Rocky Mountain Journal of Mathematics

Distributional analysis of radiation conditions for the $3+1$ wave equation

J.A. Ellison, K.A. Heinemann, and S.R. Lau

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Consider the Cauchy problem for the ordinary 3+1 wave equation. Reduction of the spatial domain to a half-space involves an exact radiation boundary condition enforced on a planar boundary. This boundary condition is most easily formulated in terms of the tangential-Fourier and time-Laplace transform of the solution. Using the Schwartz theory of distributions, we examine two other formulations: (i) the nonlocal spacetime form and (ii) its three-dimensional (tangential/time) Fourier transform. The spacetime form features a convolution between two tempered distributions.

Article information

Rocky Mountain J. Math., Volume 49, Number 1 (2019), 1-27.

First available in Project Euclid: 10 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L05: Wave equation 35L50: Initial-boundary value problems for first-order hyperbolic systems 60E05: Distributions: general theory 65M99: None of the above, but in this section

Wave equation radiation boundary conditions initial boundary value problem domain reduction distributions


Ellison, J.A.; Heinemann, K.A.; Lau, S.R. Distributional analysis of radiation conditions for the $3+1$ wave equation. Rocky Mountain J. Math. 49 (2019), no. 1, 1--27. doi:10.1216/RMJ-2019-49-1-1.

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  • M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover Publishing, Inc., New York, 1970.
  • B.B. Baker and E.T. Copson, The mathematical theory of Huygen's principle, Oxford University Press, Oxford, 1953.
  • D.A. Bizzozero, J.A. Ellison, K.A. Heinemann and S.R. Lau, Rapid evaluation of two-dimensional retarded time integrals, J. Comp. Appl. Math. 324 (2017), 118–141, with appendix at$\!$lau.
  • A. Erdélyi, ed., Tables of integral transforms, Volume 2, McGraw-Hill Book Co., Inc., New York, 1954.
  • F.J. Friedlander and M. Joshi, Introduction to the theory of distributions, Cambridge University Press, Cambridge, 1998.
  • T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Num. 8 (1999).
  • T. Hagstrom and T. Warburton, Complete radiation boundary conditions: Minimizing the long time error growth of local methods, SIAM J. Num. Anal. 47 (2009), 3678–3704.
  • T. Hagstrom, T. Warburton and D. Givoli, Radiation boundary conditions for time-dependent waves based on complete plane wave expansions, J. Comp. Appl. Math. 234 (2010), 1988–1995.
  • S.R. Lau, On partial spherical means formulas and radiation boundary conditions for the $3+1$ wave equation, Quart. Appl. Math. 68 (2010), 179–212.
  • E.M. Stein and R. Shakarchi, Real analysis: Measure theory, integration, & Hilbert space, Princeton University Press, Princeton, 2005.
  • V.H. Weston, Factorization of the wave equation in higher dimensions, J. Math. Phys. 28 (1988), 1061–1068.