Evolution equations on time-dependent intervals

A. S. Fokas, Beatrice Pelloni, Baoqiang Xia

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
43 Downloads (Pure)


We study initial boundary value problems for linear evolution partial differential equations posed on a time-dependent interval $l-1(t)<x<l-2(t)$, $0<t<T$, where $l-1(t)$ and $l-2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterize the unknown boundary values in terms of the given initial and boundary conditions. As illustrative examples we consider the heat equation and the linear Schrödinger equation. In the first case, the unknown Neumann boundary values are expressed in terms of the Dirichlet boundary values and of the initial value through the unique solution of a system of two linear integral equations with explicit kernels. In the second case, a similar result can be proved but only for a more restrictive class of boundary curves.

Original languageEnglish
Pages (from-to)1044-1060
Number of pages17
JournalIMA Journal of Applied Mathematics
Issue number5
Early online date7 Oct 2019
Publication statusPublished - Oct 2019


  • Riemann-Hilbert problem
  • initial boundary value problem
  • linear evolution PDE

ASJC Scopus subject areas

  • Applied Mathematics


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