8. The Dirichlet to Neumann Map for a Moving Boundary
Society for Industrial and Applied Mathematics eBooks · 2008
It was shown in section 1.4 that the characterization of the Dirichlet to Neumann map for the heat equation on the half-line is based on the analysis of the global relation and on the inversion of the following integral: f ^ (k) = ∫ 0 T e k2 s f (s) ds, T>0, k∈ℂ. 8.1 It turns out that the characterization of the analogous map for the heat equation on the moving boundary {l(t) 0, k∈ℂ. 8.2 The integral (8.1) can be inverted in an elementary manner using the Fourier transform. Alternatively, it can be inverted using the spectral analysis of the following eigenvalue equation for the function μ(t, k) (compare with Example 6.2 of Chapter 6): μt + k2 μ=kf (t) , 0<t<T, k∈ℂ. 8.3 The integral (8.2) apparently cannot be inverted using the Fourier transform. However, it can be inverted using the spectral analysis of the following eigenvalue equation for the function μ(t, k): μt + ( k2 −ik l ˙ (t) ) μ=kf (t) , 0<t