Hostname: page-component-5b777bbd6c-w6wnr Total loading time: 0 Render date: 2025-06-22T15:20:34.361Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetry-based algorithms to relate partial differential equations: II. Linearization by nonlocal symmetries

Published online by Cambridge University Press:  16 July 2009

G. W. Bluman  and
S. Kumei
G. W. Bluman
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, CanadaV6T 1Y4
S. Kumei
Affiliation:
Faculty of Textile Science and Technology, Shinshu University, Ueda, Nagano 386, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

An algorithm is presented to linearize nonlinear partial differential equations by non-invertible mappings. The algorithm depends on finding nonlocal symmetries of the given equations which are realized as appropriate local symmetries of a related auxiliary system. Examples include the Hopf-Cole transformation and the linearizations of a nonlinear heat conduction equation, a nonlinear telegraph equation, and the Thomas equations.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 1 , Issue 3 , September 1990 , pp. 217 - 223
Copyright
Copyright © Cambridge University Press 1990

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.)

Article purchase

Temporarily unavailable

References

Bluman, G. W. & Kumei, S. 1990 Symmetry-based algorithms to relate partial differential equations. I. Local symmetries. Preceding paper.Google Scholar
Bluman, G. W., Kumei, S. & Reid, G. J. 1988 New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811;CrossRefGoogle Scholar
Bluman, G. W. & Reid, G. J. 1988 New symmetries for ordinary differential equations. IMA J. Appl. Math. 40, 87–94.CrossRefGoogle Scholar
Cole, J. D. 1951 On a quasilinear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225–236.Google Scholar
Hasegawa, A. 1974 Propagation of wave intensity shocks in nonlinear interaction of waves and particles. Phys. Lett. 47A, 165–166.CrossRefGoogle Scholar
Hashimoto, H. 1974 Exact solutions of a certain semi-linear system of partial differential equations related to a migrating predation problem. Proc. Japan Acad. 50, 623–627.Google Scholar
Hopf, E. 1950 The partial differential equation ui+uux = μuxx. Comm. Pure Appl. Math. 3, 201–230.CrossRefGoogle Scholar
Kersten, P. H. M. 1987 Infinitesimal Symmetries: a Computational Approach. CWI Tract No. 34, Centrum voor Wiskunde en Informatica, Amsterdam.Google Scholar
Krasil'Shchik, I. S. & Vinogradov, A. M. 1984 Nonlocal symmetries and the theory of coverings: an addendum to A. M. Vinogradov's ‘Local symmetries and conservation laws’. Acta Applic.Math. 2, 79–96.CrossRefGoogle Scholar
Kumei, S. & Bluman, G. W. 1982 When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173.CrossRefGoogle Scholar
Storm, M. L. 1950 Heat conduction in simple metals. J. Appl. Phys. 22,940–951.CrossRefGoogle Scholar
Thomas, H. C. 1944 Heterogeneous ion exchange in a flowing system. J. Am. Chem. Soc. 66, 1664–1666.CrossRefGoogle Scholar
Vinogradov, A. M. & Krasil'Shchik, I. S. 1984 On the theory of nonlocal symmetries of nonlinear partial differential equations. Soy. Math. Dokl. 29, 337–341.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley, New York.Google Scholar
Yoshikawa, A. & Yamaguti, M. 1974 On further properties of solutions to a certain semi-linear system of partial differential equations. Publ. Res. Inst. Math. Sci., Kyoto Univ. 9, 577–595.Google Scholar