2019
Том 71
№ 6

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

Abstract

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.

English version (Springer): Ukrainian Mathematical Journal 66 (2014), no. 8, pp 1220-1235.

Citation Example: Barsegyan A. G., Engibaryan N. B. On One Convolution Equation in the Theory of Filtration of Random Processes // Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1092–1105.

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