Skip to main content
SpringerLink
Log in

Scalar Reaction-Diffusion Equations: Conditional Symmetry, Exact Solutions and Applications

  • Chapter
  • First Online:
Nonlinear Reaction-Diffusion Systems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2196))

Abstract

All the main results on Q-conditional symmetry (nonclassical symmetry) of the general class of nonlinear reaction-diffusion-convection equations are summarized. Although some of them were published about 25 years ago, and the others were derived in the 2000s, it is the first attempt to present an extensive review of this matter. It is shown that several well-known equations arising in applications and their direct generalizations possess conditional symmetry. Notably, the Murray, Fitzhugh–Nagumo, and Huxley equations and their natural generalizations are identified. Moreover, several exact solutions (including travelling fronts) are constructed using the conditional symmetries obtained in order to find exact solutions with a biological interpretation.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ablowitz, M., Zeppetella, A.: Explicit solutions of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41, 835–840 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ames, W.F.: Nonlinear Partial Differential Equations in Engineering. Academic Press, New York (1972)

    MATH  Google Scholar 

  3. Arrigo, D.J.: Symmetry Analysis of Differential Equations. Wiley, Hoboken, NJ (2015)

    MATH  Google Scholar 

  4. Arrigo, D.J., Hickling, F.: On the determining equations for the nonclassical reductions of the heat and Burgers’ equation. J. Math. Anal. Appl. 270, 582–589 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arrigo, D.J., Hill, J.M.: Nonclassical symmetries for nonlinear diffusion and absorption. Stud. Appl. Math. 94, 21–39 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Arrigo, D.J., Hill, J.M., Broadbridge, P.: Nonclassical symmetries reductions of the linear diffusion equation with a nonlinear source. IMA J. Appl. Math. 52, 1–24 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. 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 

  10. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)

    Book  MATH  Google Scholar 

  11. Bluman, G.W., Reid, G.J., Kumei, S.: New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2010)

    Book  MATH  Google Scholar 

  13. Britton, N.F.: Essential Mathematical Biology. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  14. Burgers, J.M.: Correlation problems in a one-dimensional model of turbulence I. Proc. Acad. Sci. Amsterdam 53, 247–260 (1950)

    MathSciNet  MATH  Google Scholar 

  15. Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974)

    Book  MATH  Google Scholar 

  16. Carini, M., Fusco, D., Manganaro, N.: Wave-like solutions for a class of parabolic models. Nonlinear Dyn. 32, 211–222 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cherniha, R.: New symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. In: Proceedings of the International Workshop “Similarity Methods”, Universität Stuttgart, Stuttgart, pp. 323–336 (1998)

    Google Scholar 

  18. Cherniha, R.: New Q-conditional symmetries and exact solutions of some reaction-diffusion-convection equations arising in mathematical biology. J. Math. Anal. Appl. 2, 783–799 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cherniha, R., Dutka, V.: Exact and numerical solutions of the generalized Fisher Equation. Rep. Math. Phys. 47, 393–411 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cherniha, R., Henkel, M.: On nonlinear partial differential equations with an infinite-dimensional conditional symmetry. J. Math. Anal. Appl. 298, 487–500 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cherniha, R., Pliukhin, O.: New conditional symmetries and exact solutions of nonlinear reaction–diffusion–convection equations. J. Phys. A Math. Theor. 40, 10049–10070 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cherniha, R., Pliukhin, O.: New conditional symmetries and exact solutions of reaction–diffusion–convection equations with exponential nonlinearities. J. Math. Anal. Appl. 403, 23–37 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cherniha, R.M., Serov, M.I.: Lie and non-lie symmetries of nonlinear diffusion equations with convection term. In: Proceedings of the 2nd International Conference “Symmetry in Nonlinear Mathematical Physics”, pp. 444–449. Institute of Mathematics, Kyiv, Ukraine (1997)

    Google Scholar 

  24. Cherniha, R.M., Serov, M.I.: Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection term. Eur. J. Appl. Math. 9, 527–542 (1998)

    Article  MATH  Google Scholar 

  25. Cherniha, R.M., Serov, M.I.: Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection term II. Eur. J. Appl. Math. 17, 597–605 (2006)

