Abstract
We determine conditions under which partial differential equations are reducible to equations with a smaller number of independent variables and show that these conditions are necessary and sufficient in the case of a single dependent variable.
Similar content being viewed by others
References
L. V. Ovsyannikov,Group Analysis of Differential Equations [in Russian], Nauka, Moscow 1978.
P. Olver,Application of Lie Groups to Differential Equations, Springer, New York 1986.
W. I. Fushchych, W. M. Shtelen, and N. I. Serov,The Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics, Naukova Dumka, Kiev 1989.
W. I. Fushchych, “Symmetry in problems of mathematical physics,” in:Algebraic-Theoretical Studies in Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1981), pp. 6–44.
W. I. Fushchych, “On symmetry and partial solutions of some multidimensional equations of mathematical physics,” in:Algebraic- Theoretical Studies in Problems of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1983), pp. 4–23.
W. I. Fushchych and N. I. Serov, “The symmetry and exact solutions of the nonlinear multidimensional Liouville, d’ Alembert, and eikonal equations,”J. Phys. A.,16, 3645–3658 (1983).
W. I. Fushchych, “On symmetry and exact solutions of multidimensional nonlinear wave equations,”Ukr. Mat. Zh.,39, No. 1, 116–123 (1987).
W. I. Fushchych, “How to expand symmetry of differential equations?,” in:Symmetry and Solutions of Nonlinear Equations of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987), pp. 4–16.
W. I. Fushchych and A. G. Nikitin,Symmetries of Maxwell’s Equations, Reidel, Dordrecht 1987.
W. I. Fushchych and M. I. Serov, “Conditional symmetry and exact solutions of the nonlinear acoustic equations,”Dokl Akad. Nauk Ukr. SSR, Ser. A., No. 10, 27–31 (1988).
W. I. Fushchych and I. M. Tsifra, “On reductions and solutions of nonlinear wave equations with broken symmetry,”J. Phys. A.,20, L45–48 (1987).
W. I. Fushchych and R. Z. Zhdanov, “Symmetry and exact solutions of nonlinear spinor equations,”Phys. Kept.,172, No. 4, 114–174 (1989).
W. I. Fushchych, W. M. Shtelen, and M. I. Serov,Symmetry Analysis and Exact Solutions of the Equations of Mathematical Physics, Kluwer, Dordrecht 1993.
P. Clarkson and M. Kruskal, “New similarity reductions of the Boussinesq equation,”J. Math. Phys.,30, No. 10, 2201–2213 (1989).
W. I. Fushchych and R. Z. Zhdanov, “Conditional symmetry and reduction of partial differential equations,”Ukr. Mat. Zh.,44, No. 7, 970–982 (1992).
W. I. Fushchych and N. I. Serov, “Conditional invariance and reduction of a nonlinear heat conduction equation,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 24–27 (1990).
Rights and permissions
About this article
Cite this article
Zhdanov, R.Z., Tsifra, I.M. Reduction of differential equations and conditional symmetry. Ukr Math J 48, 661–670 (1996). https://doi.org/10.1007/BF02384233
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02384233