Generalised Dirichelt-to-Neumann map in time dependent domains

Pelloni, B. and Fokas, A. S. (2012) Generalised Dirichelt-to-Neumann map in time dependent domains. Studies in Applied Mathematics, 129 (1). pp. 51-90. ISSN 0022-2526 doi: 10.1111/j.1467-9590.2011.00545.x

Abstract/Summary

We study the heat, linear Schrodinger and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/19700
Item Type	Article
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Massachusetts Institute of Technology
Publisher Statement	This is a preprint of an Article accepted for publication in Studies in Applied Mathematics © 2012 Massachusetts Institute of Technology.
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