156
D. Levi et al.
4 Conclusions
In this contribution we have applied our recent work on conditional symmetries of
continuous and discrete equations to a reaction–diffusion equation. We have shown
how to carry out the necessary computations for one conditional symmetry and the
difficulties which appear when the symmetry is not projective. In the solved case
we have also shown that the exact solution of the continuous equation becomes an
exact solution of the difference equation.
Acknowledgments DL has been supported by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics. MAR was supported by the Spanish MINECO under project PGC2018-094898-B-I00.
All authors thank the hospitality of CRM, Montreal (Canada).
References
1. G.W. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations
(Springer, New York, 2002)
2. G.W. Bluman, J.D. Cole, The general similarity solutions of the heat equation. J. Math. Mech.
18, 1025–1042 (1969)
3. G.W. Bluman, S. Kumei, Symmetries of Differential Equations (Springer, New York, 1989)
4. B.H. Bradshaw-Hajek, M.P. Edwards, P. Broadbridge, G.H. Williams, Nonclassical symmetry
solutions for reaction-diffusion equations with explicit spatial dependence. Nonlinear Anal. 67,
2541–2552 (2007)
5. P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq equation. J. Math.
Phys. 30 2201–2213 (1989)
6. P.G. Estévez, Non-classical symmetries and the singular manifold method: the Burgers and the
Burgers-Huxley equations. J. Phys. A: Math. Gen. 27, 2113–2127 (1994)
7. N.M. Ivanova, On Lie symmetries of a class of reaction-diffusion equations, in Proceedings of
the IV Workshop “Group Analysis of Differential Equations and Integrable Systems” (2009),
pp. 84–86
8. N.M. Ivanova, C. Sophocleous, On nonclassical symmetries of generalized Huxley equations,
in Proceedings of the V Workshop “Group Analysis of Differential Equations and Integrable
Systems” 91–98 (2009), arXiv:1010.2388v1
9. D. Levi, M.A. Rodríguez, Construction of partial difference schemes: I. The Clairaut, Schwarz,
Young theorem on the lattice. J. Phys. A Math. Theor. 46, 295203 (2013)
10. D. Levi, M.A. Rodríguez, On the construction of partial difference schemes II: discrete
variables and invariant schemes. Acta Polytech. 56, 236–244 (2014)
11. D. Levi, P. Winternitz, Nonclassical symmetry reduction: example of the Boussinesq equation.
J. Phys. A Math. Gen. 22, 2915–2924 (1989)
12. D. Levi, P. Winternitz, Continuous symmetries of difference equations. J. Phys. A Math. Gen.
39, R1 (2006)
13. D. Levi, M.A. Rodríguez, Z. Thomova, Differential equations invariant under conditional
symmetries J. Nonlinear Math. Phys. 26 281–293 (2019)
14. D. Levi, M.A. Rodríguez, Z. Thomova, The discretized Boussinesq equation and its conditional
symmetry reduction. J. Phys. A Math. Theor. 53 045201 (2019)
15. P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993)
16. H. Stephani, Differential Equations, their Solution using Symmetries (Cambridge University
Press, Cambridge, 1989)
17. R.Z. Zhdanov, I.M. Tsyfra, R.O. Popovich, A precise definition of reduction of partial
differential equations. J. Math. Anal. Appl. 238, 101–123 (1999)
D. Levi et al.
4 Conclusions
In this contribution we have applied our recent work on conditional symmetries of
continuous and discrete equations to a reaction–diffusion equation. We have shown
how to carry out the necessary computations for one conditional symmetry and the
difficulties which appear when the symmetry is not projective. In the solved case
we have also shown that the exact solution of the continuous equation becomes an
exact solution of the difference equation.
Acknowledgments DL has been supported by INFN IS-CSN4 Mathematical Methods of Nonlinear Physics. MAR was supported by the Spanish MINECO under project PGC2018-094898-B-I00.
All authors thank the hospitality of CRM, Montreal (Canada).
References
1. G.W. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations
(Springer, New York, 2002)
2. G.W. Bluman, J.D. Cole, The general similarity solutions of the heat equation. J. Math. Mech.
18, 1025–1042 (1969)
3. G.W. Bluman, S. Kumei, Symmetries of Differential Equations (Springer, New York, 1989)
4. B.H. Bradshaw-Hajek, M.P. Edwards, P. Broadbridge, G.H. Williams, Nonclassical symmetry
solutions for reaction-diffusion equations with explicit spatial dependence. Nonlinear Anal. 67,
2541–2552 (2007)
5. P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq equation. J. Math.
Phys. 30 2201–2213 (1989)
6. P.G. Estévez, Non-classical symmetries and the singular manifold method: the Burgers and the
Burgers-Huxley equations. J. Phys. A: Math. Gen. 27, 2113–2127 (1994)
7. N.M. Ivanova, On Lie symmetries of a class of reaction-diffusion equations, in Proceedings of
the IV Workshop “Group Analysis of Differential Equations and Integrable Systems” (2009),
pp. 84–86
8. N.M. Ivanova, C. Sophocleous, On nonclassical symmetries of generalized Huxley equations,
in Proceedings of the V Workshop “Group Analysis of Differential Equations and Integrable
Systems” 91–98 (2009), arXiv:1010.2388v1
9. D. Levi, M.A. Rodríguez, Construction of partial difference schemes: I. The Clairaut, Schwarz,
Young theorem on the lattice. J. Phys. A Math. Theor. 46, 295203 (2013)
10. D. Levi, M.A. Rodríguez, On the construction of partial difference schemes II: discrete
variables and invariant schemes. Acta Polytech. 56, 236–244 (2014)
11. D. Levi, P. Winternitz, Nonclassical symmetry reduction: example of the Boussinesq equation.
J. Phys. A Math. Gen. 22, 2915–2924 (1989)
12. D. Levi, P. Winternitz, Continuous symmetries of difference equations. J. Phys. A Math. Gen.
39, R1 (2006)
13. D. Levi, M.A. Rodríguez, Z. Thomova, Differential equations invariant under conditional
symmetries J. Nonlinear Math. Phys. 26 281–293 (2019)
14. D. Levi, M.A. Rodríguez, Z. Thomova, The discretized Boussinesq equation and its conditional
symmetry reduction. J. Phys. A Math. Theor. 53 045201 (2019)
15. P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993)
16. H. Stephani, Differential Equations, their Solution using Symmetries (Cambridge University
Press, Cambridge, 1989)
17. R.Z. Zhdanov, I.M. Tsyfra, R.O. Popovich, A precise definition of reduction of partial
differential equations. J. Math. Anal. Appl. 238, 101–123 (1999)