Wavelet Analysis in Solving the Cauchy Problem for the Wave Equation in Three-Dimensional Space

Abstract

We propose a new method for solving the Cauchy problem for the wave equation in three dimensional space. The method is based on continuous waveletanalysis. We show that the exact non-stationary solution of the wave equation with finite energy found in [1] at any fixed moment of time should be regarded as a mother wavelet. This solution was named in [1] as a “Gaussian wave packet”. It is a new three-dimensional axially symmetric wavelet which is given by a simple explicit formula as well as its Fourier transformation. This wavelet has an infinite number of vanishing moments. It is a smooth function, i.e. it has derivatives of any order with respect to spatial coordinates and time. We show that using the wavelet decomposition of the initial data we can find the exact formula for the solution of the Cauchy problem as a linear superposition of “Gaussian packets”.