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Order-independent method of characteristics. (English) Zbl 0689.35002

A method of characteristics is developed for any system of partial differential equations of any finite order that admits an isovector field V and an initial data map satisfying a specific transversality condition. It is shown to agree with the classical method of characteristics for a nonlinear, first-order PDE and for quasilinear systems of first-order PDE with the same principal part. The method is also applicable to systems of nonlinear, first-order PDE and to systems of higher order, where it agrees with results obtained by similarity and group invariant methods. Implementation of the characteristic method is easier than classical group invariant methods because a complete, independent system of invariants of the flow generated by the isovector (group symmetry) does not have to be computed. General solutions are obtained only when V is a Cauchy characteristic vector of the fundamental ideal; otherwise, any characteristic solution is shown to satisfy an explicit system of differential constraints. Explicit examples and comparisons with more classical methods are given.
Reviewer: D.G.B.Edelen

MSC:

35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35F25 Initial value problems for nonlinear first-order PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
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References:

[1] Ames, W. F. (1982).Nonlinear Partial Differential Equations in Engineering, Academic Press, New York. · Zbl 0509.65038
[2] Blumin, G. W., and Cole, J. D. (1974).Similarity Methods for Differential Equations, Springer-Verlag, Berlin.
[3] Cartan, E. (1984).Oeuvres Complètes, Springer-Verlag, Berlin.
[4] Chowdhury, K. L. (1986). Isovectors of a class of nonlinear diffusion equations,International Journal of Engineering Science,24, 1597-1605. · Zbl 0625.76097 · doi:10.1016/0020-7225(86)90134-5
[5] Courant, R., and Hilbert, D. (1937).Methoden der Mathematischen Physik, Band II, Springer-Verlag, Berlin. · JFM 63.0449.05
[6] Delph, T. J. (1983). Isovector fields and self-similar solutions for power law creep,International Journal of Engineering Science,21, 1061-1067. · Zbl 0516.73038 · doi:10.1016/0020-7225(83)90047-2
[7] Duff, G. F. D. (1956).Partial Differential Equations, University of Toronto Press, Toronto. · Zbl 0071.30903
[8] Edelen, D. G. B. (1980).Isovector Methods for Equations of Balance, Sitjhoff & Noordhoff, The Netherlands. · Zbl 0452.58001
[9] Edelen, D. G. B. (1983). Isovector fields for problems in the mechanics of solids and fluids,International Journal of Engineering Science,20, 803-815. · Zbl 0503.76003 · doi:10.1016/0020-7225(82)90002-7
[10] Edelen, D. G. B. (1985).Applied Exterior Calculus, Wiley-Interscience, New York. · Zbl 1101.58301
[11] Harrison, B. K., and Estabrook, F. B. (1971). Geometric approach to invariance groups and solutions of partial differential equations,Journal of Mathematical Physics,12, 653-666. · Zbl 0216.45702 · doi:10.1063/1.1665631
[12] Ibragimov, N. H. (1985).Transformation Groups Applied to Mathematical Physics, Reidel, Boston. · Zbl 0558.53040
[13] Olver, P. J. (1986).Applications of Lie Groups to Differential Equations, Springer-Verlag, Berlin. · Zbl 0588.22001
[14] Ovsiannikov, L. V. (1982).Group Analysis of Differential Equations, Academic Press, New York. · Zbl 0485.58002
[15] Papachristou, C. J., and Harrison, B. K. (in press). Isogroups of differential ideals of vector-valued differential forms: applications to partial differential equations, in press. · Zbl 0695.35156
[16] Pommaret, J. F. (1978).Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York. · Zbl 0418.35028
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