    Article  MATH  Google Scholar 

  26. Clarkson, P.A., Mansfield, E.L.: Symmetry reductions and exact solutions of a class of nonlinear heat equations. Physica D 70, 250–288 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Danilov, V.G., Maslov, V.P., Volosov, K.A.: The flow around a flat plate. In: Mathematical Modelling of Heat and Mass Transfer Processes, pp. 254–294. Springer, Dordrecht (1995)

    Google Scholar 

  28. Fisher, R.A.: The Genetical Theory of Natural Selection. University Press, Oxford (1930)

    Book  MATH  Google Scholar 

  29. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369 (1937)

    MATH  Google Scholar 

  30. Fitzhugh, R.: Impulse and physiological states in models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Article  Google Scholar 

  31. Fokas, A.S., Liu, Q.M.: Generalized conditional symmetries and exact solutions of nonintegrable equations. Theor. Math. Phys. 99, 571–582 (1994)

    Article  MATH  Google Scholar 

  32. Fokas, A.S., Yortsos, Y.C.: On the exactly solvable equation S t = [(βS + γ)−2 S x ] x + α(βS + γ)−1 S x occurring in two-phase flow in porous media. SIAM J. Appl. Math. 42, 318–332 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Fushchych, V.I., Serov, M.I., Chopyk, V.I.: Conditional invariance and nonlinear heat equations (in Russian). Proc. Acad. Sci. Ukr. 9, 17–21 (1988)

    Google Scholar 

  35. Fushchych, W.I., Shtelen, W.M., Serov, M.I., Popovych, R.O.: Q-conditional symmetry of the linear heat equation. Proc. Acad. Sci. Ukr. 12, 28–33 (1992)

    MathSciNet  Google Scholar 

  36. Fushchych, W., Shtelen, W., Serov, M.: Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1993)

    Book  MATH  Google Scholar 

  37. Gandarias, M.L.: New symmetries for a model of fast diffusion. Phys. Lett. A 286, 153–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gandarias, M.L., Romero, J.L., Diaz, J.M.: Nonclassical symmetry reductions of a porous medium equation with convection. J. Phys. A Math. Gen. 32, 1461–1473 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gilding, B.H., Kersner, R.: Travelling Waves in Nonlinear Reaction–Convection–Diffusion. Birkhauser Verlag, Basel (2004)

    Book  MATH  Google Scholar 

  40. Ibragimov, N.H., Meleshko, S.V.: Linearization of third-order ordinary differential equations by point and contact transformations. J. Math. Anal. Appl. 308, 266–289 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kamke, E.: Differentialgleichungen. Lösungmethoden and Lösungen (in German), 6th edn. B. G. Teubner, Leipzig (1959)

    Google Scholar 

  42. Kawahara, T., Tanaka, M.: Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation. Phys. Lett. A 97, 311–314 (1983)

    Article  MathSciNet  Google Scholar 

  43. Kolmogorov, A., Petrovsky, I., Piskounov, N.: Investigation of an equation of diffusion coupled with increasing of matter amount, and its application to a biological problem (in Russian). Mosc. Univ. Math. Bull. 1(6), 1–26 (1937)

    Google Scholar 

  44. Krasil’shchik, I.S., Vinogradov, A.M.: Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math. 15, 161–209 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kuang, Y., Nagy, J. D., Eikenberry S. E.: Introduction to mathematical oncology. In: Chapman & Hall/CRC Mathematical and Computational Biology Series. CRC Press, Boca Raton, FL (2016)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  47. Levi, D., Winternitz, P.: Symmetries and conditional symmetries of differential-difference equations. J. Math. Phys. 34, 3713–3730 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  48. Liu, Q.M., Fokas, A.S.: Exact interaction of solitary waves for certain nonintegrable equations. J. Math. Phys. 37, 324–345 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mansfield, E.L.: The nonclassical group analysis of the heat equation. J. Math. Anal. Appl. 231, 526–542 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  50. Murray, J.D.: Nonlinear Differential Equation Models in Biology. Clarendon Press, Oxford (1977)

    MATH  Google Scholar 

  51. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  52. Murray, J.D.: An Introduction I: Models and Biomedical Applications. Springer, Berlin (2003)

    Google Scholar 

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

    MATH  Google Scholar 

  54. Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2071 (1962)

