A Generalization of the Wiener–Hopf Method for an Equation in Two Variables with Three Unknown Functions
Abstract.
This manuscript presents an analytic solution to a generalization of the Wiener–Hopf equation in two variables and with three unknown functions. This equation arises in many applications, for example, when solving the discrete Helmholtz equation associated with scattering on a domain with perpendicular boundary. The traditional Wiener–Hopf method is suitable for problems involving boundary data on co-linear semi-infinite intervals, not for boundaries at an angle. This significant extension will enable the analytical solution to a new class of problems with more boundary configurations. Progress is made by defining an underlining manifold that links the two variables. This allows one to meromorphically continue the unknown functions on this manifold and formulate a jump condition. As a result the problem is fully solvable in terms of Cauchy-type integrals, which is surprising since this is not always possible for this type of functional equation.
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Acknowledgments.
Useful discussions with Gennady Mishuris, Michael Nieves, Andrey Shanin, and Andrey Korolkov have helped to shape this work.
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© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 29 March 2023
Accepted: 27 November 2023
Published online: 19 March 2024
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Funding Information
Royal Society Dorothy Hodgkin Research Fellowship
Dame Kathleen Ollerenshaw Fellowship
Isaac Newton Institute for Mathematical Sciences (INI): EP/R014604/1
Funding: The author is supported by a Royal Society Dorothy Hodgkin Research Fellowship and a Dame Kathleen Ollerenshaw Fellowship. This research has also been partly funded by EPSRC grant EP/W018381/1. The author would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program “Mathematical Theory and Applications of Multiple Wave Scattering” when some work on this paper was undertaken (EP/R014604/1).
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