On the group classification of variable-coefficient nonlinear diffusion-convection equations

Journal of Computational and Applied Mathematics

Volumn 197, Issue 2, 2006, Pages 322-344

  Ivanova, N M ^a   Sophocleous, C ^b  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

^b UNIVERSITY OF CYPRUS   (Cyprus)

Author keywords

Equivalence transformations; Lie symmetries; Nonlinear diffusion equations

Indexed keywords

DIFFUSION; MAPPING; MATHEMATICAL TRANSFORMATIONS;

CONVECTION-DIFFUSION EQUATION; EQUIVALENCE TRANSFORMATIONS; LIE SYMMETRIES; NONLINEAR DIFFUSION EQUATIONS;

NONLINEAR EQUATIONS;

EID: 33747329649     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2005.11.008     Document Type: Article

References (47)

1. 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
2. Berryman J.G., and Holland J.G. Nonlinear diffusion problems arising in plasma physics. Phys. Rev. Lett. 40 (1978) 1720-1722
3. Bluman G.W., and Anco S.C. Symmetry and Integration Methods for Differential Equations (2002), Springer, New York
4. - 2 (∂ u / ∂ x)] - (∂ u / ∂ t) = 0. J. Math. Phys. 21 (1980) 1019-1023
5. Bluman G.W., and Kumei S. Symmetries and Differential Equations (1989), Springer, New York
6. Broadbridge P. Integrable forms of the one-dimensional flow equation for unsaturated heterogeneous porous media. J. Math. Phys. 29 (1988) 622-627
7. Crank J. The Mathematics of Diffusion. second ed. (1979), Oxford, London
8. Crighton D.G. Basic nonlinear acoustics. In: Sette D. (Ed). Frontiers in Physical Acoustics (1986), North-Holland, Amsterdam
9. Edwards M.P. Classical symmetry reductions of nonlinear diffusion-convection equations. Phys. Lett. A 190 (1994) 149-154
10. x occurring in two-phase flow in porous media. SIAM J. Appl. Math. 42 (1982) 318-332
11. Fushchich W.I., Shtelen W.M., and Serov N.I. Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics (1993), Kluwer, Dordrecht
12. Gandarias M.L. New symmetries for a model of fast diffusion. Phys. Lett. A 286 (2001) 153-160
13. Hill J.M. Similarity solutions for nonlinear diffusion-a new integration procedure. J. Engrg. Math. 23 (1989) 141-155
14. Hill D.L., and Hill J.M. Similarity solutions for nonlinear diffusion-further exact solutions. J. Engrg. Math. 24 (1990) 109-124
15. N.H. Ibragimov (Ed.), Lie Group Analysis of Differential Equations-Symmetries, Exact Solutions and Conservation Laws, vol. 1, Chemical Rubber Company, Boca Raton, FL, 1994.
16. Ibragimov N.H. Elementary Lie Group Analysis and Ordinary Differential Equations (1999), Wiley, New York
17. Ivanova N.M., Popovych R.O., and Sophocleous C. Conservation laws of variable coefficient diffusion-convection equations. Proceedings of Tenth International Conference in Modern Group Analysis. Larnaca, Cyprus (2004) 107-113 (math-ph/0505015)
18. Kamin S., and Rosenau P. Propagation of thermal waves in an inhomogeneous medium. Comm. Pure Appl. Math. 34 (1981) 831-852
19. Kamin S., and Rosenau P. Nonlinear thermal evolution in an inhomogeneous medium. J. Math. Phys. 23 (1982) 1385-1390
20. Katkov V.L. Group classification of solutions of Hopf's equations. Zh. Prikl. Mekh. Tech. Fiz. 6 (1965) 105-106 (in Russian)
21. King J.R. Exact similarity solutions to some nonlinear diffusion equations. J. Phys. A 23 (1990) 3681-3697
22. King J.R. Exact solutions to a nonlinear diffusion equation. J. Phys. A 24 (1991) 3213-3216
23. Kingston J.G., and Sophocleous C. On form-preserving point transformations of partial differential equations. J. Phys. A 31 (1998) 1597-1619
24. Mubarakzyanov G.M. On solvable Lie algebras. Izv. Vys. Ucheb. Zaved. Matematika 32 (1963) 114-123 (in Russian)
25. Munier A., Burgan J.R., Gutierrez J., Fijalkow E., and Feix M.R. Group transformations and the nonlinear heat diffusion equation. SIAM J. Appl. Math. 40 (1981) 191-207
26. Murray J.D. Mathematical Biology I: An Introduction. third ed. (2002), Springer, New York
27. Murray J.D. Mathematical Biology II: Spatial Models and Biomedical Applications. third ed. (2003), Springer, New York
28. Olver P.J. Application of Lie Groups to Differential Equations (1986), Springer, New York
29. Oron A., and Rosenau P. Some symmetries of the nonlinear heat and wave equations. Phys. Lett. A 118 (1986) 172-176
30. Ovsiannikov L.V. Group relations of the equation of non-linear heat conductivity. Dokl. Akad. Nauk SSSR 125 (1959) 492-495 (in Russian)
31. Patera J., and Winternitz P. Subalgebras of real three- and four-dimensional Lie algebras. J. Math. Phys. 18 (1977) 1449-1455
32. Peletier L.A. Applications of Nonlinear Analysis in the Physical Sciences (1981), Pitman, London
33. Polyanin A.D., and Zaitsev V.F. Handbook of Nonlinear Equations of Mathematical Physics (2002), Fizmatlit, Moscow
34. Popovych R.O., and Ivanova N.M. New results on group classification of nonlinear diffusion-convection equations. J. Phys. A 37 (2004) 7547-7565 (math-ph/0306035)
35. Popovych R.O., and Ivanova N.M. Potential equivalence transformations for nonlinear diffusion-convection equations. J. Phys. A 38 (2005) 3145-3155 (math-ph/0402066)
36. Popovych R.O., and Ivanova N.M. Hierarchy of conservation laws of diffusion-convection equations. J. Math. Phys. 46 (2005) 043502 (math-ph/0407008)
37. R.O. Popovych, N.M. Ivanova, O.O. Vaneeva, Potential nonclassical symmetries and solutions of fast diffusion equation, math-ph/0506067.
38. Qu C., and Estevez P.G. On nonlinear diffusion equations with x-dependent convection and absorption. Nonlinear Anal. 57 (2004) 549-577
39. Rogers C., and Ames W.F. Nonlinear Boundary Value Problems in Science and Engineering (1989), Academic Press, New York
40. Rosen G. Method for the exact solution of a nonlinear diffusion-convection equation. Phys. Rev. Lett. 49 (1982) 1844-1847
41. Rosenau P., and Kamin S. Nonlinear diffusion in a finite mass medium. Comm. Pure Appl. Math. 35 (1982) 113-127
42. Sophocleous C. Classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations. Physica A 320 (2003) 169-183
43. Sophocleous C. Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations. Physica A 324 (2003) 509-529
44. Sophocleous C. Further transformation properties of generalised inhomogeneous nonlinear diffusion equations with variable coefficients. Physica A 345 (2005) 457-471
45. Touloukian Y.S., Powell P.W., Ho C.Y., and Klemens P.G. Thermodynamics Properties of Matter vol. 1 (1970), Plenum, New York
46. Wiltshire R., and El-Kafri M. Non-classical and potential symmetry analysis of Richard's equation for moisture flow in soil. J. Phys. A 37 (2004) 823-839
47. x. Int. J. Non-Linear Mech. 29 (1994) 273-278