On exact solutions of the nonlinear heat equation

1Barannyk, AF
2Barannyk, TA
3Yuryk, II
1Institute of Mathematics, Pomeranian University, Slupsk, Poland
2V.G. Korolenko Poltava National Pedagogical University
3National University of Food Technologies, Kyiv
Dopov. Nac. akad. nauk Ukr. 2019, 5:11-17
Section: Mathematics
Language: English

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).

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

1. 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
2. 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.
3. 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).
4. 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
5. 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
6. 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
7. 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