×
Popovych, R. O.; Sophocleous, C.; Vaneeva, O. O.

Exact solutions of a remarkable fin equation. (English) Zbl 1149.76047

Appl. Math. Lett. 21, No. 3, 209-214 (2008).
Summary: A model ‘remarkable’ fin equation is singled out from a class of nonlinear \((1+1)\)-dimensional fin equations. For this equation a number of exact solutions are constructed by means of using both the classical Lie algorithm and different modern techniques (functional separation of variables, generalized conditional symmetries, hidden symmetries etc.).

Cited in 8 Documents

MSC:

76R50 Diffusion
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics

Keywords:

Lie reductions; non-Lie reductions; separation of variables; generalized conditional symmetries; hidden symmetries
PDF BibTeX XML Cite
\textit{R. O. Popovych} et al., Appl. Math. Lett. 21, No. 3, 209--214 (2008; Zbl 1149.76047)
Full Text: DOI arXiv

References:

[1] Abraham-Shrauner, B.; Leach, P. G.L.; Govinder, K. S.; Ratcliff, G., Hidden and contact symmetries of ordinary differential equations, J. Phys. A, 28, 6707-6716 (1995) · Zbl 0879.58068
[2] Basarab-Horwath, P.; Lahno, V.; Zhdanov, R., The structure of Lie algebras and the classification problem for partial differential equation, Acta Appl. Math., 69, 43-94 (2001) · Zbl 1054.35002
[3] Bokhari, A. H.; Kara, A. H.; Zaman, F. D., A note on a symmetry analysis and exact solutions of a nonlinear fin equation, Appl. Math. Lett., 19, 1356-1360 (2006) · Zbl 1143.35311
[4] Dorodnitsyn, V. A., On invariant solutions of non-linear heat equation with a source, Zh. Vychis. Mat. Mat. Fiz., 22, 1393-1400 (1982), (in Russian) · Zbl 0535.35040
[5] Fokas, A. S.; Liu, Q. M., Generalized conditional symmetries and exact solutions of nonintegrable equations, Phys. Rev. Lett., 72, 263-277 (1994) · Zbl 0850.35097
[6] Fushchych, W. I.; Popovych, R. O., Symmetry reduction and exact solution of the Navier-Stokes equations, J. Nonlinear Math. Phys., 1 (1994), 75-113, 156-188;
[7] Fushchych, W. I.; Shtelen, W. M.; Serov, M. I.; Popovych, R. O., \(Q\)-conditional symmetry of the linear heat equation, Proc. Acad. Sci. Ukraine, 12, 28-33 (1992)
[8] Fushchych, W.; Zhdanov, R., Antireduction and exact solutions of nonlinear heat equations, J. Nonlinear Math. Phys., 1, 60-64 (1994) · Zbl 0954.35037
[9] Galaktionov, V. A., On new exact blow-up solutions for nonlinear heat conduction equations with source and applications, Differential Integral Equations, 3, 863-874 (1990) · Zbl 0735.35074
[10] (Ibragimov, N. H., Lie Group Analysis of Differential Equations — Symmetries, Exact Solutions and Conservation Laws, vol. 1 (1994), CRC Press: CRC Press Boca Raton, FL) · Zbl 0864.35001
[11] Kamke, E., Differentialgleichungen Lösungsmethoden und Lösungen (1971), Chelsea: Chelsea New York · Zbl 0026.31801
[12] Kapitanskiy, L. V., Group analysis of the Navier-Stokes and Euler equations in the presence of rotational symmetry and new exact solutions of these equations, Dokl. Acad. Sci. USSR, 243, 4, 901-904 (1978)
[13] Kapitanskiy, L. V., Group analysis of the Navier-Stokes equations in the presence of rotational symmetry and some new exact solutions, Zap. Nauchn. Sem. LOMI, 84, 89-107 (1979)
[14] Lie, S., Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung, Arch. Math., 6, 3, 328-368 (1881) · JFM 13.0298.01
[15] Lahno, V. I.; Spichak, S. V.; Stognii, V. I., Symmetry Analysis of Evolution Type Equations (2002), Institute of Mathematics of NAS of Ukraine: Institute of Mathematics of NAS of Ukraine Kyiv
[16] Olver, P. J., Direct reduction and differential constraints, Proc. R. Soc. Lond. Ser. A, 444, 509-523 (1994) · Zbl 0814.35003
[17] Ovsiannikov, L. V., Group properties of nonlinear heat equation, Dokl. AN SSSR, 125, 492-495 (1959), (in Russian)
[18] Ovsiannikov, L. V., Group Analysis of Differential Equations (1982), Academic Press: Academic Press New York · Zbl 0485.58002
[19] Pakdemirli, M.; Sahin, A. Z., Group classification of fin equation with variable thermal properties, Internat. J. Engrg. Sci., 42, 1875-1889 (2004) · Zbl 1211.35141
[20] Pakdemirli, M.; Sahin, A. Z., Similarity analysis of a nonlinear fin equation, Appl. Math. Lett., 19, 378-384 (2006) · Zbl 1114.80003
[21] Polyanin, A. D.; Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations (2004), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton · Zbl 1024.35001
[22] Sidorov, A. F.; Shapeev, V. P.; Yanenko, N. N., The method of differential constraints and its applications in gas dynamics, Nauka Sib. Otdel., Novosibirsk (1984), (in Russian) · Zbl 0604.76062
[23] Vaneeva, O. O.; Johnpillai, A. G.; Popovych, R. O.; Sophocleous, C., Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities, J. Math. Anal. Appl., 330, 1363-1386 (2007) · Zbl 1160.35365
[24] Vaneeva, O. O.; Johnpillai, A. G.; Popovych, R. O.; Sophocleous, C., Group analysis of nonlinear fin equations, Appl. Math. Lett. (2007), math-ph/0610006 · Zbl 1160.35365
[26] Popovych, R. O.; Vaneeva, O. O.; Ivanova, N. M., Potential nonclassical symmetries and solutions of fast diffusion equation, Phys. Lett. A, 362, 166-173 (2007) · Zbl 1197.35145
[27] Zhdanov, R. Z., Conditional Lie-Bäcklund symmetry and reduction of evolution equations, J. Phys. A, 28, 3841-3850 (1995) · Zbl 0859.35115
[28] Zhdanov, R. Z.; Lahno, V. I., Conditional symmetry of a porous medium equation, Physica D, 122, 178-186 (1998) · Zbl 0952.76087
[29] Zhdanov, R. Z.; Tsyfra, I. M.; Popovych, R. O., A precise definition of reduction of partial differential equations, J. Math. Anal. Appl., 238, 1, 101-123 (1999) · Zbl 0936.35012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.