IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i11p1038-d283102.html
   My bibliography  Save this article

Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems

Author

Listed:
  • María Ángeles Castro

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

  • Miguel Antonio García

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

  • José Antonio Martín

    (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

In recent works, exact and nonstandard finite difference schemes for scalar first order linear delay differential equations have been proposed. The aim of the present work is to extend these previous results to systems of coupled delay differential equations X ′ ( t ) = A X ( t ) + B X ( t − τ ) , where X is a vector, and A and B are commuting real matrices, in general not simultaneously diagonalizable. Based on a constructive expression for the exact solution of the vector equation, an exact scheme is obtained, and different nonstandard numerical schemes of increasing order are proposed. Dynamic consistency properties of the new nonstandard schemes are illustrated with numerical examples, and proved for a class of methods.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1038-:d:283102
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/11/1038/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/11/1038/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jódar, Lucas & Villanueva, Rafael J. & Arenas, Abraham J. & González, Gilberto C., 2008. "Nonstandard numerical methods for a mathematical model for influenza disease," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 622-633.
    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.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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).
    2. 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.
    3. Vasily E. Tarasov, 2024. "Exact Finite-Difference Calculus: Beyond Set of Entire Functions," Mathematics, MDPI, vol. 12(7), pages 1-37, March.
    4. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Abraham J. Arenas & Gilberto González-Parra & Jhon J. Naranjo & Myladis Cogollo & Nicolás De La Espriella, 2021. "Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay," Mathematics, MDPI, vol. 9(3), pages 1-21, January.
    2. 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.
    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. Tan, Zengqiang & Zhang, Chengjian, 2022. "Numerical approximation to semi-linear stiff neutral equations via implicit–explicit general linear methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 68-87.
    5. Eriqat, Tareq & El-Ajou, Ahmad & Oqielat, Moa'ath N. & Al-Zhour, Zeyad & Momani, Shaher, 2020. "A New Attractive Analytic Approach for Solutions of Linear and Nonlinear Neutral Fractional Pantograph Equations," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    6. Qin, Hongyu & Zhang, Qifeng & Wan, Shaohua, 2019. "The continuous Galerkin finite element methods for linear neutral delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 76-85.
    7. Qin, Tingting & Zhang, Chengjian, 2015. "Stable solutions of one-leg methods for a class of nonlinear functional-integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 47-57.
    8. Wang, Qi, 2015. "Numerical oscillation of neutral logistic delay differential equation," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 49-59.
    9. Posch, Olaf & Trimborn, Timo, 2013. "Numerical solution of dynamic equilibrium models under Poisson uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2602-2622.
    10. Xu, Y. & Zhao, J.J., 2008. "Stability of Runge–Kutta methods for neutral delay-integro-differential-algebraic system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 571-583.
    11. Hoang, Manh Tuan, 2022. "Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 32-56.
    12. Amat, Sergio & José Legaz, M. & Pedregal, Pablo, 2015. "A variable step-size implementation of a variational method for stiff differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 118(C), pages 49-57.
    13. Cheng, Xue & Chen, Zhong & Zhang, Qingpu, 2015. "An approximate solution for a neutral functional–differential equation with proportional delays," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 27-34.
    14. Jornet, Marc, 2021. "Exact solution to a multidimensional wave equation with delay," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    15. Zhang, Chengjian & Chen, Hao, 2010. "Asymptotic stability of block boundary value methods for delay differential-algebraic equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(1), pages 100-108.
    16. M. Motawi Khashan & Rohul Amin & Muhammed I. Syam, 2019. "A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet," Mathematics, MDPI, vol. 7(6), pages 1-12, June.
    17. 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).
    18. Olaf Posch & Timo Trimborn, 2010. "Numerical solution of continuous-time DSGE models under Poisson uncertainty," Economics Working Papers 2010-08, Department of Economics and Business Economics, Aarhus University.
    19. Tan, Zengqiang & Zhang, Chengjian, 2018. "Implicit-explicit one-leg methods for nonlinear stiff neutral equations," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 196-210.
    20. Acedo, L. & González-Parra, Gilberto & Arenas, Abraham J., 2010. "Modal series solution for an epidemic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(5), pages 1151-1157.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1038-:d:283102. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.