### Refine

#### Keywords

Piezoelectric filters are used in telecommunication to filter electrical signals. This report deals with the problem of calculating passing and damped frequency intervals for a filter with given geometrical configurations and materials. Only periodic filters, which are widely used in practice, were considered. These filters consist of periodically arranged cells. For a small amount of cells a numerical procedure to visualise the wave propagation in the filter was developed. For a big number of cells another model of the filter was obtained. In this model it is assumed that the filter occupies an infinite domain. This leads to a differential equation, with periodic coefficients, that describes propagation of the wave with a given frequency in the filter. To analyse this equation the Spectral Theory for Periodic Operators had to be employed. Different ways -- analytical and numerical -- to apply the theory were proposed and analysed.

Consider a cooling process described by a nonlinear heat equation. We are interested to recover the initial temperature from temperature measurements which are available on a part of the boundary for some time. Up to now even for the linear heat equation such a problem has been usually studied as a nonlinear ill-posed operator equation, and regularization methods involving Frechet derivatives have been applied. We propose a fast derivative-free iterative method. Numerical results are presented for the glass cooling process, where nonlinearity appears due to radiation.