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 Rn: 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.
This publication is cited in the following 20 articles:
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Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions, 2014, 1
A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Preprinty IPM im. M. V. Keldysha, 2013, 056, 31 pp.
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A. D. Bruno, “Nonpower asymptotic forms of solutions to a system of ordinary differential equations”, Dokl Math, 77:3 (2008), 325
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A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Russian Math. Surveys, 59:3 (2004), 429–480
Bruno A.D., Karulina E.S., “Expansions of solutions to the fifth Painlevé equation”, Dokl. Math., 69:2 (2004), 214–220
Bruno A.D., Goryuchkina I.V., “Expansions of solutions to the sixth Painlevé equation”, Dokl. Math., 69:2 (2004), 268–272
Bruno A.D., Shadrina T.V., “An axisymmetric boundary layer on a needle”, Dokl. Math., 69:1 (2004), 57–63
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Bruno A.D., Lunev V.V., “Invariant relations for the Fokker-Planck system”, Dokl. Math., 67:3 (2003), 416–422
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
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
Bruno, AD, “On an axially symmetric flow of a viscous incompressible fluid around a needle”, Doklady Mathematics, 66:3 (2002), 396
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