On exact solutions of the nonlinear heat equation

Authors

  • A.F. Barannyk Institute of Mathematics, Pomeranian University, Słupsk, Poland
  • T.A. Barannyk Poltava V.G. Korolenko National Pedagogical University
  • I.I. Yuryk National University of Food Technologies, Kiev

DOI:

https://doi.org/10.15407/dopovidi2019.05.011

Keywords:

exact solutions, generalized variable separation, group-theoretic methods, nonlinear heat equation

Abstract

A method for construction of exact solutions to the nonlinear heat equation ut = (F (u)ux)x + G (u)ux + H (u), which is based on the ansatz p(x) = ω1(t) φ(u) + ω2(t), is proposed. The function p(x) is a solution of the equation (p′)2 = Ap2 + B, and the functions ω1(t), ω2(t) and ϕ(u) can be found from the condition that this ansatz reduces the nonlinear heat equation to a system of two ordinary differential equations with unknown functions ω1(t) and ω2(t).

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References

Polyanin, A. D. & Zaitsev, V. F. (2004). Handbook of nonlinear partial differential equations. Boca Raton, FL: Chapman & Hall/CRC. doi: https://doi.org/10.1201/9780203489659

Galaktionov, V. A. & Svirshchevskii, S. R. (2007). Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. CRC Press. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Boca Raton, FL: Chapman & Hall/CRC.

Dorodnitsyn, V. A. (1979). Group properties and invariant solutions of an equation of nonlinear heat transport with a source or a sink. Preprint. AS USSR, Institute of Applied Mathematics; No 57. Moscow (in Russian).

Dorodnitsyn, V. A. (1982). On invariant solutions of the equation of non-linear heat conduction with a source. U.S.S.R. Comput. Math. Math. Phys., 22, No. 6, pp. 115-122. doi: https://doi.org/10.1016/0041-5553(82)90102-1

Philip, J. R. (1960). General method of exact solution of the concentration-dependent diffusion equation. Aust. J. Phys., 13, No. 1, pp. 13-20. doi: https://doi.org/10.1071/PH600001

Barannyk, A., Barannyk, T. & Yuryk, I. (2011). Separation of variables for nonlinear equations of hyperbolic and Korteweg–de Vries type. Rep. Math. Phys., 68, pp. 97-105. doi: https://doi.org/10.1016/S0034-4877(11)60029-3

Barannyk, A. F., Barannyk, T. A. & Yuryk, I. I. (2013). Generalized separation of variables for nonlinear equation. Rep. Math. Phys., 71, pp. 1-13. doi: https://doi.org/10.1016/S0034-4877(13)60018-X

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Published

21.04.2024

How to Cite

Barannyk, A., Barannyk, T., & Yuryk, I. (2024). On exact solutions of the nonlinear heat equation . Reports of the National Academy of Sciences of Ukraine, (5), 11–17. https://doi.org/10.15407/dopovidi2019.05.011

Issue

No. 5 (2019): Reports of the National Academy of Sciences of Ukraine

Section

Mathematics

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