Abstract
The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form u t =F(t,x,u,u x )u xx +G(t,x,u,u x ). We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.
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Fushchych, W. I. and Nikitin, A. G.: Symmetry of Equations of Quantum Mechanics, Allerton Press, New York, 1994.
Lie, S.: In: Gesammelte Abhandlungen, vol. 5, B. G. Teubner, Leipzig, 1924, pp. 767–773.
Lie, S.: In: Gesammelte Abhandlungen, vol. 6, Teubner, Leipzig, 1927, pp. 1–94.
Ovsjannikov, L. V.: Group Analysis of Differential Equations, Academic Press, New York, 1982.
Olver, P. J.: Applications of Lie Groups to Differential Equations, Springer, Berlin, 1986.
Fushchych, W. I., Shtelen, W. M. and Serov, N. I.: Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics, Naukova Dumka, Kiev, 1989 (English edn: Kluwer Acad. Publ., Dordrecht, 1993).
Heredero, R. H. and Olver, P. J.: Classification of invariant wave equations, J. Math. Phys. 37 (1996), 6419–6438.
Akhatov, I. S., Gazizov, R. K. and Ibragimov, N. K.: Group classification of equations of nonlinear filtration, Proc. Acad. Sci. USSR 293 (1987), 1033–1035 (in Russian).
Akhatov, I. S., Gazizov, R. K. and Ibragimov, N. K.: Nonlocal symmetries. A heuristic approach (in Russian), In: Sovremennye Problemy Matematiki. Novejshie Dostizheniya, 34, Nauka, Moscow, 1989, pp. 3–83.
Torrisi, M., Tracina, R. and Valenti, A.: A group analysis approach for a nonlinear differential system arising in diffusion phenomena, J. Math. Phys. 37 (1996), 4758–4767.
Torrisi, M. and Tracina, R.: Equivalence transformations and symmetries for a heat conduction model, Int. J. of Non-Linear Mechanics 33 (1998), 473–487.
Ibragimov, N. H., Torrisi, M. and Valenti, A.: Preliminary group classification of equation v tt = f (x, v x )v xx + g(x, v x ), J. Math. Phys. 32 (1991), 2988–2995.
Ibragimov, N. K. and Torrisi, M.: A simple method for group analysis and its applications to a model of detonation, J. Math. Phys. 33 (1992), 3931–3937.
Kingston, J. G. and Sophocleous, C.: On form-preserving point transformations of partial differential equations, J. Phys. A: Math. Gen. 31 (1998), 1595–1619.
Zhdanov, R. Z. and Lahno, V. I.: Group classification of heat conductivity equations with a nonlinear source, J. Phys. A: Math. Gen. 32 (1999), 7405–7418.
Gagnon, L. and Winternitz, P.: Symmetry classes of variable coefficient nonlinear Schrödinger equations, J. Phys. A: Math. Gen. 26 (1993), 7061–7076.
Barut, A. O. and Raczka, R.: Theory of Group Representations and Applications, PWN-Polish Scientific, Warszawa, 1977.
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
Morozov, V. V.: Classification of six-dimensional nilpotent Lie algebras (in Russian), Izv. Vys. Ucheb. Zaved. no. 5(5) (1958), 161–171.
Mubarakzyanov, G. M.: On solvable Lie algebras, Izv.Vys. Ucheb. Zaved. no. 1(32) (1963), 114–123 (in Russian).
Mubarakzyanov, G. M.: The classification of the real structure of five-dimensional Lie algebras, Izv. Vys. Ucheb. Zaved. no. 3(34) (1963), 99–105 (in Russian).
Mubarakzyanov, G. M.: The classification of six-dimensional Lie algebras with one nilpotent basis element, Izv. Vys. Ucheb. Zaved. no. 4(35) (1963), 104–116 (in Russian).
Mubarakzyanov, G. M.: Some theorems on solvable Lie algebras, Izv. Vys. Ucheb. Zaved. no. 3(55) (1966), 95–98 (in Russian).
Turkowski, P.: Solvable Lie algebras of dimensional six, J. Math. Phys. 31 (1990), 1344–1350.
Turkowski, P.: Low-dimensional real Lie algebras, J. Math. Phys. 29 (1988), 2139–2144.
Ovsiannikov, L. V.: Group properties of nonlinear heat equation, Dokl. Akad. Nauk SSSR 125(3) (1959), 492–495 (in Russian).
Dorodnitsyn, V. A.: On invariant solutions of nonlinear heat equation with a source, Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), 1393–1400 (in Russian).
Oron, A. and Rosenau, P.: Some symmetries of the nonlinear heat and wave equations, Phys. Lett. A 118 (1986), 172–176.
Edwards, M. P.: Classical symmetry reductions of nonlinear diffusion-convection equations, Phys. Lett. A 190 (1994), 149–154.
Cherniha, R. and Serov, M.: Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, European J. Appl. Math. 9 (1998), 527–542.
Gandarias, M. L.: Classical point symmetries of a porous medium equation, J. Phys. A: Math. Gen. 29 (1996), 607–633.
Sokolov, V. V.: On symmetries of evolution equations, Uspekhi Mat. Nauk 43 (1988), 133–163 (in Russian).
Magadeev, B. A.: On group classification of nonlinear evolution equations, Algebra i Analiz 5 (1993), 141–156 (in Russian).
King, J. R.: Exact results for the nonlinear diffusion equations \(\frac{{\partial u}}{{\partial t}} = \frac{\partial }{{\partial x}} ({u^{ - 4/3} \frac{{\partial u}}{{\partial x}}}) and \frac{{\partial u}}{{\partial t}} = \frac{\partial }{{\partial x}} ({u^{ - 2/3} \frac{{\partial u}}{{\partial x}}} )\) J. Phys. A: Math. Gen. 24 (1991), 5721–5745.
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Basarab-Horwath, P., Lahno, V. & Zhdanov, R. The Structure of Lie Algebras and the Classification Problem for Partial Differential Equations. Acta Applicandae Mathematicae 69, 43–94 (2001). https://doi.org/10.1023/A:1012667617936
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DOI: https://doi.org/10.1023/A:1012667617936