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Introduction to Group Analysis and Invariant Solutions of Integro-Differential Equations

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Symmetries of Integro-Differential Equations

Part of the book series: Lecture Notes in Physics ((LNP,volume 806))

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Abstract

In this chapter an introduction into applications of group analysis to equations with nonlocal operators, in particular, to integro-differential equations is given. The most known integro-differential equations are kinetic equations which form a mathematical basis in the kinetic theories of rarefied gases, plasma, radiation transfer, coagulation. Since these equations are directly associated with fundamental physical laws, there is special interest in studies of their solutions.

The first section of this chapter contains a retrospective survey of different methods for constructing symmetries and finding invariant solutions of such equations. The presentation of the methods is carried out using simple model equations of small dimensionality, allowing the reader to follow the calculations in detail. In the next section, the classical scheme of the construction of determining equations of an admitted Lie group is generalized for equations with nonlocal operators. In the concluding sections of this chapter, the developed regular method of obtaining admitted Lie groups is illustrated by applications to some known integro-differential equations.

The method is a technique which I have applied twice.

Maxim of a traditional professor in mathematics.

G. Polya

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Notes

  1. 1.

    The complete study of local Maxwellian solutions of (2.2.1) done in [21].

  2. 2.

    Some generalizations of the Bobylev approach were also done in [24].

  3. 3.

    More impressive simplification the Fourier transform gives for the full Boltzmann equation with Maxwell molecules: the five-fold collision integral is reduced to a two-fold integral [7]. Unfortunately for other power-like molecular potentials Fourier transform does not give simplifications [33].

  4. 4.

    The invariance of the Boltzmann equation with isotropic Maxwell molecular model with respect to semigroup (2.2.31) was discovered in [50] but it was not used by the author for constructing invariant solutions and for a long time this result was lost.

  5. 5.

    The authors of BKW-mode [42] used a much more long and intricate approach.

  6. 6.

    It is worth to note that the Fourier transform of the Boltzmann equation and the explicit solution rediscovered in [6, 7, 42] were first derived in unknown MS thesis of R. Krupp (see Ref. [15]).

  7. 7.

    This Lie group coincides with the group obtained by using the scaling conjecture (compare with (2.2.51)).

  8. 8.

    This equation coincides with the equation obtained in [66] for the moment generating function of power moments of the original distribution function.

  9. 9.

    Complete calculations using the regular method are presented in the next section.

  10. 10.

    This solution is usually considered as invariant solution with respect to transformations corresponding to the subalgebra {X 2−X 3+c −1 X 1} which is similar to {X 1+X 3}.

  11. 11.

    See Chap. 1 for details.

  12. 12.

    See also [11].

  13. 13.

    Definitions of an admitted Lie group of partial differential equations are discussed in [47], Chap. 6, Sect. 1, [55], Sect. 2.6, [35], Sect. 1.3, [36] (see also Chap. 1), Sect. 9.2, [62], [49], Sect. 6.1 and references therein.

  14. 14.

    According to the Cartan–Kähler theorem, after a finite number of prolongations the system (S) becomes either involutive or incompatible. Therefore, from the theory of compatibility point of view, there is no necessity for infinite prolongations of the system (S).

  15. 15.

    For the sake of simplicity only a Lie group of point transformations is discussed. For tangent transformations the study is similar.

  16. 16.

    There are some trivial examples of such applications for integro-differential equations.

  17. 17.

    See, for example, [9].

  18. 18.

    This solution is usually considered as invariant solution with respect to the subalgebra {X 2−X 3+c −1 X 1} which is similar to {X 1+X 3}.

  19. 19.

    These conditions are boundary conditions, rather than initial conditions.

  20. 20.

    They are called degenerate kernels.

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Authors and Affiliations

  1. Inst. Computational Technologies, Russian Academy of Sciences, Pr. Lavrentjeva 6, 630090, Novosibirsk, Russia

    Prof. Yurii N. Grigoriev

  2. Dept. Mathematics & Science, Blekinge Institute of Technology, 371 79, Karlskrona, Sweden

    Prof. Nail H. Ibragimov

  3. Inst. Mathematical Modelling, Russian Academy of Sciences, Miusskaya Square 4A, 125047, Moscow, Russia

    Dr. Vladimir F. Kovalev

  4. School of Mathematics, Institute of Science, Suranaree University of Technology (SUT), 30000, Nakhon Ratchasima, Thailand

    Sergey V. Meleshko

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  1. Prof. Yurii N. Grigoriev
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  2. Prof. Nail H. Ibragimov
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  3. Dr. Vladimir F. Kovalev
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  4. Sergey V. Meleshko
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Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V. (2010). Introduction to Group Analysis and Invariant Solutions of Integro-Differential Equations. In: Symmetries of Integro-Differential Equations. Lecture Notes in Physics, vol 806. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3797-8_2

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