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Applying an iterative method numerically to solve n × n matrix Wiener–Hopf equations with exponential factors

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Priddin, Matthew J 
Kisil, Anastasia V 
Ayton, Lorna J 


This paper presents a generalization of a recent iterative approach to solving a class of 2 × 2 matrix Wiener–Hopf equations involving exponential factors. We extend the method to square matrices of arbitrary dimension n, as arise in mixed boundary value problems with n junctions. To demonstrate the method, we consider the classical problem of scattering a plane wave by a set of collinear plates. The results are compared to other known methods. We describe an effective implementation using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy.



Riemann–Hilbert problem, Wiener–Hopf equations, iterative methods, n-partboundaries, scattering

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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

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The Royal Society


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EPSRC (EP/N509620/1)
Engineering and Physical Sciences Research Council (EP/R014604/1)
Engineering and Physical Sciences Research Council (EP/P015980/1)
Engineering and Physical Sciences Research Council (EP/N509620/1)
EPSRC (1936262)
This work was supported by EPSRC DTP grant no. EP/N509620/1 (M.J.P.), the Sultan Qaboos Research Fellowship at Corpus Christi College at University of Cambridge (A.V.K.) and by EPSRC early career fellowship grant no. EP/P015980/1 (L.J.A.). The authors thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the WHT programme where some work on this paper was undertaken (EPSRC grant no. EP/R014604/1).