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Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

Author

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  • 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
    Department of Applied Mathematics, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia)

Abstract

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:90-:d:305689
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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. Zhang, Shun-Li & Lou, S.Y., 2004. "Derivative-dependent functional separable solutions for the KdV-type equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(3), pages 430-444.
    3. Bradshaw-Hajek, B.H. & Moitsheki, R.J., 2015. "Symmetry solutions for reaction–diffusion equations with spatially dependent diffusivity," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 30-38.
    4. 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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    Citations

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    Cited by:

    1. 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.
    2. 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.

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