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Lobachevskii Journal of Mathematics

, Volume 31, Issue 2, pp 100–122 | Cite as

Group analysis of variable coefficient diffusion-convection equations. I. Enhanced group classification

  • N. M. Ivanova
  • R. O. Popovych
  • C. Sophocleous
Article

Abstract

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1 + 1)-dimensional nonlinear diffusion-convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.

Key words and phrases

Lie summetry group classification equivalence transformations form-preserving transformations diffusion-convection equations 

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References

  1. 1.
    I. Sh. Akhatov, R. K. Gazizov and N. Kh. Ibragimov, Group Classification of Equation of Nonlinear Filtration, Dokl. AN SSSR 293, 1033 (1987).MathSciNetGoogle Scholar
  2. 2.
    I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, Nonlocal Symmetries. A Heuristic Approach, Itogi Nauki i Tekhniki, Current Problems in Mathematics. Newest Results 34, 3 (1989) (Russian, translated in J. Soviet Math. 55, 1401 (1991)).zbMATHMathSciNetGoogle Scholar
  3. 3.
    W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov, R. K. Gazizov, N. H. Ibragimov, and S. R. Svirshchevskiy, CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1, Symmetries, Exact Solutions and Conservation Laws, Ed. by N. H. Ibragimov (CRC Press, Boca Raton, FL, 1994).Google Scholar
  4. 4.
    Barenblatt G.I. On automodel motions of compressible fluid in a porous medium, Prikl. Mat. Mekh. 16, 679 (1952).zbMATHMathSciNetGoogle Scholar
  5. 5.
    P. Basarab-Horwath, V. Lahno, and R. Zhdanov, The structure of Lie algebras and the classification problem for partial differential equation, Acta Appl. Math. 69, 43 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18, 1025 (1969).zbMATHMathSciNetGoogle Scholar
  7. 7.
    G.W. Bluman and S. Kumei, Symmetries and differential equations (Springer, New York, 1989).zbMATHGoogle Scholar
  8. 8.
    G.W. Bluman, G. J. Reid, and S. Kumei, New classes of symmetries for partial differential equations, J.Math. Phys. 29, 806 (1988).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. W. Bluman and Z. Yan, Nonclassical potential solutions of partial differential equations, Euro. J. Appl.Math. 16, 239 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. V. Borovskikh, Group classification of the eikonal equations for a three-dimensional nonhomogeneous medium, Mat. Sb. 195(4), 23 (2004) (in Russian); translation in Sb. Math. 195 (3), 479 (2004).MathSciNetGoogle Scholar
  11. 11.
    A. V. Borovskikh, The two-dimensional eikonal equation, Sibirsk. Mat. Zh. 47, 993 (2006).zbMATHMathSciNetGoogle Scholar
  12. 12.
    V. M. Boyko and V. O. Popovych, Group classification of Galilei-invariant higher-orders equations, Proceedings of Institute of Mathematics of NAS of Ukraine 36, 45 (2001).Google Scholar
  13. 13.
    J. T. Chayes, S. J. Osher, and J. V. Ralston, On singular diffusion equations with applications to self-organized criticality, Comm. Pure Appl. Math. 46, 1363 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Cherniha and M. Serov, Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, European J. Appl. Math. 9, 527 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comp. Phys. Comm. 176, 48 (2007).CrossRefMathSciNetGoogle Scholar
  16. 16.
    S. Dimas and D. Tsoubelis, SYM: A new symmetry — finding package for Mathematica, Proceedings of Tenth International Conference in Modern Group Analysis (Larnaca, Cyprus, 2004), p. 64.Google Scholar
  17. 17.
    V. A. Dorodnitsyn, On invariant solutions of non-linear heat equation with a sourse, Zhurn. Vych. Matemat. Matemat. Fiziki 22, 1393 (1982) (in Russian).MathSciNetGoogle Scholar
  18. 18.
    M. P. Edwards, Classical symmetry reductions of nonlinear diffusion-convection equations, Phys. Lett. A. 190, 149 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    W. I. Fushchych and I. M. Tsyfra, On a reduction and solutions of the nonlinear wave equations with broken symmetry, J. Phys. A: Math. Gen. 20, L45 (1987).CrossRefGoogle Scholar
  20. 20.
    M. L. Gandarias, Classical point symmetries of a porous medium equation, J. Phys. A: Math. Gen. 29, 607 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    M. L. Gandarias, New symmetries for a model of fast diffusion, Phys. Let. A 286, 153 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    L. Gagnon and P. Winternitz, Exact solutions of the cubic and quintic non-linear Schrodinger equation for a cylindrical geometry, Phys. Rev. A 39, 296 (1989).CrossRefMathSciNetGoogle Scholar
  23. 23.
    R. K. Gazizov, Contact transformations of equations of the type of nonlinear filtration, in Physicochemical Hydrodynamics (Inter-University scientific collection, Bashkir State University, Ufa, 1987), p. 38.Google Scholar
  24. 24.
    P. G. De Gennes, Wetting: statics and dynamics, Rev. Mod. Phys. 57, 827 (1985).CrossRefGoogle Scholar
