Advertisement

Journal of Mathematical Sciences

, Volume 149, Issue 6, pp 1648–1657 | Cite as

Smooth solutions to some differential-difference equations of neutral type

  • V. B. Cherepennikov
  • P. G. Ermolaeva
Article
  • 17 Downloads

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 of 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 ∈ [t 0 − 1, t 4] such that the continuous solution x(t), t > t 0, generated by g(t) possesses the required smoothness at 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 Δ(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.

Keywords

Linear System Smooth Solution Initial Function Exact Analytical Solution Neutral Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations [in Russian], Nauka, Moscow (1991).MATHGoogle Scholar
  2. 2.
    R. Bellman and K. Cooke, Differential-Difference Equations, Academic Press, London-New York (1963).MATHGoogle Scholar
  3. 3.
    V. B. Cherepennikov, “Polynomial quasi-solutions to linear systems of differential-difference equations,” Russian Math. (Iz. VUZ), 43, No. 10, 46–55 (2000).MathSciNetGoogle Scholar
  4. 4.
    V. B. Cherepennikov, “The inverse initial-value problem for some linear systems of differential-difference equations,” Russian Math. (Iz. VUZ), 48, No. 6, 56–68 (2005).MathSciNetGoogle Scholar
  5. 5.
    V. B. Cherepennikov, “Polynomial quasi-solutions of linear differential-difference equations,” Proc. Int. Conf. “CDDE-2005 Conference on Differential and Difference Equations,” Gdansk (2005), to appear.Google Scholar
  6. 6.
    J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York (1977).MATHGoogle Scholar
  7. 7.
    A. D. Myshkis, Linear Differential Equations with Retarded Argument [in Russian], Gostekhizdat, Moscow-Leningrad (1951).MATHGoogle Scholar
  8. 8.
    E. Pinni, Ordinary Differential Difference Equations, Univ. of Calif. Press, Berkeley-Los Angeles (1958).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.The Institute for System Dynamics and Control TheorySiberian branch of the RASIrkutskRussia

Personalised recommendations