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Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Author

Listed:
  • Vsevolod G. Sorokin

    (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, Bldg 1, 119526 Moscow, Russia)

  • Andrei V. Vyazmin

    (Chemical Engineering and Biotechnology Department, Moscow Polytechnic University, 38 Bolshaya Semenovskaya St., 107023 Moscow, Russia
    The Lomonosov Institute of Fine Chemical Technologies, MIREA—Russian Technological University, 86 Vernadsky Avenue, 119571 Moscow, Russia)

Abstract

The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems.

Suggested Citation

  • Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1886-:d:828948
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    References listed on IDEAS

    as
    1. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    2. Gilberto González-Parra & Sharmin Sultana & Abraham J. Arenas, 2022. "Mathematical Modeling of Toxoplasmosis Considering a Time Delay in the Infectivity of Oocysts," Mathematics, MDPI, vol. 10(3), pages 1-20, January.
    3. Noé Chan Chí & Eric ÁvilaVales & Gerardo García Almeida, 2012. "Analysis of a HBV Model with Diffusion and Time Delay," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-25, November.
    4. Andrei D. Polyanin & Vsevolod G. Sorokin, 2021. "Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy," Mathematics, MDPI, vol. 9(5), pages 1-22, March.
    5. Wei Yang, 2021. "Modeling COVID-19 Pandemic with Hierarchical Quarantine and Time Delay," Dynamic Games and Applications, Springer, vol. 11(4), pages 892-914, December.
    6. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    7. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    8. Xuebing Zhang & Honglan Zhu, 2019. "Hopf Bifurcation and Chaos of a Delayed Finance System," Complexity, Hindawi, vol. 2019, pages 1-18, January.
    9. De Cesare, Luigi & Sportelli, Mario, 2005. "A dynamic IS-LM model with delayed taxation revenues," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 233-244.
    10. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    11. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    12. Polyanin, Andrei D., 2019. "Functional separable solutions of nonlinear reaction–diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 282-292.
    13. Bai, Zhenguo & Wu, Shi-Liang, 2015. "Traveling waves in a delayed SIR epidemic model with nonlinear incidence," Applied Mathematics and Computation, Elsevier, vol. 263(C), pages 221-232.
    14. Jia, Yunfeng, 2020. "Bifurcation and pattern formation of a tumor–immune model with time-delay and diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 92-108.
    15. Lekomtsev, Andrey & Pimenov, Vladimir, 2015. "Convergence of the scheme with weights for the numerical solution of a heat conduction equation with delay for the case of variable coefficient of heat conductivity," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 83-93.
    16. Ardak Kashkynbayev & Fathalla A. Rihan, 2021. "Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay," Mathematics, MDPI, vol. 9(15), pages 1-16, August.
    17. Li, Wan-Tong & Yan, Xiang-Ping & Zhang, Cun-Hua, 2008. "Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 227-237.
    18. Dmitry Lukyanenko & Tatyana Yeleskina & Igor Prigorniy & Temur Isaev & Andrey Borzunov & Maxim Shishlenin, 2021. "Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay," Mathematics, MDPI, vol. 9(4), pages 1-12, February.
    19. Li, Jing & Sun, Gui-Quan & Jin, Zhen, 2014. "Pattern formation of an epidemic model with time delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 403(C), pages 100-109.
    20. Lu, Jun Guo, 2008. "Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 116-125.
    21. Bellen, Alfredo & Zennaro, Marino, 2013. "Numerical Methods for Delay Differential Equations," OUP Catalogue, Oxford University Press, number 9780199671373.
    22. Zhu, Cheng-Cheng & Zhu, Jiang, 2021. "Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    23. Liu, Pan-Ping, 2015. "Periodic solutions in an epidemic model with diffusion and delay," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 275-291.
    24. Andrei D. Polyanin & Alexei I. Zhurov, 2022. "Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions," Mathematics, MDPI, vol. 10(9), pages 1-28, May.
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    Cited by:

    1. Alexandra Kashchenko & Sergey Kashchenko, 2023. "Relaxation Oscillations in the Logistic Equation with Delay and Modified Nonlinearity," Mathematics, MDPI, vol. 11(7), pages 1-18, April.
    2. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay," Mathematics, MDPI, vol. 11(14), pages 1-19, July.
    3. Andrei D. Polyanin & Alexander V. Aksenov, 2024. "Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions," Mathematics, MDPI, vol. 12(13), pages 1-29, July.
    4. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays," Mathematics, MDPI, vol. 11(3), pages 1-25, January.

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