  25. 25.
    A. K. Head, LIE, a PC program for Lie analysis of differential equations, Comput. Phys. Comm. 77, 241 (1993) (see also http://www.cmst.csiro.au/LIE/LIE.htm).zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    W. Hereman, Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath Bull. 1, 45 (1994).zbMATHMathSciNetGoogle Scholar
  27. 27.
    W. Hereman, Review of symbolic software for Lie symmetry analysis. Algorithms and software for symbolic analysis of nonlinear systems, Math. Comput. Modelling 25(8), 115 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    N. M. Ivanova, R. O. Popovych and H. Eshraghi, On symmetry properties of nonlinear Schrödinger equations with potentials, Proc. of Third Summer School on Mathematical Physics (Zlatibor, Serbia and Montenegro, 2004); Sveske Fiz. Nauka 18 (A1), 451 (2005).Google Scholar
  29. 29.
    N. M. Ivanova, R.O. Popovych, and C. Sophocleous, Conservation laws of variable coefficient diffusion-convection equations, Proceedings of Tenth International Conference in Modern Group Analysis (Larnaca, Cyprus, 2004), p. 107.Google Scholar
  30. 30.
    N. M. Ivanova, R. O. Popovych, and C. Sophocleous, Group analysis of variable coefficient diffusion-convection equations. II (Contractions and exact solutions), 2007, arXiv: 0710.3049, 19 p.Google Scholar
  31. 31.
    N. M. Ivanova, R. O. Popovych, and C. Sophocleous, Group analysis of variable coefficient diffusion-convection equations. III (Conservation laws), 2007, arXiv: 0710.3053, 26 p.Google Scholar
  32. 32.
    N. M. Ivanova, R. O. Popovych, and C. Sophocleous, Group analysis of variable coefficient diffusion-convection equations. IV (Potential symmetries), 2007, arXiv: 0710.4251, 14 p.Google Scholar
  33. 33.
    N. M. Ivanova, R. O. Popovych, C. Sophocleous, and O. O. Vaneeva, Conservation laws and potential symmetries for certain evolution equations, Physica A 388, 343 (2009), arXiv: 0806.1698.CrossRefGoogle Scholar
  34. 34.
    N. M. Ivanova and C. Sophocleous, On the group classification of variable coefficient nonlinear diffusion-convection equations, J. Comput. Appl. Math. 197, 322 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    V. L. Katkov, Group classification of solutions of Hopf’s equations (Russian), Zh. Prikl. Mekh. Tech. Fiz. 6, 105 (1965).MathSciNetGoogle Scholar
  36. 36.
    J. G. Kingston and C. Sophocleous, On point transformations of a generalised Burgers equation, Phys. Lett. A 155, 15 (1991).CrossRefMathSciNetGoogle Scholar
  37. 37.
    J. G. Kingston and C. Sophocleous, On form-preserving point transformations of partial differential equations, J. Phys. A: Math. Gen. 31, 1597 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    J. G. Kingston and C. Sophocleous, Symmetries and form-preserving transformations of one-dimensional wave equations with dissipation, Int. J. Non-Lin. Mech. 36, 987 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    S. P. Kurdyumov, S. A. Posashkov, and A. V. Sinilo, On the invariant solutions of the heat equation with the coefficient of heat conduction allowing the widest group of transformations, Preprint№110, Moscow (Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, 1989).Google Scholar
  40. 40.
    V. I. Lahno, S. V. Spichak, and V. I. Stognii, Symmetry analysis of evolution type equations (Kyiv: Institute of Mathematics of NAS of Ukraine, 2002).Google Scholar
  41. 41.
    S. Lie, 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, p. 473–508. (Translation by N. H. Ibragimov of Arch. for Math., Bd. VI, Heft 3, 328–368, Kristiania 1881.)Google Scholar
  42. 42.
    I. G. Lisle, Equivalence transformations for classes of differential equations (Thesis, University of British Columbia, 1992) (http://www.ise.canberra.edu.au/mathstat/StaffPages/LisleDissertation.pdf). (See also I. G. Lisle and G. J. Reid, 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)
  43. 43.
    S. V. Meleshko, Group classification of equations of two-dimensional gas motions, Prikl. Mat. Mekh. 58, 56 1994, (in Russian); translation in J. Appl.Math. Mech. 58, 629 (1994).MathSciNetGoogle Scholar
  44. 44.
    A. G. Nikitin and R. O. Popovych, Group classification of nonlinear Schrödinger equations, Ukr. Math. J. 53, 1053 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    P. Olver, Applications of Lie groups to differential equations (Springer-Verlag, New York, 1986).zbMATHGoogle Scholar
  46. 46.
    A. Oron and P. Rosenau, Some symmetries of the nonlinear heat and wave equations, Phys. Lett. A 118, 172 (1986).CrossRefMathSciNetGoogle Scholar
  47. 47.
    L. V. Ovsiannikov, Group properties of nonlinear heat equation, Dokl. AN SSSR 125, 492 (1959) (in Russian).Google Scholar
  48. 48.
    L. V. Ovsiannikov, Group analysis of differential equations (Academic Press, New York, 1982).zbMATHGoogle Scholar
  49. 49.
