Article

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

11/2000; DOI: 10.1137/S0036139995287673
Source: CiteSeer

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.

0 Bookmarks
 · 
80 Views
  • [Show abstract] [Hide abstract]
    ABSTRACT: We present a solution for the interaction of normally incident linear waves with a submerged elastic plate of semi-infinite extent, where the water has finite depth. While the problem has been solved previously by the eigenfunction-matching method, the present study shows that this problem is also amenable to the more analytical, and extremely efficient, Wiener–Hopf (WH) and residue calculus (RC) methods. We also show that the WH and RC solutions are actually equivalent for problems of this type, a result which applies to many other problems in linear wave theory. (e.g., the much-studied floating elastic plate scattering problem, or acoustic wave propagation in a duct where one wall has an abrupt change in properties.) We present numerical results and a detailed convergence study, and discuss as well the scattering by a submerged rigid dock, particularly the radiation condition beneath the dock.
    Journal of Engineering Mathematics 01/2012; 75(1). · 1.08 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In the present paper we provide an analytical solution for pricing discrete barrier options in the Black-Scholes framework. We reduce the valuation problem to a Wiener-Hopf equation that can be solved analytically. We are able to give explicit expressions for the Greeks of the contract. The results from our formulae are compared with those from other numerical methods available in the literature. Very good agreement is obtained, although evaluation using the present method is substantially quicker than the alternative methods presented. JEL Classification: G13, C63 Mathematics Subject Classification (1991): 45E10, 44A10, 60H30, 60G51.
  • [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we present an analytical solution to the problem of sound radiation from semi-infinite coaxial cylinders, as a model for rearward noise emission by aeroengines. The cylinders carry uniform subsonic flows, whose Mach numbers may differ from each other and from that of the external flow. The incident field takes the form of a downstream-going acoustic mode in either the outer cylinder (the bypass flow) or the inner cylinder (the jet). The key geometrical ingredient of our problem is that the two open ends are staggered by a finite axial distance, so that the inner cylinder can be either buried upstream inside the outer cylinder, or can protrude downstream beyond the end of the outer cylinder (sometimes called the ‘half-cowl’ configuration). The solution is found by solving a matrix Wiener–Hopf equation, which involves the factorization of a certain matrix in the form − = +, with ± analytic, invertible and with algebraic behaviour at infinity in the upper and lower halves of the complex Fourier plane respectively. It turns out that the method of solution is different for the buried and protruding cases. In the buried case the well-known pole removal technique can be applied to a certain meromorphic function (denoted k11), but in the protruding case the corresponding function k22 is no longer meromorphic. Progress is made, however, by using a Padé representation of k22 to yield a meromorphic problem which can then be solved using the pole removal technique as before. A range of results is presented, for both buried and protruding systems and with and without mean flow, and it becomes clear that the stagger of the two open ends can have a very significant effect on the far-field noise. We also obtain reasonable agreement between our predictions and some experimental results. One particular noise mechanism we identify in the presence of mean shear is the way in which a Kelvin–Helmholtz instability mode launched from the upstream trailing edge can be scattered into sound by its interaction with the downstream edge, provided that the separation between the edges is sufficiently large in a way which we identify.
    Journal of Fluid Mechanics 10/2008; 613:275 - 307. · 2.29 Impact Factor

Full-text

Download
1 Download
Available from