Equations of convolution type. The Laplace transform
South Ural State University, 454080, Chelyabinsk, Lenin pr. 76, +7-351-267-99-04
We discuss application of the Laplace transform to solving integral equations of convolution type with imprecise input data. 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 the measurement errors form a stationary process with zero mean (the absence of a systematic measurement error) and a given correlation function, we obtain the essential characteristics of the error in the reconstructed signal.
It is demonstrated that numerical implementation of the Laplace method connected with the restoration of the Laplace original from its image, technically significantly complicates the regularization procedure due to the impossibility of using the Mellin- Bromwich inversion formula.