Skip to main content
SpringerLink
Log in

Consistent approximate Q-conditional symmetries of PDEs: application to a hyperbolic reaction-diffusion-convection equation

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

Within the theoretical framework of a recently introduced approach to approximate Lie symmetries of differential equations containing small terms, which is consistent with the principles of perturbative analysis, we define accordingly approximate Q-conditional symmetries of partial differential equations. The approach is illustrated by considering the hyperbolic version of a reaction-diffusion-convection equation. By looking for its first order approximate Q-conditional symmetries, we are able to explicitly determine a large set of non-trivial approximate solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Lie, S., Engel, F.: Theorie der transformationsgruppen. Teubner, Leipzig, Germany (1888)

    Google Scholar 

  2. Lie, S.: Vorlesungen über differentialgleichungen mit bekannten infinitesimalen transformationen. Teubner, Leipzig, Germany (1891)

    MATH  Google Scholar 

  3. Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)

    MATH  Google Scholar 

  4. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. D. Reidel Publishing Company, Dordrecht (1985)

    Book  MATH  Google Scholar 

  5. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)

    Book  MATH  Google Scholar 

  6. Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, New York (1995)

    Book  MATH  Google Scholar 

  7. Ibragimov, N.H. (Eds.): CRC Handbook of Lie Group Analysis of Differential Equations (three volumes). CRC Press, Boca Raton (1994, 1995, 1996)

  8. Baumann, G.: Symmetry Analysis of Differential Equations with Mathematica. Springer, New York (2000)

    Book  MATH  Google Scholar 

  9. Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002)

    MATH  Google Scholar 

  10. Meleshko, S.V.: Methods for Constructing Exact Solutions of Partial Differential Equations. Springer, New York (2005)

    MATH  Google Scholar 

  11. Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2009)

    MATH  Google Scholar 

  12. Bordag, L.A.: Geometrical Properties of Differential Equations. Applications of the Lie Group Analysis in Financial Mathematics. World Scientific, Singapore (2015)

    Book  MATH  Google Scholar 

  13. Oliveri, F.: Lie symmetries of differential equations: classical results and recent contributions. Symmetry 2, 658–706 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Donato, A., Oliveri, F.: Linearization procedure of nonlinear first order systems of PDE’s by means of canonical variables related to Lie groups of point transformations. J. Math. Anal. Appl. 188, 552–568 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Donato, A., Oliveri, F.: When nonautonomous equations are equivalent to autonomous ones. Appl. Anal. 58, 313–323 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Donato, A., Oliveri, F.: How to build up variable transformations allowing one to map nonlinear hyperbolic equations into autonomous or linear ones. Transp. Th. Stat. Phys. 25, 303–322 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gorgone, M., Oliveri, F.: Nonlinear first order partial differential equations reducible to first order homogeneous and autonomous quasilinear ones. Ricerche di Matematica 66, 51–63 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gorgone, M., Oliveri, F.: Nonlinear first order PDEs reducible to autonomous form polynomially homogeneous in the derivatives. J. Geom. Phys. 113, 53–64 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Oliveri, F.: General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form. Int. J. Non-Linear Mech. 47, 53–60 (2012)

    Article  Google Scholar 

  20. Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1969)

    MathSciNet  MATH  Google Scholar 

  21. Fushchich, W.I.: How to extend symmetry of differential equation? In: Fushchych, W.I., Galitsyn, A.S., Demenin, A.N., Nikitin, A.G. (eds.) Symmetry and Solutions of Nonlinear Equations of Mathematical Physics. Inst. Math. Acad. Sci. Ukra., 4–16 (1987)

