The Fredholm Factorization Method Directly Applied to Generalized Wiener-Hopf Equations

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• Guido Lombardi (Politecnico di Torino)
• Thursday 04 July 2024, 11:45-12:15
• Seminar Room 1, Newton Institute.

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WHTW02 - WHT Follow on: the applications, generalisation and implementation of the Wiener-Hopf Method

Authors: Vito Daniele, Guido Lombardi, Politecnico di Torino, Italy, e-mail: vito.daniele@polito.it, guido.lombardi@polito.it   In this work we present a new and comprehensive theory for the solution of Generalized Wiener-Hopf equations (GWHEs). We recall that GWH Es have plus and minus unknowns that are defined into different complex planes but related together. In particular, this kind of Wiener-Hopf equations arise while studying diffraction problems from angular regions, such as in acoustics, electromagnetics and elasticity. The effectiveness of this technique has already been demonstrated in the analysis of electromagnetic scattering from wedge problems immersed in isotropic media resorting to solution methods ranging from closed form factorization to approximate factorization called Fredholm factorization technique. The technique is combined with special complex mapping to transform GWH Es into Classical Wiener-Hopf equations. The same technique has been also extended to mixed types of canonical regions i.e. with rectangular an angular shapes. We believe that this mathematical technique significantly expands the possibilities for spectral analysis of problems involving angular regions filled with complex arbitrary linear media, in particular in electromagentics. We observe that the GWH Es in arbitrary linear media usually report physical unknowns defined into multiple complex planes (more than 2) as the physical problem usually contains more than one propagation constant. Traditional effective spectral methods to study diffraction problem such as Sommerfeld-Malyuzhinets (SM) method and the Kontorovich-Lebedev (KL) takes benefit from the use of spectral complex angular plane derived from the Sommerfeld integral theory, which has also been successfully applied in the Generalized Wiener-Hopf method. However, the definition of this complex plane is possible in problem with one propagation constant. In the present work, we apply for the first times direct Fredholm factorization to GWH Es avoiding introduction of spectral mapping. In particular the method is effective for problems with more than one propagation constant where the spectral mapping cannot be introduced, and other techniques are ineffective. During the presentation we will show the effectiveness of applying the technique and its generality.  

This talk is part of the Isaac Newton Institute Seminar Series series.

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