wiener-Hopf factorisation in Daniele--Khrapkov class. The second part of the paper concerns the class of matrix functions which can be exactly or approximately reduced to the factorisation of the Daniele--Khrapkov matrices. The results of the paper are demonstrated by numerical examples with partial indices $$\{1,-1\}$$, $$\{0,0\}$$ and $$\{-1,-1\}$$.
Full-text · Article · Apr 2015 · Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences
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• "If G cannot be represented as (2) then some numerical [9] or approximate (e.g. [10]) methods can be applied. "
##### Article: Wiener-Hopf matrix factorization using ordinary differential equations in the commutative case
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ABSTRACT: A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a Riemann-Hilbert problem using the Hurd's method. Secondly, the Riemann-Hilbert problem is embedded into a family of Riemann-Hilbert problems indexed by a variable b taking values on a half-line. A linear ordinary differential equation (ODE1) with respect to b is derived. The coefficient of this equation remains unknown at this step. Finally, the coefficient of the ODE1 is computed. For this, it is proven that it obeys a non-linear ordinary differential equation (ODE2) on a half-line. Thus, the numerical procedure of matrix factorization becomes reduced to two runs of solving of ordinary differential equations on a half-line: first ODE2 for the coefficient of ODE1, and then ODE1 for the unknown function. The efficiency of the new method is demonstrated on some examples.
Preview · Article · Jan 2013