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Symmetries of Equations with Nonlocal Terms

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Symmetries, Differential Equations and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 266))

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Abstract

An approach for applying group analysis to equations with nonlocal terms is given in the presentation. Similar to the theory of partial differential equations, for invariant solutions of equations with nonlocal terms the number of the independent variables is reduced. The presentation consists of reviewing results obtained by the author with his colleagues related with applications of the group analysis to equations with nonlocal terms such as: integro-differential equations, delay differential equations and stochastic differential equations. The proposed approach can also be applied for defining a Lie group of equivalence, contact and Lie–Bäcklund transformations for equations with nonlocal terms. The presentation is devoted to review the results where the author took a part.

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Notes

  1. 1.

    One of the well-known examples of noninvolutive system of partial differential equations is the system of the Navier–Stokes equations. Although it should be mentioned that the solution of the determining equations of the original system of the Navier–Stokes equations and of the system extended by the equation making the system of the Navier–Stokes equations to be involutive do not change the admitted group.

  2. 2.

    Details can be found in [1, 2].

  3. 3.

    Notice that if Z is constant, then using an equivalence transform it can be reduced to zero.

  4. 4.

    Group analysis also was applied to other viscoelastic and nonlocal elastic models [20,21,22].

  5. 5.

    The determining equations are derived by equating the integrands of each of the integrals. It requires justification. Necessary and sufficient conditions for this justification were obtained recently in [35].

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Acknowledgements

This research was supported by Russian Science Foundation Grant No 18-11-00238 ‘Hydrodynamics-type equations: symmetries, conservation laws, invariant difference schemes’.

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Authors and Affiliations

  1. School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, Thailand

    Sergey V. Meleshko

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  1. Sergey V. Meleshko
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Correspondence to Sergey V. Meleshko .

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Editors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA, USA

    Victor G. Kac

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Peter J. Olver

  3. Department of Mathematics and Statistics, Université de Montréal, Montréal, QC, Canada

    Pavel Winternitz

  4. Civil Engineering Dept, Div.of Mechanics, Istanbul Technical University, Civil Engineering Department, Div.of Mechanics, Maslak (Istanbul), Turkey

    Teoman Özer

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Meleshko, S.V. (2018). Symmetries of Equations with Nonlocal Terms. In: Kac, V., Olver, P., Winternitz, P., Özer, T. (eds) Symmetries, Differential Equations and Applications. Springer Proceedings in Mathematics & Statistics, vol 266. Springer, Cham. https://doi.org/10.1007/978-3-030-01376-9_6

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