Abstract
A PDE modeling a reaction–diffusion physical system is discretized using its conditional symmetries. Discretization is carried out using two specific conditional symmetries. Explicit solutions of the difference equation are constructed when the symmetry is projective.
Keywords
- Symmetry
- Integrable systems
- Difference equations
- Conditional symmetry
- Invariant discretization
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References
G.W. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations (Springer, New York, 2002)
G.W. Bluman, J.D. Cole, The general similarity solutions of the heat equation. J. Math. Mech. 18, 1025–1042 (1969)
G.W. Bluman, S. Kumei, Symmetries of Differential Equations (Springer, New York, 1989)
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)
P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq equation. J. Math. Phys. 30 2201–2213 (1989)
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)
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
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
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)
D. Levi, M.A. Rodríguez, On the construction of partial difference schemes II: discrete variables and invariant schemes. Acta Polytech. 56, 236–244 (2014)
D. Levi, P. Winternitz, Nonclassical symmetry reduction: example of the Boussinesq equation. J. Phys. A Math. Gen. 22, 2915–2924 (1989)
D. Levi, P. Winternitz, Continuous symmetries of difference equations. J. Phys. A Math. Gen. 39, R1 (2006)
D. Levi, M.A. Rodríguez, Z. Thomova, Differential equations invariant under conditional symmetries J. Nonlinear Math. Phys. 26 281–293 (2019)
D. Levi, M.A. Rodríguez, Z. Thomova, The discretized Boussinesq equation and its conditional symmetry reduction. J. Phys. A Math. Theor. 53 045201 (2019)
P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993)
H. Stephani, Differential Equations, their Solution using Symmetries (Cambridge University Press, Cambridge, 1989)
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)
Acknowledgements
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).
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Levi, D., Rodríguez, M.A., Thomova, Z. (2021). Conditional Discretization of a Generalized Reaction–Diffusion Equation. In: Paranjape, M.B., MacKenzie, R., Thomova, Z., Winternitz, P., Witczak-Krempa, W. (eds) Quantum Theory and Symmetries. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-55777-5_14
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DOI: https://doi.org/10.1007/978-3-030-55777-5_14
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