Numerical Analysis and Applications

, Volume 6, Issue 3, pp 236–246 | Cite as

Numerical analytical method of studying some linear functional differential equations

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Abstract

This paper presents the results of studying a scalar linear functional differential equation of a delay type (t) = a(t)x(t − 1) + b(t)x(t/q) + f(t), q > 1. Primary attention is given to the original problem with the initial point, when the initial condition is specified at the initial point, and the classical solution, whose substitution into the original equation transforms it into an identity, is sought. The method of polynomial quasi-solutions, based on representation of an unknown function x(t) as a polynomial of degree N, is applied as the method of investigation. Substitution of this function into the original equation yields a residual Δ(t) = O(t N ), for which an accurate analytical representation is obtained. In this case, the polynomial quasi-solution is understood as an exact solution in the form of a polynomial of degree N, disturbed because of the residual of the original initial problem. Theorems of existence of polynomial quasi-solutions for the considered linear functional differential equation and exact polynomial solutions have been proved. Results of a numerical experiment are presented.

Keywords

functional differential equations initial-value problem exact solutions polynomial quasi-solutions 

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References

  1. 1.
    Bernoulli, J., Meditationes. Dechordis vibrantibis …, in Commentarial Academia Scientiarum Imperialis Petropolitanae, Coll. Work, 1728, vol. 3, pp. 198–221.Google Scholar
  2. 2.
    Myshkis, A.D., Lineinye differentsial’nye uravneniya s zapazdyvayushchim argumentom (Linear Differential Equations with a Delay Argument), Moscow: Gostekhizdat, 1951.Google Scholar
  3. 3.
    Bellman, R. and Cooke, K.L., Differential-Difference Equations, New York: Academic Press, 1963.MATHGoogle Scholar
  4. 4.
    Azbelev, N.V., Maksimov, V.P., and Rakhmatullina, L.F., Vvedenie v teoriyu funktsional’nodifferentsial’nykh uravnenii (Introduction to the Theory of Functional Differential Equations), Moscow: Nauka, 1991.Google Scholar
  5. 5.
    Cherepennikov, V.B., Polynomial Quasi-Solutions of Linear Systems of Differential-Difference Equations, Izv. Vuz., Ser. Mat., 1999, no. 10, pp. 49–58.Google Scholar
  6. 6.
    Cherepennikov, V.B. and Ermolaeva, P.G., Polynomial Quasisolutions of Linear Differential Difference Equations, Opuscula Math., 2006, vol. 26, no. 3, pp. 431–443.MathSciNetMATHGoogle Scholar
  7. 7.
    Cherepennikov, V.B. and Ermolaeva, P.G., Smooth Solutions to the Initial-Value Problem for Some Differential-Difference Equations, Sib. Zh. Vych. Mat., 2010, vol. 13, no. 2, pp. 213–226.MATHGoogle Scholar
  8. 8.
    Cherepennikov, V.B., On Analytical Solutions of Some Systems of Functional Differential Equations, Diff. Urav., 1990, vol. 26, no. 6, pp. 1094–1095.Google Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Melentiev Energy Systems Institute, Siberian BranchRussian Academy of SciencesIrkutskRussia

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