Advertisement

Symmetries of Equations with Nonlocal Terms

  • Sergey V. MeleshkoEmail author
Conference paper
  • 438 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)

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.

Keywords

Lie group Symmetry Invariant solution Integro-differential equation Delay differential equation Stochastic differential equations 

Notes

Acknowledgements

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

References

  1. 1.
    Meleshko, S.V.: Methods for Constructing Exact Solutions of Partial Differential Equations. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York (2005)Google Scholar
  2. 2.
    Grigoriev, YuN, Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V.: Symmetries of Integro-Differential Equations and their Applications in Mechanics and Plasma Physics. Lecture Notes in Physics, vol. 806. Springer, Berlin (2010)CrossRefGoogle Scholar
  3. 3.
    Grigoriev, Yu.N., Meleshko, S.V.: Investigation of invariant solutions of the Boltzmann kinetic equation and its models (1986). Preprint of Institute of Theoretical and Applied MechanicsGoogle Scholar
  4. 4.
    Bobylev, A.V., Ibragimov, N.H.: Relationships between the symmetry properties of the equations of gas kinetics and hydrodynamics. J. Math. Model. 1(3), 100–109 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Taranov, V.B.: On symmetry of one-dimensional high frequency motion of noncollision plasma. J. Tech. Phys. 46(6), 1271–1277 (1976)Google Scholar
  6. 6.
    Bunimovich, A.I., Krasnoslobodtsev, A.V.: Invariant-group solutions of kinetic equations. Mechan. Jzydkosti i gasa 4, 135–140 (1982)Google Scholar
  7. 7.
    Bunimovich, A.I., Krasnoslobodtsev, A.V.: On some invariant transformations of kinetic equations. Vestn. Mosc. State Univ. Ser. 1. Mat. Mech. 4, 69–72 (1983)Google Scholar
  8. 8.
    Grigoriev, Yu.N., Meleshko, S.V.: Group analysis of the integro-differential Boltzman equation. Dokl. AS USSR 297(2), 323–327 (1987)Google Scholar
  9. 9.
    Lin, F., Flood, A.E., Meleshko, S.V.: Exact solutions of population balance equations for aggregation, breakage and growth processes. Appl. Math. Comput. 307(15), 193–203 (2017)MathSciNetGoogle Scholar
  10. 10.
    Lin, F., Flood, A.E., Meleshko, S.V.: Exact solutions of the population balance equation involving aggregation, breakage and growth processes and particle transport in one dimension. Commun. Nonlinear Sci. Numer. Simul. 59, 255–271 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Meleshko, S.V.: On group properties of viscoelastic media motion equations. Modelirovanie v mechanike 2(19)(4), 116–126 (1988)Google Scholar
  12. 12.
    Zhou, L.Q., Meleshko, S.V.: Group analysis of integro-differential equations describing stress relaxation behavior of one-dimensional viscoelastic materials. Int. J. Non-Linear Mech. 77, 223–231 (2015)CrossRefGoogle Scholar
  13. 13.
    Zhou, L.Q., Meleshko, S.V.: Invariant and partially invariant solutions for a linear thermoviscoelasticity. Contin. Mech. Thermodyn. 29, 207–224 (2017)Google Scholar
  14. 14.
    Zhou, L.Q., Meleshko, S.V.: Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory. J. Appl. Mech. Tech. Phys. 58(4), 22–45 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Long, F.-S., Karnbanjong, A., Suriyawichitseranee, A., Grigoriev, Y.N., Meleshko, S.V.: Application of a Lie group admitted by a homogeneous equation for group classification of a corresponding inhomogeneous equation. Commun. Nonlinear Sci. Numer. Simul. 48, 350–360 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bobylev, A.V.: On exact solutions of the Boltzmann equation. Dokl. AS USSR 225(6), 1296–1299 (1975)MathSciNetGoogle Scholar
  17. 17.
    Karnbanjong, A., Suriyawichitseranee, A., Grigoriev, Yu.N., Meleshko, S.V.: Preliminary group classification of the full Boltzmann equation with a source term. AIP Conf. Proc. (030062) (2017)Google Scholar
  18. 18.
    Meleshko, S.V., Grigoriev, Yu.N., Karnbanjong, A., Suriyawichitseranee, A.: Invariant solutions in explicit form of the Boltzmann equation with a source term. J. Phys.: Conf. Ser. (012063) (2017)Google Scholar
  19. 19.
