
Smooth solutions to some differentialdifference 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 differentialdifference equation of neutral type
$$
dx(t)/dt+p(t)dx(t1)/dt=a(t)x(t1)+b(t)x(t)+f(t).
$$
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 $t\in[t_01,t_4]$ 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 initialvalue 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 initialvalue condition are considered as quasisolutions 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 quasisolutions. Since polynomial quasisolutions 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 differentialdifference equations of neutral type”, Proceedings of the Fourth International Conference on Differential and FunctionalDifferential 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 differentialdifference equations of neutral type
\inbook Proceedings of the Fourth International Conference on Differential and FunctionalDifferential Equations (Moscow, August 1421, 2005). Part~3
\serial CMFD
\yr 2006
\vol 17
\pages 7887
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd58}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2336460}
\transl
\jour Journal of Mathematical Sciences
\yr 2008
\vol 149
\issue 6
\pages 16481657
\crossref{https://doi.org/10.1007/s109580080087z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2s2.040549135938}
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