Application of Wavelet Transform to Urysohn-Type Equations

Abstract

This paper deals with convolution-type Urysohn equations of the first kind. Finding a solution for such equations is an ill-posed problem. For it to be solved, regularization algorithms and the continuous wavelet transform are used. Similar to the Fourier transform, the continuous wavelet transform is applied to convolution-type equations (based on the Fourier and wavelet transforms) and to Urysohn equations with unknown shift. The wavelet transform is preferable for the cases with approximated right-hand sides and for type 1 equations. We demonstrated that the application of the wavelet transform to Urysohn-type equations with unknown shift translates into a solution of a nonlinear equation with an oscillating kernel. Depending on the availability of a priori information, a combination of regularization and iterative algorithms with the use of close equations are effective for solving convolution-type equations based on the continuous wavelet transform and Urysohn equation.