  22. Fushchich, W.I., Tsyfra, I.M.: On a reduction and solutions of the nonlinear wave equations with broken symmetry. J. Phys. A: Math. Gen. 20, L45–L48 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Arrigo, D.J., Broadbridge, P., Hill, J.M.: Nonclassical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal. J. Math. Phys. 34, 4692–4703 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Clarkson, P.A., Kruskal, M.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  25. Clarkson, P.A., Mansfield, E.L.: Symmetry reductions and exact solutions of nonlinear heat equations. Physica D 10, 250–288 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Levi, D., Winternitz, P.: Nonclassical symmetry reduction: example of the Boussinesq equation. J. Phys. A: Math. Gen. 22, 2915–2924 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nucci, M.C.: Nonclassical symmetries and Bäcklund transformations. J. Math. Anal. Appl. 178, 294–300 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nucci, M.C., Clarkson, P.A.: The nonclassical method is more general than the direct method for symmetry reductions. An example of the FitzHugh-Nagumo equation. Phys. Lett. A 164, 49–56 (1992)

    Article  MathSciNet  Google Scholar 

  29. Olver, P.J., Rosenau, P.: The construction of special solutions to partial differential equations. Phys. Lett. 144A, 107–112 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Olver, P.J., Rosenau, P.: Group invariant solutions of differential equations. SIAM J. Appl. Math. 47, 263–278 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Saccomandi, G.: A personal overview on the reduction methods for partial differential equations. Note Mat. 23, 217–248 (2004/2005)

  32. Cherniha, R.: New Q-conditional symmetries and exact solutions of some reaction-diffusion-convection equations arising in mathematical biology. J. Math. Anal. Appl. 326, 783–799 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cherniha, R.: Conditional symmetries for systems of PDEs: new definitions and their application for reaction-diffusion systems. J. Phys. A: Math. Theor. 41, 405207 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Cherniha, R., Davydovych, V.: Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system. Math. Comp. Model. 54, 1238–1251 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Cherniha, R., Davydovych, V.: Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities. Commun. Nonlinear Sci. Numer. Simulat. 17, 3177–3188 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Cherniha, R., Pliukhin, O.: New conditional symmetries and exact solutions of reaction-diffusion systems with power diffusivities. J. Phys. A: Math. Theor. 41, 185208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Baikov, V.A., Gazizov, R.I., Ibragimov, N.K.: Approximate symmetries. Mat. Sb. 136, 435–450 (1988), English Transl. in Math. USSR Sb. 64, 427-441 (1989)

  38. Ibragimov, N.H., Kovalev, V.K.: Approximate and Renormgroup Symmetries. Higher Education Press, Springer-Verlag GmbH, Beijing, Berlin-Heidelberg (2009)

    Book  MATH  Google Scholar 

  39. Baikov, V.A., Kordyukova, S.A.: Approximate symmetries of the Boussinesq equation. Quaestiones Mathematicae 26, 1–14 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Dolapçi, I.T., Pakdemirli, M.: Approximate symmetries of creeping flow equations of a second grade fluid. Int. J. Non-Linear Mech. 39, 1603–1618 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gazizov, R.K., Ibragimov, N.H., Lukashchuk, V.O.: Integration of ordinary differential equation with a small parameter via approximate symmetries: reduction of approximate symmetry algebra to a canonical form. Lobachevskii J. Math. 31, 141–151 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gazizov, R.K., Ibragimov, N.H.: Approximate symmetries and solutions of the Kompaneets equation. J. Appl. Mech. Techn. Phys. 55, 220–224 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Gan, Y., Qu, C.: Approximate conservation laws of perturbed partial differential equations. Nonlinear Dyn. 61, 217–228 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ibragimov, N.H., Ünal, G., Jogréus, C.: Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems. J. Math. Anal. Appl. 297, 152–168 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kara, A.H., Mahomed, F.M., Qadir, A.: Approximate symmetries and conservation laws of the geodesic equations for the Schwarzschild metric. Nonlinear Dyn. 51, 183–188 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kovalev, V.F.: Approximate transformation groups and renormgroup symmetries. Nonlinear Dyn. 22, 73–83 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. Pakdemirli, M., Yürüsoy, M., Dolapçi, I.T.: Comparison of approximate symmetry methods for differential equations. Acta Appl. Math. 80, 243–271 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wiltshire, R.J.: Perturbed Lie symmetry and systems of non-linear diffusion equations. J. Nonlinear Math. Phys. 3, 130–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wiltshire, R.J.: Two approaches to the calculation of approximate symmetry exemplified using a system of advection-diffusion equations. J. Comp. Appl. Math. 197, 287–301 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)

