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On One Convolution Equation in the Theory of Filtration of Random Processes

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Ukrainian Mathematical Journal Aims and scope

We study the problems of analytic theory and the numerical-analytic solution of the integral convolution equation of the second kind

$$ \begin{array}{cc}\hfill {\varepsilon}^2f(x)+{\displaystyle \underset{0}{\overset{r}{\int }}K\left(x-t\right)f(t)dt=g(x),}\hfill & \hfill x\in \left[0,r\right)\hfill \end{array}, $$

where

$$ \begin{array}{cccc}\hfill \varepsilon >0,\hfill & \hfill r\le \infty, \hfill & \hfill K\in {L}_1\left(-\infty, \infty \right),\hfill & \hfill K(x)={\displaystyle \underset{a}{\overset{b}{\int }}{e}^{-\left|x\right|s}d\sigma (s)\ge 0.}\hfill \end{array} $$

The factorization approach is used and developed. The key role in this approach is played by the V. Ambartsumyan nonlinear equation.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 8, pp. 1092–1105, August, 2014.

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Engibaryan, N.B., Barsegyan, A.G. On One Convolution Equation in the Theory of Filtration of Random Processes. Ukr Math J 66, 1220–1235 (2015). https://doi.org/10.1007/s11253-015-1004-5

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  • DOI: https://doi.org/10.1007/s11253-015-1004-5

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