Advertisement

Letters in Mathematical Physics

, Volume 108, Issue 4, pp 1109–1135 | Cite as

The Riemann–Hilbert approach to the Helmholtz equation in a quarter-plane: Neumann, Robin and Dirichlet boundary conditions

  • Alexander Its
  • Elizabeth Its
Article

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.

Keywords

Helmholtz equation Riemann–Hilbert problem Lax pair 

Mathematics Subject Classification

35J25 35Q15 31B20 

Notes

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.

References

  1. 1.
    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)Google Scholar
  2. 2.
    Fokas, A.S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. Lond. Ser. A 453, 1411–1443 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fokas, A.S.: A Unified Approach to Boundary Value Problems. CBMS-NSF Regional Conference Series in Applied Mathematics 78. SIAM, Philadelphia, PA (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Its, E.: Lax pair and the Riemann–Hilbert Method for solving diffraction and scattering problems in geophysics. In: Proceedings of the SAGEEP (2001)Google Scholar
  7. 7.
    Its, E.: Riemann–Hilbert approach to the elastodynamic equation in a quarter-space, Part I. Pr07-05 www.math.iupui.edu
  8. 8.
    Its, A., Its, E., Kaplunov, J.: Riemann–Hilbert approach to the elastodynamic equation. Part I. Lett. Math. Phys. 96(1–3), 53–83 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Spence, E.A.: Boundary value problems for linear elliptic PDEs. Ph.D. thesis, University of Cambridge (2010)Google Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndiana University – Purdue University IndianapolisIndianapolisUSA
  2. 2.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations