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 CMFD, 2006, Volume 17, Pages 78–87 (Mi cmfd58)

Smooth solutions to some differential-difference equations of neutral type

V. B. Cherepennikov, P. G. Ermolaeva

Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: The paper is devoted to the scalar linear differential-difference equation of neutral type
$$dx(t)/dt+p(t)dx(t-1)/dt=a(t)x(t-1)+b(t)x(t)+f(t).$$
We study the existence and methods for finding solutions possessing required smoothness on intervals of length greater than 1.
The following two settings are considered:
(1) To find an initial function $g(t)$ defined on the initial set $t\in[t_0-1,t_4]$ such that the continuous solution $x(t)$, $t>t_0$, generated by $g(t)$ possesses required smoothness at the points divisible by the delay time. For the investigation, we apply the inverse initial-value problem method.
(2) Let $a(t), b(t), p(t),$ and $f(t)$ be polynomials and let the initial value $x(0)=x_0$ be assigned at the initial point $t=0$. Polynomials satisfying the initial-value condition are considered as quasi-solutions to the original equation. After substitution of a polynomial of degree $N$ for $x(t)$ in the original equation, there appears a residual $\Delta(t)=O(t^N)$, for which sharp estimates are obtained by the method of polynomial quasi-solutions. Since polynomial quasi-solutions may contain free parameters, the problem of minimization of the residual on some interval can be considered on the basis of variational criteria.

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English version:
Journal of Mathematical Sciences, 2008, 149:6, 1648–1657

Bibliographic databases:

UDC: 517.929

Citation: V. B. Cherepennikov, P. G. Ermolaeva, “Smooth solutions to some differential-difference equations of neutral type”, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, CMFD, 17, PFUR, M., 2006, 78–87; Journal of Mathematical Sciences, 149:6 (2008), 1648–1657

Citation in format AMSBIB
\Bibitem{CheErm06} \by V.~B.~Cherepennikov, P.~G.~Ermolaeva \paper Smooth solutions to some differential-difference equations of neutral type \inbook Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14--21, 2005). Part~3 \serial CMFD \yr 2006 \vol 17 \pages 78--87 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd58} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2336460} \transl \jour Journal of Mathematical Sciences \yr 2008 \vol 149 \issue 6 \pages 1648--1657 \crossref{https://doi.org/10.1007/s10958-008-0087-z} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-40549135938}