Advertisement

Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 786–789 | Cite as

Smooth Solutions to Some Differential-Difference Equations

  • V. B. Cherepennikov
Article
  • 7 Downloads

Abstract

In this paper, we consider a scalar linear differential-difference equation (LDDE) of neutral type \( \overset{\cdot }{x} \)(t) + p(t)\( \overset{\cdot }{x} \)(t − 1) = a(t)x(t − 1) + f(t). We examine the initial-value problem with an initial function in the case where the initial condition is given on an initial set. We use the method of polynomial quasisolutions based on the representation of the unknown function x(t) in the form of a polynomial of degree N. Substituting this function in the original equation we obtain the discrepancy Δ(t) = O(t N ), for which an exact analytic representation is obtained. We prove that if a polynomial quasisolution of degree N is taken as an initial function, then the smoothness of the solution generated by this initial functions at connection points is no less than N.

Keywords and phrases

differential-difference equation initial-value problem with an initial function polynomial quasisolution smooth solution 

AMS Subject Classification

34K15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. D. Myshkis, Linear Differential Equations with Delayed Arguments [in Russian], Gostekhizdat, Moscow–Leningrad (1951).Google Scholar
  2. 2.
    V. Cherepennikov, Numerical and Analytical Methods of the Study of Linear Functional-Differential Equations [in Russian], Novosibirsk (2013).Google Scholar
  3. 3.
    V. Cherepennikov and P. Ermolaeva, “Polynomial quasisolutions of linear differential-difference equations,” Opuscula Math., 26, No. 3, 431–443 (2006).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Melentiev Energy System InstituteSiberian Branch of the RASIrkutskRussia

Personalised recommendations