IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i3p516-d1039771.html
   My bibliography  Save this article

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Author

Listed:
  • Andrei D. Polyanin

    (Ishlinsky Institute for Problems in Mechanics RAS, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia)

  • Vsevolod G. Sorokin

    (Ishlinsky Institute for Problems in Mechanics RAS, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia)

Abstract

The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:516-:d:1039771
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/3/516/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/3/516/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Tang, Changyang & Zhang, Chengjian, 2021. "A fully discrete θ-method for solving semi-linear reaction–diffusion equations with time-variable delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 179(C), pages 48-56.
    2. Lobo, Jervin Zen & Valaulikar, Y.S., 2020. "Group analysis of the one dimensional wave equation with delay," Applied Mathematics and Computation, Elsevier, vol. 378(C).
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. Xuebing Zhang & Honglan Zhu, 2019. "Hopf Bifurcation and Chaos of a Delayed Finance System," Complexity, Hindawi, vol. 2019, pages 1-18, January.
    15. 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.
    16. 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).
    17. Alexander V. Aksenov & Andrei D. Polyanin, 2021. "Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions," Mathematics, MDPI, vol. 9(4), pages 1-31, February.
    18. 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.
    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. 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.

    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. 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.
    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 & 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.
    4. 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.
    5. Zhang, Zizhen & Kundu, Soumen & Tripathi, Jai Prakash & Bugalia, Sarita, 2020. "Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    6. Cristian Ghiu & Constantin Udriste, 2022. "Solutions for Multitime Reaction–Diffusion PDE," Mathematics, MDPI, vol. 10(19), pages 1-12, October.
    7. Yingkang Xie & Zhen Wang & Bo Meng, 2019. "Stability and Bifurcation of a Delayed Time-Fractional Order Business Cycle Model with a General Liquidity Preference Function and Investment Function," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    8. Omran, A.K. & Zaky, M.A. & Hendy, A.S. & Pimenov, V.G., 2022. "An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 218-239.
    9. Xin Ai & Xinyu Liu & Yuting Ding & Han Li, 2022. "Dynamic Analysis of a COVID-19 Vaccination Model with a Positive Feedback Mechanism and Time-Delay," Mathematics, MDPI, vol. 10(9), pages 1-24, May.
    10. Rajpal, Akanksha & Bhatia, Sumit Kaur & Hiremath, Kirankumar R., 2022. "Inspecting the stability of non-linear IS-LM model with dual time delay," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    11. Yunhan Huang & Quanyan Zhu, 2022. "Game-Theoretic Frameworks for Epidemic Spreading and Human Decision-Making: A Review," Dynamic Games and Applications, Springer, vol. 12(1), pages 7-48, March.
    12. Yüzbaşı, Şuayip & Yıldırım, Gamze, 2022. "A collocation method to solve the parabolic-type partial integro-differential equations via Pell–Lucas polynomials," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    13. Chen, Zhang, 2009. "Dynamic analysis of reaction–diffusion Cohen–Grossberg neural networks with varying delay and Robin boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1724-1730.
    14. Alexander V. Aksenov & Andrei D. Polyanin, 2021. "Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions," Mathematics, MDPI, vol. 9(4), pages 1-31, February.
    15. Jian Zhao & Zhenyue Chen & Jingqi Tu & Yunmei Zhao & Yiqun Dong, 2022. "Application of LSTM Approach for Predicting the Fission Swelling Behavior within a CERCER Composite Fuel," Energies, MDPI, vol. 15(23), pages 1-14, November.
    16. M. Syed Ali & Gani Stamov & Ivanka Stamova & Tarek F. Ibrahim & Arafa A. Dawood & Fathea M. Osman Birkea, 2023. "Global Asymptotic Stability and Synchronization of Fractional-Order Reaction–Diffusion Fuzzy BAM Neural Networks with Distributed Delays via Hybrid Feedback Controllers," Mathematics, MDPI, vol. 11(20), pages 1-24, October.
    17. Chen, Wei & Yu, Yongguang & Hai, Xudong & Ren, Guojian, 2022. "Adaptive quasi-synchronization control of heterogeneous fractional-order coupled neural networks with reaction-diffusion," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    18. 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.
    19. Hayrengul Sadik & Abdujelil Abdurahman & Rukeya Tohti, 2023. "Fixed-Time Synchronization of Reaction-Diffusion Fuzzy Neural Networks with Stochastic Perturbations," Mathematics, MDPI, vol. 11(6), pages 1-15, March.
    20. Li, Tingting & Guo, Youming, 2022. "Optimal control and cost-effectiveness analysis of a new COVID-19 model for Omicron strain," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 606(C).

    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:11:y:2023:i:3:p:516-:d:1039771. 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.