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On the Construction of Exact Numerical Schemes for Linear Delay Models

Author

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  • Carlos Julio Mayorga

    (Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain
    Department of Mathematics, National Polytechnic School, Quito P.O. Box 17-01-2759, Ecuador)

  • María Ángeles Castro

    (Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain)

  • Antonio Sirvent

    (Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain)

  • Francisco Rodríguez

    (Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain
    Multidisciplinary Institute for Environmental Studies (IMEM), University of Alicante, Apdo. 99, 03080 Alicante, Spain)

Abstract

Exact numerical schemes have previously been obtained for some linear retarded delay differential equations and systems. Those schemes were derived from explicit expressions of the exact solutions, and were expressed in the form of perturbed difference systems, involving the values at previous delay intervals. In this work, we propose to directly obtain expressions of the same type for the fundamental solutions of linear delay differential equations, by considering vector equations with vector components corresponding to delay-lagged values at previous intervals. From these expressions for the fundamental solutions, exact numerical schemes for arbitrary initial functions can be proposed, and they may also facilitate obtaining explicit exact solutions. We apply this approach to obtain an exact numerical scheme for the first order linear neutral equation x ′ ( t ) − γ x ′ ( t − τ ) = α x ( t ) + β x ( t − τ ) , with the general initial condition x ( t ) = φ ( t ) for − τ ≤ t ≤ 0 . The resulting expression reduces to those previously published for the corresponding retarded equations when γ = 0 .

Suggested Citation

  • 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.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1836-:d:1122073
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    References listed on IDEAS

    as
    1. Hoang, Manh Tuan, 2023. "Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 291-314.
    2. Wood, Daniel T. & Kojouharov, Hristo V. & Dimitrov, Dobromir T., 2017. "Universal approaches to approximate biological systems with nonstandard finite difference methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 133(C), pages 337-350.
    3. 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.
    4. 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.
    5. 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.
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