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 Uspekhi Mat. Nauk, 2000, Volume 55, Issue 1(331), Pages 3–44 (Mi umn248)

Self-similar solutions and power geometry

A. D. Bruno

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: The prime application of the ideas and algorithms of power geometry is in the study of parameter-free partial differential equations. To each differential monomial we assign a point in $\mathbb R^n$: the vector exponent of this monomial. To a differential equation corresponds its support, which is the set of vector exponents of the monomials in the equation. The forms of self-similar solutions of an equation can be calculated from the support using the methods of linear algebra. The equations of a combustion process, with or without sources, are used as examples. For a quasihomogeneous ordinary differential equation, this approach enables one to reduce the order and to simplify some boundary-value problems. Next, generalizations are made to systems of differential equations. Moreover, we suggest a classification of levels of complexity for problems in power geometry. This classification contains four levels and is based on the complexity of the geometric objects corresponding to a give problem (in the space of exponents). We give a comparative survey of these objects and of the methods based on them for studying solutions of systems of algebraic equations, ordinary differential equations, and partial differential equations. We list some publications in which the methods of power geometry have been effectively applied.

DOI: https://doi.org/10.4213/rm248

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English version:
Russian Mathematical Surveys, 2000, 55:1, 1–42

Bibliographic databases:

UDC: 517.9
MSC: Primary 35B99, 34A34; Secondary 14M25, 34C20, 52B20, 80A25

Citation: A. D. Bruno, “Self-similar solutions and power geometry”, Uspekhi Mat. Nauk, 55:1(331) (2000), 3–44; Russian Math. Surveys, 55:1 (2000), 1–42

Citation in format AMSBIB
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• https://doi.org/10.4213/rm248
• http://mi.mathnet.ru/eng/umn/v55/i1/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bruno, AD, “On an axially symmetric flow of a viscous incompressible fluid around a needle”, Doklady Mathematics, 66:3 (2002), 396
2. Andrianov I.V., Awrejcewicz J., Barantsev R.G., “Asymptotic approaches in mechanics: New parameters and procedures”, Appl. Mech. Rev., 56:1 (2003), 87
3. Bruno A.D., Lunev V.V., “Invariant relations for the Fokker-Planck system”, Dokl. Math., 67:3 (2003), 416–422
4. Bruno A.D., “Power geometry as a new calculus”, Analysis and Applications - Isaac 2001, International Society for Analysis, Applications and Computation, 10, 2003, 51–71
5. Weissbac M., Isensee E., Brunotte J., Sommer C., “The use of powerful machines in different soil tillage systems”, Conservation Agriculture: Environment, Farmers Experiences, Innovations, Socio-Economy, Policy, 2003, 367–373
6. A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Russian Math. Surveys, 59:3 (2004), 429–480
7. Shamrovskii A.D., Andrianov I.V., Awrejcewicz J., “Asymptotic-group analysis of algebraic equations”, Math. Probl. Eng., 2004, no. 5, 411–451
8. Bruno A.D., Karulina E.S., “Expansions of solutions to the fifth Painlevé equation”, Dokl. Math., 69:2 (2004), 214–220
9. Bruno A.D., Goryuchkina I.V., “Expansions of solutions to the sixth Painlevé equation”, Dokl. Math., 69:2 (2004), 268–272
10. Bruno A.D., Shadrina T.V., “An axisymmetric boundary layer on a needle”, Dokl. Math., 69:1 (2004), 57–63
11. A. D. Bruno, “Power asymptotics of solutions to an ODE system”, Dokl Math, 74:2 (2006), 712
12. A. D. Bruno, “Power-logarithmic expansions of solutions to a system of ordinary differential equations”, Dokl Math, 77:2 (2008), 215
13. A. D. Bruno, “Nonpower asymptotic forms of solutions to a system of ordinary differential equations”, Dokl Math, 77:3 (2008), 325
14. Leiter M.P., Gascon S., Martinez-Jarreta B., “Making Sense of Work Life: A Structural Model of Burnout”, Journal of Applied Social Psychology, 40:1 (2010), 57–75
15. A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Preprinty IPM im. M. V. Keldysha, 2013, 056, 31 pp.
16. N. I. Sidnyaev, N. M. Gordeeva, “The asymptotic theory of flows for the near wake of an axisymmetric body”, J. Appl. Industr. Math., 9:1 (2015), 110–118
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