Talks given at NuHAG events

New exact integral representations of solutions of the wave equation based on continuous wavelet analysis

  Maria Perel
    Department of Mathematical Physics, Physics Faculty, St. Petersburg University

  given at  strobl07 (22.06.07 09:30)
  id:  627
  length:  25min
  status:  accepted
  type:  talk
Solutions of the wave equation with constant coefficients in a three-dimensional space are presented as superpositions of its localized solutions, which we call physical wavelets, following G.Kaiser. The talk consists of two parts.

The first one is based on my works with M.S.Sidorenko [1-4]. The space of square integrable solutions is decomposed into a direct sum of two subspaces. In each subspace, the formalism of continuous wavelet analysis is developed. The choice of the mother physical wavelet is discussed. A family of wavelets is constructed from this wavelet with the help of transformations of dilation, spatial translation and rotation. Next the wavelet transform of an arbitrary solution from the subspace is defined. An isometry and a reconstruction formula are proved.
The comparison of results with the results of G. Kaiser is given.

The second part contains an integral representation for solutions of the boundary-initial value problem for the wave equation in a half-space. Physical wavelets are constructed by dilation, translation and Lorentz transformations from four mother physical wavelets [5].

[1] Perel M.V. and Sidorenko M.S., 2003, Wavelet Analysis in Solving the Cauchy Problem for the Wave Equation in Three-Dimensional Space In:
Mathematical and numerical aspects of wave propagation: Waves 2003, Ed G C Cohen, E Heikkola, P Jolly and P Neittaanmaki (Springer-Verlag) pp 794-798

[2] Perel M.V. and Sidorenko M.S., 2006, Wavelet analysis for the solution of the wave equation, In: Proc. of the Int. Conf. DAYS on DIFFRACTION 2006, Ed. I V Andronov (SPbU), pp 208-217

[3] Perel M.V. and Sidorenko M.S., 2007, New physical wavelet 'Gaussian wave packet', Journal of Physics A: Mathematical and Theoretical 40, pp 3441-3461,

[4] Perel M.V. and Sidorenko M.S., Wavelet-based integral representation for solutions of wave and Klein-Gordon equations, to appear

[5] Perel M.V. Wavelet-analysis in solving the boundary-initial value problem for the wave equation in a half-space, to appear

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