Journal of Mathematical Sciences

, Volume 243, Issue 1, pp 63–72 | Cite as

Generalization of the Cauchy–Poisson Method and the Construction of Timoshenko-Type Equations

  • I. T. Selezov

We consider a generalization of the Cauchy–Poisson method to the n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. The presented investigation generalizes and supplements the results obtained earlier. In the Euclidean space, we introduce certain restrictions for the derivatives. The principle of hyperbolic degeneracy in terms of parameters is formulated and its realization in the form of necessary and sufficient conditions is presented. In a special case of fourdimensional space (in which the operators are preserved up to the sixth order), we obtain a generalized hyperbolic equation for the bending vibrations of plates with coefficients that depend only on the Poisson ratio. This equation includes, as special cases, the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As the development of Maxwell's and Einstein's investigations of the propagation of perturbations with finite velocity in continuous media, we can mention the nontrivial construction of Timoshenko’s equation for the bending vibrations of a beam.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Weber, General Relativity and Gravitational Waves, Interscience Publ., New York (1961).zbMATHGoogle Scholar
  2. 2.
    B. I. Davydov, “Diffusion equation with regard for the molecular velocity,” Dokl. Akad. Nauk SSSR,2, No. 7, 474–475 (1935).Google Scholar
  3. 3.
    A. S. Kalashnikov, “The concept of a finite rate of propagation of a perturbation,” Uspekhi Mat. Nauk,34, No. 2, 199–200 (1979); English translation: Russ. Math. Surveys,34, No. 2, 235–236 (1979).MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge (1979).Google Scholar
  5. 5.
    A. S. Monin, “On the diffusion with finite velocity,” Izv. Akad. Nauk SSSR. Ser. Geofiz., No. 3, 234–248 (1955).MathSciNetGoogle Scholar
  6. 6.
    Ya. S. Podstrigach, “Temperature field in a system of solids coupled with the help of a thin intermediate layer,” Inzh.-Fiz. Zh.,6, No. 10, 129–136 (1963).Google Scholar
  7. 7.
    I. T. Selezov and Iu. G. Krivonos, Wave Hyperbolic Models of the Propagation of Perturbations [in Russian], Naukova Dumka, Kiev (2015).Google Scholar
  8. 8.
    I. T. Selezov and Iu. G. Krivonos, “Modeling medicine propagation in tissue: Generalized statement,” Kibernet. Sist. Anal.,53, No. 4, 50–58 (2017); English translation: Cybernet. Syst. Analysis,53, No. 4, 535–542 (2017), DOI 10.1007/s10559-017-9955-1.MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. A. Fock, “Solution of one problem of the theory of diffusion by the method of finite differences and its application to the diffusion of light,” Trudy Gos. Optich. Inst.,4, Issue 34, 1–32 (1926); (Sec. 13. Connection with Differential Equations and the Expression for Diffusion, pp. 29–31.)Google Scholar
  10. 10.
    C. Cattaneo, “Sulla conduzione del calore,” Atti Semin. Mat. Fis. della Univ. Modena,3, No. 3, 83–101 (1948).MathSciNetzbMATHGoogle Scholar
  11. 11.
    A. L. Cauchy, “Sur l’équilibre et le mouvement d’une lame solide,” in: A. L. Cauchy, Exercices de Mathematiques, 3, Paris (1828), pp. 245–326,
  12. 12.
    E. Cosserat and F. Cosserat, Théorie de Corps Déformables, Hermann, Paris (1909).zbMATHGoogle Scholar
  13. 13.
    R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publ., New York (1962), Vols. 1, 2.Google Scholar
  14. 14.
    R. W. Davies, “The connection between the Smoluchowski equation and the Kramers–Chandrasekhar equation,” Phys. Rev.,93, No. 6, 1169–1170 (1954).MathSciNetCrossRefGoogle Scholar
  15. 15.
    N. Dunford and J. T. Schwartz, Linear Operators. Part II. Spectral Theory. Self-Adjoint Operators in Hilbert Space, Wiley, New York (1963).zbMATHGoogle Scholar
  16. 16.
    A. Einstein, The Meaning of Relativity, Princeton Univ. Press, Princeton (1950).Google Scholar
  17. 17.
    G. Kirchhoff, “Über das gleichgewicht und die Bewegung einer elastischen Scheibe,” J. Reine Angew. Math.,40, No. 1, 51–88 (1850).MathSciNetGoogle Scholar
  18. 18.
    P. K. Kythe, Fundamental Solutions for Differential Operators and Applications, Birkhäuser, Boston (1996).CrossRefGoogle Scholar
  19. 19.
    A. V. Luikov, “Application of irreversible thermodynamics methods to investigation of heat and mass transfer,” Int. J. Heat Mass Transfer,9, No. 2, 139–152 (1966).CrossRefGoogle Scholar
  20. 20.
    J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Phil. Trans. R. Soc. London,155, 459–512 (1865).CrossRefGoogle Scholar
  21. 21.
    J. C. Maxwell, “On the dynamical theory of gases,” Phil. Trans. R. Soc. London,157, 49–88 (1867).CrossRefGoogle Scholar
  22. 22.
    S. D. Poisson, “Mémoire sur l’équilibre et le mouvement des corps élastiques,” Mém. Acad. Roy. Sci.,8, 357–570 (1829).Google Scholar
  23. 23.
    L. Rayleigh, “On the free vibrations of an infinite plate of homogeneous isotropic elastic matter,” Proc. London Math. Soc.,20, 225–234 (1889).MathSciNetzbMATHGoogle Scholar
  24. 24.
    I. Selezov, “Degenerated hyperbolic approximations of the wave theory of elastic plates,” in: Differential Operators and Related Topics: Proc. M. Krein Internat. Conf. on Operator Theory and Applications (Odessa, Ukraine, August 18–22, 1997), Vol. 1, Operator Theory: Advances and Applications,117, Birkhäuser, Basel (2000), pp. 339–354.CrossRefGoogle Scholar
  25. 25.
    I. Selezov, “Extended models of sedimentation in coastal zone,” Vibrat. Phys. Sys.,26, 243–250 (2014).Google Scholar
  26. 26.
    S. P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bars,” Phil. Mag., Ser. 6,41, No. 245, 744–746 (1921), Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • I. T. Selezov
    • 1
  1. 1.Institute of Hydromechanics, Ukrainian National Academy of SciencesKievUkraine

Personalised recommendations