    Article  Google Scholar 

  55. Nucci, M.C.: Symmetries of linear, C-integrable, S-integrable and nonintegrable equations and dynamical systems. In: Nonlinear Evolution Equations and Dynamical Systems, pp. 374–381. World Scientific Publishing, River Edge (1992)

    Google Scholar 

  56. Nucci, M.C.: Iterations of the non-classical symmetries method and conditional Lie-Bäcklund symmetries. J. Phys. A Math. Gen. 29, 8117–8122 (1996)

    Article  MATH  Google Scholar 

  57. 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)

    MathSciNet  Google Scholar 

  58. Okubo, A., Levin, S.A.: Diffusion and Ecological Problems. Modern Perspectives, 2nd edn. Springer, Berlin (2001)

    Google Scholar 

  59. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  61. Olver, P.J., Vorob’ev, E.M.: Nonclassical and conditional symmetries. In: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, pp. 291–328. CRC Press, Boca Raton (1996)

    Google Scholar 

  62. Oron, A., Rosenau, P.: Some symmetries of the nonlinear heat and wave equations. Phys. Lett. A 118, 172–176 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  63. Ovsiannikov, L.V.: Group properties of nonlinear heat equation (in Russian). Dokl. AN SSSR 125, 492–495 (1959)

    Google Scholar 

  64. Ovsiannikov, L.V.: The Group Analysis of Differential Equations. Academic, New York (1980)

    MATH  Google Scholar 

  65. Popovych, R.O.: Reduction operators of linear second-order parabolic equations. J. Phys. A Math. Theor. 41, 185202 (31pp) (2008)

    Google Scholar 

  66. Pucci, E.: Similarity reductions of partial differential equations. J. Phys. A Math. Gen. 25, 2631–2640 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  67. Pucci, E., Saccomandi, G.: On the weak symmetry groups of partial differential equations. J. Math. Anal. Appl. 163, 588–598 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  68. Pucci, E., Saccomandi, G.: Evolution equations, invariant surface conditions and functional separation of variables. Physica D. 139, 28–47 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  69. Qu, C.: Group classification and generalized conditional symmetry reduction of the nonlinear diffusion–convection equation with a nonlinear source. Stud. Appl. Math. 99, 107–136 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  70. Qu, C., Estévez, P.G.: On nonlinear diffusion equations with x-dependent convection and absorption. Nonlin. Anal. TMA 57, 549–577 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  72. Serov, M.I.: Conditional invariance and exact solutions of the nonlinear equation. Ukr. Math. J. 42, 1216–1222 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  73. Waniewski, J.: Theoretical Foundations for Modeling of Membrane Transport in Medicine and Biomedical Engineering. Institute of Computer Science Polish Academy of Sciences, Warsaw (2015)

    Google Scholar 

  74. Webb, G.M.: Painlev\(\acute{e}\) analysis of a coupled Burgers’ heat equation system, and nonclassical similarity reductions of the heat equation. Physica D 41, 208–218 (1990)

    Article  MathSciNet  Google Scholar 

  75. Witelski, T.P.: Merging traveling waves for the porous-Fisher’s equation. Appl. Math. Lett. 8, 57–62 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  76. Zhdanov, R.Z.: Conditional Lie-Backlund symmetry and reduction of evolution equations. J. Phys. A Math. Gen. 28, 3841–3850 (1995)

    Article  MATH  Google Scholar 

  77. Zhdanov, R.Z., Lahno, V.I.: Conditional symmetry of a porous medium equation. Physica D 122, 178–186 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  78. Zhdanov, R.Z., Tsyfra, I.M., Popovych, R.O.: A precise definition of reduction of partial differential equations. J. Math. Anal. Appl. 238, 101–123 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Science, Kyiv, Ukraine

    Roman Cherniha & Vasyl’ Davydovych

Authors
  1. Roman Cherniha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vasyl’ Davydovych
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Cherniha, R., Davydovych, V. (2017). Scalar Reaction-Diffusion Equations: Conditional Symmetry, Exact Solutions and Applications. In: Nonlinear Reaction-Diffusion Systems. Lecture Notes in Mathematics, vol 2196. Springer, Cham. https://doi.org/10.1007/978-3-319-65467-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation