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Asymptotic stability analysis of θ-methods for functional differential equations

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Summary

Stability analysis of θ-methods for functional differential equations based on the test equation

$$\begin{gathered} y'(t) = ay(t - \lambda ) + by(t),t > 0 \hfill \\ y(t) = \psi (t),t \in \left[ { - \lambda ,0} \right] \hfill \\ \end{gathered} $$

λ>0, is presented. It is known thaty(t)→0 ast→∞ if and only if |b|<−a and we investigate whether this property is inherited by the numerical solution approximatingy.

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References

  1. Baker, C.T.H., Makroglou, A., Short, E.: Regions of stability in the numerical treatment of Volterra integro-differential equations. SIAM J. Numer. Anal.16, 890–910 (1979)

    Article  Google Scholar 

  2. Barwell, V.K.: Special stability problems for functional differential equations. BIT15, 130–135 (1975)

    Google Scholar 

  3. Bellman, R., Cooke, K.L.: Differential difference equations. New York: Academic Press 1963

    Google Scholar 

  4. Brayton, R.K., Willoughby, R.A.: On the numerical integration of a symmetric system of a difference-differential equations. J. Math. Anal. Appl.18, 182–189 (1967)

    Article  Google Scholar 

  5. Brunner, H., Lambert, J.D.: Stability of numerical methods for Volterra integro-differential equations. Computing12 75–89 (1974)

    Google Scholar 

  6. Cryer, C.W.: A new class of highly stable methods:,A o-stable methods. BIT13, 153–159 (1973)

    Google Scholar 

  7. Cryer, C.W.: Highly stable multistep methods for retarded differential equations. SIAM J. Numer. Anal.11, 788–797 (1974)

    Article  Google Scholar 

  8. Cryer, C.W., Tavernini, L.: The numerical solution of Volterra functional differential equations by Euler's method. SIAM J. Numer. Anal.9, 105–129 (1972)

    Article  Google Scholar 

  9. Fox, L., Mayers, D.F., Ockendon, J.R., Tayler, A.B.: On a functional differential equation. J. Inst. Math. Appl.8, 271–307 (1971)

    Google Scholar 

  10. Guinn, T.: On optimal control problems with variable delay. Manuscript. The University of New Mexico, Albuquerque 1979

    Google Scholar 

  11. Hill, D.R.: A new class of one-step methods for the solution of Volterra functional differential equations. BIT14, 298–305 (1974)

    Google Scholar 

  12. Jackiewicz, Z.: Convergence of multistep methods for Volterra functional-differential equations. Numer. Math.32, 307–332 (1979)

    Google Scholar 

  13. Kato, T., McLeod, J.B.: The functional-differential equationy′(x)=ay(λx)+by(x). Bull. Amer. Math. Soc.77, 891–937 (1971)

    Google Scholar 

  14. Marden, M.: Geometry of polynomials. Providence, Rhode Island: American Mathematical Society 1966

    Google Scholar 

  15. Matthys, J.: A-stable linear multistep methods for Volterra integro-differential equations. Numer. Math.27, 85–94 (1976)

    Google Scholar 

  16. Miller, J.J.H.: On the location of zeros of certain classes of polynomials with application to numerical analysis. J. Inst. Math. Appl.8, 397–406 (1971)

    Google Scholar 

  17. Tavernini, L.: One-step methods for the numerical solution of Volterra functional differential equations. SIAM J. Numer. Anal.8, 786–795 (1971)

    Article  Google Scholar 

  18. Tavernini, L.: Linear multistep methods for the numerical solution of Volterra functional differential equations. Applicable Anal.1, 169–185 (1973)

    Google Scholar 

  19. Tavernini, L.: The approximate solution of Volterra differential systems with state-dependent time lags. SIAM J. Numer. Anal.15, 1039–1052 (1978)

    Article  Google Scholar 

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  1. Department of Mathematics, University of Arkansas, 72701, Fayetteville, AR, USA

    Zdzislaw Jackiewicz

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  1. Zdzislaw Jackiewicz
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Jackiewicz, Z. Asymptotic stability analysis of θ-methods for functional differential equations. Numer. Math. 43, 389–396 (1984). https://doi.org/10.1007/BF01390181

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