The wavelet transform of exponential growth generalized functions and their application to the solution of the heat equation
The task of the article is to determine and study the wavelet transform for a wider class of functions, in particular, for the exponential growth generalized functions. The object of the research is exponential growth generalized functions defined on a real axis. The subject of the research is the wavelet transform of exponential growth generalized functions. The purpose of this research is to study the properties of the wavelet transform on the space of exponential growth generalized functions. In the introduction, it is noted the information about the results of studying the wavelet transform obtained earlier. In the main part, the wavelet transform is defined on the space of exponential growth generalized functions. Examples of the Laplace transform of the wavelet transform of exponential growth generalized functions are given. It is considered some properties of the Laplace transform of the wavelet transform on the space of exponential growth generalized functions, namely, the Laplace transform of the wavelet transform of the kth-order derivative of the generalized function and the Laplace transform over the wavelet transform of the convolution of two generalized functions. With the uses the first and second order Sobolev wavelet the wavelet transform of the exponential growth generalized function is constructed for the example. The figures show the results of the transform. The schemes are constructed for solving the heat equation with the generalized function on the right-hand side. The efficiency of the obtained schemes is demonstrated by the example of the solution of the Cauchy problem for the heat equation with the Dirac function on the right-hand side. In the conclusion, the results received are described. These results develop the theory of wavelet and find the application in area of theory of the wavelet transform not only, but and theory of the exponential growth generalized functions.