Equations of Convolution Type with Random Data

Abstract

The possibility of using the Laplace transform to solve integral equations of convolution type with imprecise initial data is being discussed. Theoretically, the possibility of reducing the integral equation to an algebraic equation should greatly simplify the procedure for its solution. However, the measurement errors present in the actual measuring process cause the need to filter the interference in the frequency domain. Assuming that measurement errors can be described with a stationary random process with zero mean (the absence of systematic measurement errors) and a given correlation function, the main characteristics of the error in the signal under regeneration are obtained.

It is shown that technically, numerical implementation of the Laplace method, connected with the restoration of the Laplace original from its image, significantly complicates the procedure of its regularization due to impossibility of using the Mellin–Bromwich inversion formula.