    MATH  Google Scholar 

  51. Fushchich, W.I., Shtelen, W.H.: On approximate symmetry and approximate solutions of the non-linear wave equation with a small parameter. J. Phys. A: Math. Gen. 22, 887–890 (1989)

    Article  MATH  Google Scholar 

  52. Diatta, B., Wafo Soh, C., Khalique, C.M.: Approximate symmetries and solutions of the hyperbolic heat equation. Appl. Math. Comp. 205, 263–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. Euler, N., Shulga, M.W., Steeb, W.H.: Approximate symmetries and approximate solutions for a multi-dimensional Landau-Ginzburg equation. J. Phys. A: Math. Gen. 25, 1095–1103 (1992)

    Article  Google Scholar 

  54. Euler, W.H., Euler, N., Köhler, A.: On the construction of approximate solutions for a multidimensional nonlinear heat equation. J. Phys. A: Math. Gen. 27, 2083–2092 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  55. Euler, N., Euler, M.: Symmetry properties of the approximations of multidimensional generalized Van der Pol equations. J. Nonlinear Math. Phys. 1, 41–59 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  56. Di Salvo, R., Gorgone, M., Oliveri, F.: A consistent approach to approximate Lie symmetries of differential equations. Nonlinear Dyn. 91, 371–386 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  57. Gorgone, M.: Approximately invariant solutions of creeping flow equations. Int. J. Non-Linear Mech. 105, 212–220 (2018)

    Article  Google Scholar 

  58. Mahomed, F.M., Qu, C.: Approximate conditional symmetries for partial differential equations. J. Phys. A: Math. Gen. 33, 343–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  59. Shih, M., Momoniat, E., Mahomed, F.M.: Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation. J. Math. Phys. 46, 023503 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  60. Gorgone, M., Oliveri, F.: Approximate Q-conditional symmetries of partial differential equations. Electron. J. Differ. Equ. 25, 133–147 (2018)

    MathSciNet  MATH  Google Scholar 

  61. Murray, J.D.: Mathematical Biology. Spatial Models and Biomedical Applications. Springer-Verlag, Berlin, II (2003)

    MATH  Google Scholar 

  62. Plyukhin, O.H.: Conditional symmetries and exact solutions of one reaction-diffusion-convection equation. Nonlinear Oscillations 10, 381–394 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  63. Oliveri, F.: Relie: a Reduce package for Lie group analysis of differential equations. Submitted (2020) [source code of the package available upon request to the author]

  64. Hearn, A.C.: Reduce Users’ Manual Version 3.8. Santa Monica, CA, USA (2004)

  65. Manno, G., Oliveri, F., Vitolo, R.: On differential equations characterized by their Lie point symmetries. J. Math. Anal. Appl. 332, 767–786 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  66. Manno, G., Oliveri, F., Vitolo, R.: Differential equations uniquely determined by algebras of point symmetries. Theor. Math. Phys. 151, 843–850 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  67. Manno, G., Oliveri, F., Saccomandi, G., Vitolo, R.: Ordinary differential equations described by their Lie symmetry algebra. J. Geom. Phys. 85, 2–15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  68. Gorgone, M., Oliveri, F.: Lie remarkable partial dierential equations characterized by Lie algebras ofpoint symmetries. J. Geom. Phys. 144, 314–323 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Work partly supported by GNFM of “Istituto Nazionale di Alta Matematica”, and by local grants of the University of Messina.

Author information

Authors and Affiliations

  1. Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences (MIFT), University of Messina, viale F.  Stagno d’Alcontres 31, 98166, Messina, Italy

    Matteo Gorgone & Francesco Oliveri

Authors
  1. Matteo Gorgone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Oliveri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Oliveri.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorgone, M., Oliveri, F. Consistent approximate Q-conditional symmetries of PDEs: application to a hyperbolic reaction-diffusion-convection equation. Z. Angew. Math. Phys. 72, 119 (2021). https://doi.org/10.1007/s00033-021-01554-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01554-2

Keywords

Mathematics Subject Classification

Search

Navigation