Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-03T04:08:16.040Z Has data issue: false hasContentIssue false
  • English
  • Français

On invariant subspaces for nonlinear finite-difference operators

Published online by Cambridge University Press:  14 November 2011

Victor A. Galaktionov
Victor A. Galaktionov
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K. and Keldysh Institute of Applied Mathematics, Miusskaya Sq., 4, 125047 Moscow, Russia e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 6 , 1998 , pp. 1293 - 1308
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bakirov, M. I. and Dorodnitsyn, V. A.. Invariant difference model for the semilinear heat transfer equation. Internal. J. Modern Phys. A, Math. Gen. 30 (1997), 8139–55.Google Scholar
2Dorodnitsyn, V. A.. Symmetry of finite-difference equations. In CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1: Symmetries, Exact Solutions and Conservation Laws, 365–403 (New York: CRC Press, 1993).Google Scholar
3Galaktionov, V. A.. On new exact blow-up solutions for nonlinear heat conduction equations with source and application. Differential and Integral Equations 3 (1990), 863–74.Google Scholar
4Galaktionov, V. A.. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 225–46.CrossRefGoogle Scholar
5Galaktionov, V. A.. Quasilinear heat equations with first-order sign-invariants and new explicit solutions. Nonlinear Anal. 23 (1994), 1595–621.CrossRefGoogle Scholar
6Galaktionov, V. A., Dorodnitsyn, V. A., Elenin, G. G., Kurdyumov, S. P. and Samarskii, A. A.. Quasilinear heat conduction equation with source: blow-up, localization, symmetry, exact solutions, asymptotics, structures. In Modern Math. Problems. New Achievements, Vol. 28, 95–206 (Moscow: VINITI AN SSSR, 1986 (in Russian); English translation: J. Soviet Math. 41 (1988), 1222–92).Google Scholar
7Galaktionov, V. A. and Posashkov, S. A.. On new explicit solutions of parabolic equations with quadratic nonlinearities. Comput. Math. Math. Phys. 29 (1989), 112–19.Google Scholar
8Galaktionov, V. A. and Posashkov, S. A.. Examples of nonsymmetric extinction and blow-up for quasilinear heat equations. Differential and Integral Equations 8 (1995), 87–103.CrossRefGoogle Scholar
9Galaktionov, V. A. and Posashkov, S. A.. New explicit solutions of quasilinear heat equations with general first-order sign-invariants. Phys. D 99 (1996), 217–36.CrossRefGoogle Scholar
10Galaktionov, V. A., Posashkov, S. A. and Svirshchevskii, S. R.. On invariant sets and explicit solutions of nonlinear evolution equations with quadratic nonlinearities. Differential and Integral Equations 8 (1995), 1997–2024.Google Scholar
11Ibragimov, N. H.. Transformations Groups in Mathematical Physics (Moscow: Nauka, 1983 (in Russian); English translation: Dordrecht: D. Reidel, 1985).Google Scholar
12Kalashnikov, A. S.. Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russian Math. Surveys 42 (1987), 169–222.CrossRefGoogle Scholar
13Olver, P. J.. Applications of Lie Groups in Differential Equations (New York: Springer, 1986).Google Scholar
14Ovsjannikov, L. V.. Group Analysis of Differential Equations (Moscow: Nauka, 1978 (in Russian); English translation: New York: Academic Press, 1982).Google Scholar
15Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. and Mikhailov, A. P.. Blow-up in Quasilinear Parabolic Equations (Moscow: Nauka, 1987 (in Russian); English translation: Berlin/New York: Walter de Gruyter, 1995).Google Scholar