Potential equivalence transformations for nonlinear diffusion-convection equations

Journal of Physics A: Mathematical and General

Volumn 38, Issue 14, 2005, Pages 3145-3155

  Popovych, Roman O ^a   Ivanova, Nataliya M ^a  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17044412509     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/38/14/006     Document Type: Article

References (42)

1. Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1987 Group classification of equation of nonlinear filtration Dokl. Acad. Nauk SSSR 293 1033-5 (in Russian) Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1991 Nonlocal symmetries. A heuristic approach. (Russian) Translated in J. Sov. Math. 55 1401-50 Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, vol 34 (Russian), pp 3-83, 195, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989
2. Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1987 Group classification of equation of nonlinear filtration Dokl. Acad. Nauk SSSR 293 1033-5 (in Russian) Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1991 Nonlocal symmetries. A heuristic approach. (Russian) Translated in J. Sov. Math. 55 1401-50 Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, vol 34 (Russian), pp 3-83, 195, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989
3. Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1987 Group classification of equation of nonlinear filtration Dokl. Acad. Nauk SSSR 293 1033-5 (in Russian) Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1991 Nonlocal symmetries. A heuristic approach. (Russian) Translated in J. Sov. Math. 55 1401-50 Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, vol 34 (Russian), pp 3-83, 195, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989
4. Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1987 Group classification of equation of nonlinear filtration Dokl. Acad. Nauk SSSR 293 1033-5 (in Russian) Akhatov I Sh, Gazizov R K and Ibragimov N Kh 1991 Nonlocal symmetries. A heuristic approach. (Russian) Translated in J. Sov. Math. 55 1401-50 Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, vol 34 (Russian), pp 3-83, 195, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989
5. Barenblatt G I 1952 On some unsteady motions of a liquid and gas in a porous medium Prikl. Mat. Mekh. 16 67-78
6. Barenblatt G I 1952 On automodel motions of compressible fluid in a porous medium Prikl. Mat. Mekh. 16 679-98
7. Basarab-Horwath P, Lahno V and Zhdanov R 2001 The structure of Lie algebras and the classification problem for partial differential equation Acta Appl. Math. 69 43-94
8. Bluman G W 1993 Potential symmetries and equivalent conservation laws Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (Acireale, 1992) ed N H Ibragimov, M Torrisi and A Valenti (Dordrecht: Kluwer) pp 71-84
9. -2(∂u/∂x)] - ∂u/∂t = 0 J. Math. Phys. 21 1019-23
10. Bluman G W and Kumei S 1989 Symmetries and Differential Equations (New York: Springer)
11. Bluman G W, Reid G J and Kumei S 1988 New classes of symmetries for partial differential equations J. Math. Phys. 29 806-11
12. Cherniha R and Serov M 1998 Symmetries ansätze and exact solutions of nonlinear second-order evolution equations with convection terms Eur. J. Appl. Math. 9 527-42
13. Cole J D 1951 On a quasilinear parabolic equation used in aerodinamics Q. Appl. Math. 9 225-36
14. Edwards M P 1994 Classifical symmetry reductions of nonlinear diffusion-convection equations Phys. Lett. A 190 149-54
15. x occurring in two-phase flow in porous media SIAM J. Appl. Math. 42 318-32
16. Forsyth A R 1906 The Theory of Differential Equations vol 6 (Cambridge: Cambridge University Press)
17. Fushchych W I, Serov M I and Amerov T K 1992 On nonlocal ansätzes for a nonlinear one-dimensional heat equation Proc. Acad. Sci. of Ukraine No. 1 pp 26-30
18. Gandarias M L 2001 New symmetries for a model of fast diffusion Phys. Lett. A 286 153-60
19. Head A K 1993 LIE, a PC program for Lie analysis of differential equations Comput. Phys. Commun. 77 241-8 (see also http://www.cmst.csiro.au/LIE/ LIE.htm)
20. xx Commun. Pure Appl. Math. 3 201-30
21. Kamin S and Rosenau P 1982 Nonlinear thermal evolution in an inhomogeneous medium J. Math. Phys. 23 1385-90
22. Katkov V L 1968 Exact solutions of some problems of convection Prikl. Mat. Mech. 32 482-6
23. Lie S 1881 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 pp 473-508 (Translation by N H Ibragimov 1881 Arch. Math., Bd. VI, Heft 3, Kristiania, pp 328-68)
24. Lie S 1881 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 pp 473-508 (Translation by N H Ibragimov 1881 Arch. Math., Bd. VI, Heft 3, Kristiania, pp 328-68)
25. Lisle I G 1992 Equivalence transformations for classes of differential equations Thesis University of British Columbia (http://www.ise.canberra.edu.au/ mathstat/StaffPages/LisleDissertation.pdf) ( see also Lisle I G and Reid G J Symmetry classification using invariant moving frames ORCCA Technical Report TR-00-08 (University of Western Ontario), http://www.orcca.on.ca/TechReports/ 2000/TR-00-08.html submitted to Foundations of Computational Mathematics)
26. Lisle I G 1992 Equivalence transformations for classes of differential equations Thesis University of British Columbia (http://www.ise.canberra.edu.au/ mathstat/StaffPages/LisleDissertation.pdf) ( see also Lisle I G and Reid G J Symmetry classification using invariant moving frames ORCCA Technical Report TR-00-08 (University of Western Ontario), http://www.orcca.on.ca/TechReports/ 2000/TR-00-08.html submitted to Foundations of Computational Mathematics)
27. Oron A and Rosenau P 1986 Some symmetries of the nonlinear heat and wave equations Phys. Lett. A 118 172-6
28. Ovsiannikov L V 1982 Group Analysis of Differential Equations (New York: Academic)
29. Ovsiannikov L V 1959 Group properties of nonlinear heat equation Dokl. Acad. Nauk SSSR 125 492-5 (in Russian)
30. Philip J R 1980 Quasi-analytic and analytic approaches to unsaturated flow Flow and Transport in the Natural Environment: Advances and Applications ed W L Steffen and O T Denmead (Berlin: Springer)
31. Polyanin A D and Zaitsev V F 2002 Handbook of Nonlinear Equations of Mathematical Physics (Moscow: Fizmatlit)
32. Popovych R O and Ivanova N M 2004 New results on group classification of nonlinear diffusion-convection equations J. Phys. A: Math. Gen. 37 7547-65 (see also Preprint math-ph/0306035.)
33. Pukhnachov V V 1996 Nonlocal symmetries in nonlinear heat equations Energy Methods in Continuum Mechanics (Oviedo, 1994) (Dordrecht: Kluwer) pp 75-99
34. Richard's L A 1931 Capillary conduction of liquids through porous mediums Physics 1 318-33
35. Sophocleous C 1996 Potential symmetries of nonlinear diffusion-convection equations J. Phys. A: Math. Gen. 29 6951-9
36. Sophocleous C 2000 Potential symmetries of inhomogeneous nonlinear diffusion equations Bull. Aust. Math. Soc. 61 507-21
37. Sophocleous C 2003 Classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations Physica A 320 169-83
38. Sposito G 1990 Lie group invariance of Richard's equation fluids Hierarchical Porous Medium ed J Cushman (New York: Academic) pp 327-47
39. Storm M L 1951 Heat conduction in simple metals J. Appl. Phys. 22 940-51
40. Strampp W 1982 Backlund transformations for diffusion equations Physica D 6 113-8
41. Wiltshire R and El-Kafri M 2004 Non-classical and potential symmetry analysis of Richard's equation for moisture flow in soil J. Phys. A: Math. Gen. 37 823-39
42. x Int. J. Nonlinear Mech. 29 273-8