Exact solutions of a remarkable fin equation

Applied Mathematics Letters

Volumn 21, Issue 3, 2008, Pages 209-214

  Popovych, R O ^a,b   Sophocleous, C ^c   Vaneeva, O O ^a  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

^b UNIVERSITY OF VIENNA   (Austria)

^c UNIVERSITY OF CYPRUS   (Cyprus)

Author keywords

Exact solutions; Fin equations; Lie reductions; Non Lie reductions; Nonclassical symmetries

Indexed keywords

ALGORITHMS; MATHEMATICAL MODELS; NUMBER THEORY;

FIN EQUATIONS; LIE ALGORITHMS; MODERN TECHNIQUES;

NONLINEAR EQUATIONS;

EID: 38049070316     PISSN: 08939659     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.aml.2007.03.009     Document Type: Article

References (29)

1. Abraham-Shrauner B., Leach P.G.L., Govinder K.S., and Ratcliff G. Hidden and contact symmetries of ordinary differential equations. J. Phys. A 28 (1995) 6707-6716
2. Basarab-Horwath P., Lahno V., and Zhdanov R. The structure of Lie algebras and the classification problem for partial differential equation. Acta Appl. Math. 69 (2001) 43-94
3. Bokhari A.H., Kara A.H., and Zaman F.D. A note on a symmetry analysis and exact solutions of a nonlinear fin equation. Appl. Math. Lett. 19 (2006) 1356-1360
4. Dorodnitsyn V.A. On invariant solutions of non-linear heat equation with a source. Zh. Vychis. Mat. Mat. Fiz. 22 (1982) 1393-1400 (in Russian)
5. Fokas A.S., and Liu Q.M. Generalized conditional symmetries and exact solutions of nonintegrable equations. Phys. Rev. Lett. 72 (1994) 263-277
6. Fushchych W.I., and Popovych R.O. Symmetry reduction and exact solution of the Navier-Stokes equations. J. Nonlinear Math. Phys. 1 (1994). 75-113, 156-188; math-ph/0207016
7. Fushchych W.I., Shtelen W.M., Serov M.I., and Popovych R.O. Q-conditional symmetry of the linear heat equation. Proc. Acad. Sci. Ukraine 12 (1992) 28-33
8. Fushchych W., and Zhdanov R. Antireduction and exact solutions of nonlinear heat equations. J. Nonlinear Math. Phys. 1 (1994) 60-64
9. Galaktionov V.A. On new exact blow-up solutions for nonlinear heat conduction equations with source and applications. Differential Integral Equations 3 (1990) 863-874
10. In: Ibragimov N.H. (Ed). Lie Group Analysis of Differential Equations - Symmetries, Exact Solutions and Conservation Laws vol. 1 (1994), CRC Press, Boca Raton, FL
11. Kamke E. Differentialgleichungen Lösungsmethoden und Lösungen (1971), Chelsea, New York
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 (1978) 901-904
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 (1979) 89-107
14. Lie S. Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung. Arch. Math. 6 3 (1881) 328-368. Translation by N.H. Ibragimov: Lie S. On integration of a Class of Linear Partial Differential Equations by Means of Definite Integrals, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 2, 1994, pp. 473-508.
15. Lahno V.I., Spichak S.V., and Stognii V.I. Symmetry Analysis of Evolution Type Equations (2002), Institute of Mathematics of NAS of Ukraine, Kyiv
16. Olver P.J. Direct reduction and differential constraints. Proc. R. Soc. Lond. Ser. A 444 (1994) 509-523
17. Ovsiannikov L.V. Group properties of nonlinear heat equation. Dokl. AN SSSR 125 (1959) 492-495 (in Russian)
18. Ovsiannikov L.V. Group Analysis of Differential Equations (1982), Academic Press, New York
19. Pakdemirli M., and Sahin A.Z. Group classification of fin equation with variable thermal properties. Internat. J. Engrg. Sci. 42 (2004) 1875-1889
20. Pakdemirli M., and Sahin A.Z. Similarity analysis of a nonlinear fin equation. Appl. Math. Lett. 19 (2006) 378-384
21. Polyanin A.D., and Zaitsev V.F. Handbook of Nonlinear Partial Differential Equations (2004), Chapman & Hall/CRC, Boca Raton
22. Sidorov A.F., Shapeev V.P., and Yanenko N.N. The method of differential constraints and its applications in gas dynamics. Nauka Sib. Otdel., Novosibirsk (1984) (in Russian)
23. Vaneeva O.O., Johnpillai A.G., Popovych R.O., and Sophocleous C. Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities. J. Math. Anal. Appl. 330 (2007) 1363-1386. math-ph/0605081
24. Vaneeva O.O., Johnpillai A.G., Popovych R.O., and Sophocleous C. Group analysis of nonlinear fin equations. Appl. Math. Lett. (2007). http://arxiv.org/math-ph/0610006 10.1016/j.aml.2007.02.023 math-ph/0610006
25. R.O. Popovych, On a class of Q-conditional symmetries and solutions of evolution equations, in: Symmetry and Analytic Methods in Mathematical Physics, in: Proceedings of Institute of Mathematics, vol. 19, Kyiv, 1998, pp. 194-199 (in Ukrainian)
26. Popovych R.O., Vaneeva O.O., and Ivanova N.M. Potential nonclassical symmetries and solutions of fast diffusion equation. Phys. Lett. A 362 (2007) 166-173. math-ph/0506067
27. Zhdanov R.Z. Conditional Lie-Bäcklund symmetry and reduction of evolution equations. J. Phys. A 28 (1995) 3841-3850
28. Zhdanov R.Z., and Lahno V.I. Conditional symmetry of a porous medium equation. Physica D 122 (1998) 178-186
29. Zhdanov R.Z., Tsyfra I.M., and Popovych R.O. A precise definition of reduction of partial differential equations. J. Math. Anal. Appl. 238 1 (1999) 101-123. math-ph/0207023