Provider: Ingenta Connect
Database: Ingenta Connect
Content: application/x-research-info-systems
TY - ABST
AU - Fokas, A. S.
AU - Pelloni, B.
TI - Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains
JO - Studies in Applied Mathematics
PY - 2012-07-01T00:00:00///
VL - 129
IS - 1
SP - 51
EP - 90
N2 - We study the heat, linear SchrÃ¶dinger (LS), 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.
UR - https://www.ingentaconnect.com/content/bpl/sapm/2012/00000129/00000001/art00003
M3 - doi:10.1111/j.1467-9590.2011.00545.x
UR - https://doi.org/10.1111/j.1467-9590.2011.00545.x
ER -