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Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Author

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  • 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

We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u = u ( x , t ) , also contain the same functions with dilated or contracted arguments of the form w = u ( p x , t ) , w = u ( x , q t ) , and w = u ( p x , q t ) , where p and q are the free scaling parameters (for equations with proportional delay we have 0 < p < 1 , 0 < q < 1 ). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:511-:d:508727
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    References listed on IDEAS

    as
    1. Andrei D. Polyanin, 2019. "Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions," Mathematics, MDPI, vol. 7(5), pages 1-19, April.
    2. Andrei D. Polyanin, 2020. "Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations," Mathematics, MDPI, vol. 8(1), pages 1-38, January.
    3. 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.
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    Cited by:

    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 & 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.
    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 & 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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