    C. Pallikaros and C. Sophocleous, On point transformations of generalized nonlinear diffusion equations, J. Phys. A: Math. Gen. 28, 6459 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    R. O. Popovych, Classification of admissible transformations of differential equations, Collection of Works of Institute of Mathematics, Kyiv 3(2), 239 (2006).zbMATHGoogle Scholar
  51. 51.
    R. O. Popovych and R. M. Cherniha, Complete classification of Lie symmetries of systems of two-dimensional Laplace equations, Proceedings of Institute of Mathematics of NAS of Ukraine, 36, 212 (2001).MathSciNetGoogle Scholar
  52. 52.
    R. O. Popovych and H. Eshraghi, Admissible point transformations of nonlinear Schrödinger equations, Proceedings of Tenth International Conference in Modern Group Analysis (Larnaca, Cyprus, 2004), p. 167.Google Scholar
  53. 53.
    R. O. Popovych and N. M. Ivanova, New results on group classification of nonlinear diffusion-convection equations, J. Phys. A: Math. Gen. 37, 7547 (2004), arXiv: math-ph/0306035.zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    R. O. Popovych, N. M. Ivanova and H. Eshraghi, Lie Symmetries of (1 + 1-dimensional cubic Schrödinger equation with potential, Proc. of Inst. of Math. of NAS of Ukraine, 50, 219 (2004), arXiv: math-ph/0310039.MathSciNetGoogle Scholar
  55. 55.
    R. O. Popovych, N. M. Ivanova, and H. Eshraghi, Group classification of (1 + 1)-dimensional Schrödinger equations with potentials and power nonlinearities, J.Math. Phys. 45, 3049 (2004), arXiv: math-ph/0311039.zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    R. O. Popovych and N. M. Ivanova, Potential equivalence transformations for nonlinear diffusion-convection equations, J. Phys. A: Math. Gen. 38, 3145 (2005), arXiv: math-ph/0402066.zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    R. O. Popovych, M. Kunzinger, and H. Eshraghi, Admissible point transformations and normalized classes of nonlinear Schrödinger equations, Acta Appl.Math. 109, 315 (2010).zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    R. O. Popovych, O. O. Vaneeva, and N. M. Ivanova, Potential nonclassical symmetries and solutions of fast diffusion equation, Phys. Lett. A 362, 166 (2007), arXiv: math-ph/0506067.CrossRefMathSciNetGoogle Scholar
  59. 59.
    L. A. Richard’s, Capillary conduction of liquids through porous mediums, Physics 1, 318 (1931).CrossRefGoogle Scholar
  60. 60.
    C. Sophocleous, Potential symmetries of nonlinear diffusion-convection equations, J. Phys. A: Math. Gen. 29, 6951 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    C. Sophocleous, Potential symmetries of inhomogeneous nonlinear diffusion equations, Bull. Austral. Math. Soc. 61, 507 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    C. Sophocleous, Classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations, Physica A, 320, 169 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    O. O. Vaneeva, A. G. Johnpillai, R. O. Popovych, and C. Sophocleous, Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities, J. Math. Anal. Appl. 330, 1363 (2007), arXiv: math-ph/0605081.zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    O. O. Vaneeva, R. O. Popovych, and C. Sophocleous, Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion equations with a power source, Acta Appl.Math. 106, 1 (2009), arXiv: 0708.3457.zbMATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    O. F. Vasilenko and I. A. Yehorchenko, Group classification of multidimensional nonlinear wave equations, Proceedings of Institute of Mathematics of NAS of Ukraine 36, 63 (2001).Google Scholar
  66. 66.
    A. D. Wittkopf, Algorithms and Implementations for Differential Elimination (Ph.D. thesis, Simon Fraser University, 2004).Google Scholar
  67. 67.
    T. Wolf, An efficiency improved program LIEPDE for determining Lie-symmetries of PDEs, Proc. of Modern Group Analysis, (Catania, Italy, Oct. 1992, Kluwer Acad. Publ., 1993), p. 377.Google Scholar
  68. 68.
    C.M. Yung, K. Verburg, and P. Baveye, Group classification and symmetry reductions of the non-linear diffusion-convection equation u t = (D(u)u x)xK’(u)u x, Int. J. Non-Lin. Mech. 29, 273 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    R.Z. Zhdanov and V.I. Lahno, Group classification of heat conductivity equations with a nonlinear source, J. Phys. A: Math. Gen. 32, 7405 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    R. Z. Zhdanov, I. M. Tsyfra, and R. O. Popovych, A precise definition of reduction of partial differential equations, J. Math. Anal. Appl. 238, 101 (1999), arXiv: math-ph/0207023.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • N. M. Ivanova
    • 1
    • 2
  • R. O. Popovych
    • 1
    • 3
  • C. Sophocleous
    • 2
  1. 1.Institute of Mathematics of NAS of UkraineKyivUkraine
  2. 2.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus
  3. 3.Fakultät für MathematikUniversität WienWienAustria

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