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.
|Number of pages||17|
|Journal||IMA Journal of Applied Mathematics|
|Early online date||7 Oct 2019|
|Publication status||Published - Oct 2019|
- Riemann-Hilbert problem
- initial boundary value problem
- linear evolution PDE
ASJC Scopus subject areas
- Applied Mathematics
FingerprintDive into the research topics of 'Evolution equations on time-dependent intervals'. Together they form a unique fingerprint.
- School of Mathematical & Computer Sciences - Professor
- School of Mathematical & Computer Sciences, Mathematics - Professor
Person: Academic (Research & Teaching)