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The Fredholm factorization technique of Generalized Wiener-Hopf Equations in Wave scattering problems

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Recently, we have proposed the Generalized Wiener-Hopf Technique (GWHT) that is a novel and effective spectral technique to solve scattering problems constituted of isolated impenetrable and penetrable wedges. The Wiener-Hopf (WH) method is a well-established technique to solve problems in all branches of engineering, mathematical physics and applied mathematics. In our opinion, the GWHT together with the SM technique and the methods based on the KL transform completes the spectral techniques capable to handle isolated wedge problems.

Recently, the GWHT is able to further extend the class of solvable problems, in particular for the capability to handle complex scattering problems constituted of angular and rectangular/layer shapes.

The GWHT can now easily formulate complex scattering problems in terms of Generalized Wiener-Hopf equations (GWHEs). Although, in general, the relevant GWH Es of the problems cannot be solved in closed form, this limit has been successfully overcome by resorting to the Fredholm Factorization. The Fredholm factorization is a semi-analytical method that provides very accurate approximate solutions of GWH Es of a given problem. Its efficiency is based on the reduction of the classical factorization problem to system of Fredholm integral equations of second kind, by eliminating some of the WH unknowns via contour integration. The benefit of the semi-analytical solution is that the solution can be analyzed in terms of field components via inverse spectral transformation and asymptotics.

The application of the GWHT consists of four steps:

  1. Deduction of GWH Es in spectral domain possibly with the help of equivalent network modelling,
  2. Approximate solution via Fredholm factorization,
  3. Analytic continuation of the approximate solution,
  4. Evaluation of field with physical interpretation.

In steps 1-2 network modelling orders and systematizes the procedure to obtain the spectral equations for complex problems avoiding redundancy. Moreover, in practice, steps 2 and 3 substitute the fundamental procedure of the classical WH technique, i.e. 1) the factorization of the kernel, 2) the computation of solution via decomposition and 3) the application of Liouville’s Theorem.

This talk is part of the Waves Group (DAMTP) series.

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