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Diffraction by wedges: higher order boundary conditions, integral transforms, vector Riemann-Hilbert problems, and Riemann surfaces

Presented by: 
Yuri Antipov
Friday 1st November 2019 - 13:30 to 14:30
INI Seminar Room 1
Acoustic and electromagnetic diffraction by a wedge is modeled by one and two Helmholtz equations coupled by boundary conditions. When the wedge walls are membranes or elastic plates, the impedance boundary conditions have derivatives of the third or fifth order, respectively. A new method of integral transforms for right-angled wedges is proposed. It is based on application of two Laplace transforms. The main feature of the method is that the second integral transform parameter is a specific root of the characteristic polynomial of the ordinary differential operator resulting from the transformed PDE by the first Laplace transform. For convex domains (concave obstacles), the problems reduce to scalar and order-2 vector Riemann-Hilbert problems. When the wedge is concave (a convex obstacle), the acoustic problem is transformed into an order-3 Riemann-Hilbert problem. The order-2 and 3 vector Riemann-Hilbert problems are solved by recasting them as scalar Riemann-Hilbert problems on Riemann surfaces. Exact solutions of the problems are determined. Existence and uniqueness issues are discussed.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons