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Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Author

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  • Alexander V. Aksenov

    (Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 1 Leninskiye Gory, Main Building, 119234 Moscow, Russia
    Department of Applied Mathematics, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia
    Keldysh Institute of Applied Mathematics RAS, Miusskaya Square, 125047 Moscow, Russia)

  • Andrei D. Polyanin

    (Ishlinsky Institute for Problems in Mechanics RAS, 101 Vernadsky Avenue, Bldg 1, 119526 Moscow, Russia
    Department of Applied Mathematics, Bauman Moscow State Technical University, 5 Second Baumanskaya Street, 105005 Moscow, Russia
    Chemical Engineering and Biotechnology Department, Moscow Polytechnic University, 38 Bolshaya Semenovskaya st., 107023 Moscow, Russia)

Abstract

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u = u ( x , t ) , these equations contain the same function at a past time, w = u ( x , t − τ ) , where τ > 0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u = u ( x , t ) , also contain the same functions with dilated or contracted arguments, w = u ( p x , q t ) , where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

Suggested Citation

  • 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.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:345-:d:496305
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    References listed on IDEAS

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    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. 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.
    3. 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.
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    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.
    2. 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.
    3. Cristian Ghiu & Constantin Udriste, 2022. "Solutions for Multitime Reaction–Diffusion PDE," Mathematics, MDPI, vol. 10(19), pages 1-12, October.
    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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