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

  • A. G. Barsegyan
  • N. B. Engibaryan

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.
Published
25.08.2014
How to Cite
BarsegyanA. G., and EngibaryanN. B. “On One Convolution Equation in the Theory of Filtration of Random Processes”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, no. 8, Aug. 2014, pp. 1092–1105, http://umj.imath.kiev.ua/index.php/umj/article/view/2201.
Section
Research articles