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

This article is cited in 16 scientific papers (total in 16 papers)

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

Full text: PDF file (431 kB)
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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
Received: 17.12.1999

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    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  zmath  isi  elib
    2. Andrianov I.V., Awrejcewicz J., Barantsev R.G., “Asymptotic approaches in mechanics: New parameters and procedures”, Appl. Mech. Rev., 56:1 (2003), 87  crossref  adsnasa  elib  scopus
    3. Bruno A.D., Lunev V.V., “Invariant relations for the Fokker-Planck system”, Dokl. Math., 67:3 (2003), 416–422  mathscinet  isi
    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  crossref  mathscinet  zmath  isi
    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  isi  scopus  scopus
    6. A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Russian Math. Surveys, 59:3 (2004), 429–480  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Shamrovskii A.D., Andrianov I.V., Awrejcewicz J., “Asymptotic-group analysis of algebraic equations”, Math. Probl. Eng., 2004, no. 5, 411–451  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. Bruno A.D., Karulina E.S., “Expansions of solutions to the fifth Painlevé equation”, Dokl. Math., 69:2 (2004), 214–220  mathnet  mathscinet  zmath  isi
    9. Bruno A.D., Goryuchkina I.V., “Expansions of solutions to the sixth Painlevé equation”, Dokl. Math., 69:2 (2004), 268–272  mathnet  mathscinet  zmath  isi
    10. Bruno A.D., Shadrina T.V., “An axisymmetric boundary layer on a needle”, Dokl. Math., 69:1 (2004), 57–63  mathnet  mathscinet  mathscinet  zmath  isi  elib
    11. A. D. Bruno, “Power asymptotics of solutions to an ODE system”, Dokl Math, 74:2 (2006), 712  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. A. D. Bruno, “Power-logarithmic expansions of solutions to a system of ordinary differential equations”, Dokl Math, 77:2 (2008), 215  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    13. A. D. Bruno, “Nonpower asymptotic forms of solutions to a system of ordinary differential equations”, Dokl Math, 77:3 (2008), 325  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  isi
    15. A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Preprinty IPM im. M. V. Keldysha, 2013, 056, 31 pp.  mathnet
    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  mathnet  crossref  mathscinet
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