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Exact and nonstandard numerical schemes for linear delay differential models

Author

Listed:
  • García, M.A.
  • Castro, M.A.
  • Martín, J.A.
  • Rodríguez, F.

Abstract

Delay differential models present characteristic dynamical properties that should ideally be preserved when computing numerical approximate solutions. In this work, exact numerical schemes for a general linear delay differential model, as well as for the special case of a pure delay model, are obtained. Based on these exact schemes, a family of nonstandard methods, of increasing order of accuracy and simple computational properties, is proposed. Dynamic consistency of the new nonstandard methods are proved, and illustrated with numerical examples, for asymptotic stability, positive preserving properties, and oscillation behaviour.

Suggested Citation

  • García, M.A. & Castro, M.A. & Martín, J.A. & Rodríguez, F., 2018. "Exact and nonstandard numerical schemes for linear delay differential models," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 337-345.
    • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:337-345
    DOI: 10.1016/j.amc.2018.06.029
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    References listed on IDEAS

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    1. Li, Dongfang & Zhang, Chengjian, 2016. "Construction of high-order Runge–Kutta methods which preserve delay-dependent stability of DDEs," Applied Mathematics and Computation, Elsevier, vol. 280(C), pages 168-179.
    2. Garba, S.M. & Gumel, A.B. & Hassan, A.S. & Lubuma, J.M.-S., 2015. "Switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 388-403.
    3. Bellen, Alfredo & Zennaro, Marino, 2003. "Numerical Methods for Delay Differential Equations," OUP Catalogue, Oxford University Press, number 9780198506546.
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    Cited by:

    1. Julia Calatayud & Juan Carlos Cortés & Marc Jornet & Francisco Rodríguez, 2020. "Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay," Mathematics, MDPI, vol. 8(9), pages 1-17, August.
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    3. Carlos Julio Mayorga & María Ángeles Castro & Antonio Sirvent & Francisco Rodríguez, 2023. "On the Construction of Exact Numerical Schemes for Linear Delay Models," Mathematics, MDPI, vol. 11(8), pages 1-9, April.
    4. Kerr, Gilbert & González-Parra, Gilberto & Sherman, Michele, 2022. "A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    5. Sharmin Sultana & Gilberto González-Parra & Abraham J. Arenas, 2023. "Mathematical Modeling of Toxoplasmosis in Cats with Two Time Delays under Environmental Effects," Mathematics, MDPI, vol. 11(16), pages 1-20, August.
    6. María Ángeles Castro & Miguel Antonio García & José Antonio Martín & Francisco Rodríguez, 2019. "Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems," Mathematics, MDPI, vol. 7(11), pages 1-14, November.
    7. Jornet, Marc, 2021. "Exact solution to a multidimensional wave equation with delay," Applied Mathematics and Computation, Elsevier, vol. 409(C).

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