Abstract
We revisit the Helmholtz equation in a quarter-plane in the framework of the Riemann–Hilbert approach to linear boundary value problems suggested in late 1990s by A. Fokas. We show the role of the Sommerfeld radiation condition in Fokas’ scheme.
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Notes
Although it would take some extra work, one can show that the needed properties of the solution follow from the setting of problem (2.1)–(2.4). Hence, to arrive to our final formulae one only needs to assume the existence of the solution. This, in fact, would automatically mean the uniqueness of the solution of the boundary value problem we are studying.
Strictly speaking, we have to be a little bit careful with the behavior of the integrand near \(\zeta = 0\), but the presence of the exponential factor \(e^{E(\zeta )}\) takes care of the convergence of the corresponding integrals.
References
Fokas, A.S.: Integrability: from d’Alembert to Lax. In: Proceedings of the Conference in Honor of the 70th Birthdays of Peter D. Lax and Louis Nirenberg Held in Venice, 1996. Recent Advances in Partial Differential Equations, Venice 1996, Proceedings of Symposia in Applied Mathematics, vol. 54, pp. 131–161. American Mathematical Society, Providence, RI (1998)
Fokas, A.S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. Lond. Ser. A 453, 1411–1443 (1997)
Fokas, A.S.: Lax pairs and a new spectral method for linear and integrable nonlinear PDEs. Selecta Math. (N.S.) 4(1), 31–68 (1998)
Fokas, A.S.: A Unified Approach to Boundary Value Problems. CBMS-NSF Regional Conference Series in Applied Mathematics 78. SIAM, Philadelphia, PA (2008)
Fokas, A.S.: Two-dimensional linear partial differential equations in a convex polygon. R. Soc. Lord. Proc. Ser. A Math. Phys. Eng. Sci. 457(2006), 371–393 (2001)
Its, E.: Lax pair and the Riemann–Hilbert Method for solving diffraction and scattering problems in geophysics. In: Proceedings of the SAGEEP (2001)
Its, E.: Riemann–Hilbert approach to the elastodynamic equation in a quarter-space, Part I. Pr07-05 www.math.iupui.edu
Its, A., Its, E., Kaplunov, J.: Riemann–Hilbert approach to the elastodynamic equation. Part I. Lett. Math. Phys. 96(1–3), 53–83 (2011)
Lyalinov, M., Zhu, N.Y.: Scattering of Wedges and Cones with Impedance Boundary Conditions. Mario Boella Series on Electromagnetism in Information & Communication. SciTech-IET, Edison, NJ (2012)
Spence, E.A.: Boundary value problems for linear elliptic PDEs. Ph.D. thesis, University of Cambridge (2010)
Smitheman, S.A., Spence, E.A., Fokas, A.S.: A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon. IMA J. Numer. Anal. 30(4), 1184–1205 (2010)
Acknowledgements
This work was partially supported by the National Science Foundation (NSF) under Grant No. DMS-0203104, by the Russian Science Foundation Grant No.17-11-01126, and by a Grant of the London Mathematical Society. The authors are grateful to M. Lyalinov for very useful discussions.
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Its, A., Its, E. The Riemann–Hilbert approach to the Helmholtz equation in a quarter-plane: Neumann, Robin and Dirichlet boundary conditions. Lett Math Phys 108, 1109–1135 (2018). https://doi.org/10.1007/s11005-017-1030-3
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DOI: https://doi.org/10.1007/s11005-017-1030-3