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Applications of Group-Theoretical Methods in Hydrodynamics

  • Book
  • © 1998

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Authors:
  1. V. K. Andreev

    1. Institute for Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia

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  2. O. V. Kaptsov

    1. Institute for Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia

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  3. V. V. Pukhnachov

    1. Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

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  4. A. A. Rodionov

    1. Institute for Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia

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Part of the book series: Mathematics and Its Applications (MAIA, volume 450)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xii
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  2. Group-Theoretic Classification of the Equations of Motion of a Homogeneous or Inhomogeneous Inviscid Fluid in the Presence of Planar and Rotational Symmetry

    • V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov
    Pages 1-67

  3. Exact Solutions to the Nonstationary Euler Equations in the Presence of Planar and Rotational Symmetry

    • V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov
    Pages 68-127

  4. Nonlinear Diffusion Equations and Invariant Manifolds

    • V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov
    Pages 128-152

  5. The Method of Defining Equations

    • V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov
    Pages 153-180

  6. Stationary Vortex Structures in an Ideal Fluid

    • V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov
    Pages 181-240

  7. Group-Theoretic Properties of the Equations of Motion for a Viscous Heat Conducting Liquid

    • V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov
    Pages 241-270

  8. Exact Solutions to the Equations of Dynamics for a Viscous Liquid

    • V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, A. A. Rodionov
    Pages 271-382

  9. Back Matter

    Pages 383-396
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Keywords

About this book

It was long ago that group analysis of differential equations became a powerful tool for studying nonlinear equations and boundary value problems. This analysis was especially fruitful in application to the basic equations of mechanics and physics because the invariance principles are already involved in their derivation. It is in no way a coincidence that the equations of hydrodynamics served as the first object for applying the new ideas and methods of group analysis which were developed by 1. V. Ovsyannikov and his school. The authors rank themselves as disciples of the school. The present monograph deals mainly with group-theoretic classification of the equations of hydrodynamics in the presence of planar and rotational symmetry and also with construction of exact solutions and their physical interpretation. It is worth noting that the concept of exact solution to a differential equation is not defined rigorously; different authors understand it in different ways. The concept of exact solution expands along with the progress of mathematics (solu­ tions in elementary functions, in quadratures, and in special functions; solutions in the form of convergent series with effectively computable terms; solutions whose searching reduces to integrating ordinary differential equations; etc. ). We consider it justifiable to enrich the set of exact solutions with rank one and rank two in­ variant and partially invariant solutions to the equations of hydrodynamics.

Authors and Affiliations

  • Institute for Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russia

    V. K. Andreev, O. V. Kaptsov, A. A. Rodionov

  • Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    V. V. Pukhnachov

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