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Open AccessArticle

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

1
Ishlinsky Institute for Problems in Mechanics RAS, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia
2
Department of Applied Mathematics, Bauman Moscow State Technical University, 5 Second Baumanskaya Street, 105005 Moscow, Russia
3
Department of Applied Mathematics, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia
Mathematics 2020, 8(1), 90; https://doi.org/10.3390/math8010090
Received: 11 December 2019 / Revised: 28 December 2019 / Accepted: 30 December 2019 / Published: 6 January 2020
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. View Full-Text
Keywords: functional separation of variables; generalized separation of variables; exact solutions; nonlinear reaction-diffusion equations; nonlinear partial differential equations; equations of mathematical physics; splitting principle; nonclassical method of symmetry reductions; invariant surface condition; differential constraints functional separation of variables; generalized separation of variables; exact solutions; nonlinear reaction-diffusion equations; nonlinear partial differential equations; equations of mathematical physics; splitting principle; nonclassical method of symmetry reductions; invariant surface condition; differential constraints
MDPI and ACS Style

Polyanin, A.D. Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations. Mathematics 2020, 8, 90.

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