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The one-dimensional wave equation with general boundary conditions

Abstract

We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions α j u′(j) + β j u(j) + γ 1–j u(1 − j) = 0, (j = 0, 1) generates a cosine family on L p(0, 1) for every \({p\,{\in}\,[1,\infty)}\). Here α j , β j and γ j are complex numbers satisfying α 0, α 1 ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.

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Correspondence to Mahamadi Warma.

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Alvarez-Pardo, E., Warma, M. The one-dimensional wave equation with general boundary conditions. Arch. Math. 96, 177–186 (2011). https://doi.org/10.1007/s00013-010-0209-y

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Mathematics Subject Classification (2000)

  • 35L05
  • 35L90
  • 47D06
  • 47D09

Keywords

  • Wave equation
  • Cosine and sine families
  • Nonlocal Robin boundary conditions