    Lin, F., Flood, A.E., Meleshko, S.V.: Exact solutions of population balance equation. Commun. Nonlinear Sci. Numer. Simul. 36, 378–390 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Senashov, S.I.: Group classification of viscoelastic bar equation. Modelirovanie v mechanike 4(21)(1), 69–72 (1990)Google Scholar
  21. 21.
    Özer, T.: On the symmetry group properties of equations of nonlocal elasticity. Mech. Res. Commun. 26(6), 725–733 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Özer, T.: Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity. Int. J. Eng. Sci. 44(18), 2193–2211 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rudenko, O.V., Soluyan, S.I.: Theoretical Foundations of Nonlinear Acoustics. Plenum Consultants Bureau, New York (1977)CrossRefGoogle Scholar
  24. 24.
    Rudenko, O.V., Soluyan, S.I., Khokhlov, R.V.: Problems of the theory of nonlinear acoustics. Sov. Phys. Acoust. 20(3), 356–359 (1974)Google Scholar
  25. 25.
    Ibragimov, N.H., Meleshko, S.V., Rudenko, O.V.: Group analysis of evolutionary integro-differential equations describing nonlinear waves: general model. J. Phys. A: Math. Theor. 44(315201) (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sarvazyan, A.P., Rudenko, O.V., Swanson, S.D., Fowlkes, J.B., Emelianov, S.Y.: Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics. Ultrasound Med. Biol. 24(9), 1419–1435 (1998)CrossRefGoogle Scholar
  27. 27.
    Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V., Bychenkov, V.Yu.: Group analysis of kinetic equations in a nonlinear thermal transport problem. Int. J. Non-Linear Mech. 71, 1–7 (2015)CrossRefGoogle Scholar
  28. 28.
    Long, F.-S., Meleshko, S.V.: On the complete group classification of the one-dimensional nonlinear Klein-Gordon equation with a delay. Math. Methods Appl. Sci. 39(12), 3255–3270 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Long, F.-S., Meleshko, S.V.: Symmetry analysis of the nonlinear two-dimensional Klein-Gordon equation with a time-varying delay. Math. Methods Appl. Sci. 40(13), 4658–4673 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Dorodnitsyn, V.A., Kozlov, R., Meleshko, S.V., Winternitz, P.: Lie group classification of first-order delay ordinary differential equations. J. Phys. A: Math. Theor. (in press)Google Scholar
  31. 31.
    Dorodnitsyn, V.A., Kozlov, R., Meleshko, S.V., Winternitz, P.: Linear or linearizable first-order delay ordinary differential equations and their lie point symmetries. J. Phys. A: Math. Theor. (in press)Google Scholar
  32. 32.
    Gonzalez-Lopez, A., Kamran, N., Olver, P.J.: Lie algebras of differential operators in two complex variables. Am. J. Math. 114, 1163–1185 (1992)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Gonzalez-Lopez, A., Kamran, N., Olver, P.J.: Lie algebras of vector fields in the real plane. Proc. Lond. Math. Soc. 64, 339–368 (1992)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Pue-on, P., Meleshko, S.V.: Group classification of second-order delay ordinary differential equation. Commun. Nonlinear Sci. Numer. Simul. 15, 1444–1453 (2010).  https://doi.org/10.1016/j.cnsns.2009.06.013MathSciNetCrossRefGoogle Scholar
  35. 35.
    Nasyrov, F.S.: Local Times, Symmetric Integrals and Stochastic Analysis. Fizmatlit, Moscow (2011)Google Scholar
  36. 36.
    Meleshko, S.V., Sumrum, O., Schulz, E.: Application of group analysis to stochastic equations of fluid dynamics. J. Appl. Mech. Tech. Phys. 54(1), 21–33 (2013)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Grigoriev, Yu.N., Meleshko, S.V., Suriyawichitseranee, A.: On group classification of the spatially homogeneous and isotropic Boltzmann equation with sources ii. Int. J. Non-Linear Mech. 61, 15–18 (2014)CrossRefGoogle Scholar
  38. 38.
    Nasyrov, F.S., Abdullin, M.A., Meleshko, S.V.: On new approach to group analysis of one-dimensional stochastic differential equations. J. Appl. Mech. Tech. Phys. 55(2), 1–9 (2014)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Mkhize, T.G., Govinder, K., Moyo, S., Meleshko, S.V.: Linearization criteria for systems of two second-order stochastic ordinary differential equations. Appl. Math. Comput. 301, 25–35 (2017)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics, Institute of ScienceSuranaree University of TechnologyNakhon RatchasimaThailand

Personalised recommendations