Skip to main content
SpringerLink
Log in

Pseudodifferential Local Interaction Equation for Moving Objects

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

The general stochastic nature of the pseudodifferential superdiffusion equation is established. Developing Holtsmark’s idea, we show that the Green’s function of the Cauchy problem for this equation is the probability distribution of the local interaction force of moving objects in a system with power-law interaction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Petrowsky, I., Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen, Math. Sb., 1937, vol. 2, no. 5, pp. 815–870.

    MATH  Google Scholar 

  2. Shilov, G.E., On well-posedness conditions for the Cauchy problem for systems of partial differential equations with constant coefficients, Usp. Mat. Nauk, 1955, vol. 10, no. 4, pp. 89–100.

    MathSciNet  Google Scholar 

  3. Zhitomirskii, Ya.I., The Cauchy problem for some types of G.E. Shilov-parabolic systems of linear partial differential equations with continuous coefficients, Izv. Akad. Nauk SSSR. Ser. Mat., 1959, vol. 23, pp. 925–932.

    MathSciNet  Google Scholar 

  4. Litovchenko, V.A. and Dovzhytska, I.M., Stabilization of solutions to Shilov-type parabolic systems with nonnegative genus, Sib. Math. J., 2014, vol. 55, no. 2, pp. 276–283.

    Article  MathSciNet  Google Scholar 

  5. Shirota, T., Оn Cauchy problem for linear partial differential equations with variable coefficients, Osaka Math. J., 1957, vol. 8, no. 1, pp. 43–59.

    MathSciNet  MATH  Google Scholar 

  6. Wiener, N., Differential space, J. Math. Phys., 1923, vol. 2, pp. 131–174.

    Article  MathSciNet  Google Scholar 

  7. Samko, S.G., Kilbas, A.A., and Marichev, O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya (Fractional Integrals and Derivatives and Some Applications), Minsk: Nauka Tekh., 1987.

    Google Scholar 

  8. Uchaikin, V.V., Metod drobnykh proizvodnykh (Method of Fractional Derivatives), Ul’yanovsk: Artishok, 2008.

    Google Scholar 

  9. Eidel’man, S.D. and Drin’, Ya.M., Necessary and sufficient conditions for stabilization of the solution of the Cauchy problem for parabolic pseudodifferential equations, in Priblizhennye metody matematicheskogo analiza (Approximate Methods of Mathematical Analysis), Kiev: KGPI, 1974, pp. 60–69.

    Google Scholar 

  10. Drin’, Ya.M., Study of one class of parabolic pseudodifferential operators in spaces of Hölder functions, Dop. AN URSR. Ser. A, 1974, no. 1, pp. 19–21.

  11. Fedoryuk, M.V., Asymptotics of the Green’s function of a pseudodifferential parabolic equation, Differ. Uravn., 1978, vol. 14, no. 7, pp. 1296–1301.

    MathSciNet  Google Scholar 

  12. Schneider, W.R., Stable distributions: Fox function representation and generalization, Lect. Not. Phys., 1986, vol. 262, pp. 497–511.

    Article  MathSciNet  Google Scholar 

  13. Kochubei, A.N., Parabolic pseudodifferential equations, hypersingular integrals, and Markov processes, Math. USSR-Izv., 1989, vol. 33, no. 2, pp. 233–259.

    Article  MathSciNet  Google Scholar 

  14. Litovchenko, V.A., Cauchy problem with Riesz operator of fractional differentiation, Ukr. Math. J., 2005, vol. 57, no. 12, pp. 1936–1957.

    Article  MathSciNet  Google Scholar 

  15. Lévy, P., Calcul des Probabilités, Paris: Gauthier-Villars, 1925.

  16. Zolotarev, V.M., Odnomernye ustoichivye raspredeleniya (One-Dimensional Stable Distributions), Moscow: Nauka, 1983.

    MATH  Google Scholar 

  17. Mandelbrot, B., The Pareto–Lévy law and the distribution of income, Int. Econ. Rev., 1960, vol. 1, pp. 79–106.

    Article  Google Scholar 

  18. Sobel’man, I.I., Vvedenie v teoriyu atomnykh spektrov (Introduction to the Theory of Atomic Spectra), Moscow: Fizmatgiz, 1963.

    Google Scholar 

  19. Kac, M., Probability and Related Topics in Physical Sciences, London–New York: Interscience, 1957. Translated under the title: Veroyatnost’ i smezhnye voprosy v fizike, Moscow: Mir, 1965.

    Google Scholar 

  20. Feller, W., An Introduction to Probability Theory and Its Applications, New York: John Wiley & Sons, 1957. Translated under the title: Vvedenie v teoriyu veroyatnostei i ee prilozheniya. T. 2 , Moscow: Mir, 1984.

    MATH  Google Scholar 

  21. Nikiforov, A.F., Novikov, V.G., and Uvarov, V.B., Kvantovo-statisticheskie modeli vysokotemperaturnoi plazmy i metody rascheta rosselandovykh probegov i uravnenii sostoyaniya (Quantum-Statistical Models of High-Temperature Plasma and Methods for Calculating Rosseland Ranges and Equations of State), Moscow: Fizmatlit, 2000.

    Google Scholar 

  22. Agekyan, T.A., Teoriya veroyatnostei dlya astronomov i fizikov (Probability Theory for Astronomers and Physicists), Moscow: Nauka, 1974.

    Google Scholar 

  23. Holtsmark, J., Über die Verbreiterung von Spektrallinien, Ann. der Phys., 1919, vol. 58, pp. 577–630.

    Article  Google Scholar 

  24. Chandrasekhar, S., Stochastic problems in physics and astronomy, Rev. Mod. Phys., 1943, vol. 15, no. 1, pp. 1–89.

    Article  MathSciNet  Google Scholar 

  25. Schwartz, L., Théorie des distributions. Vol. 1 , Paris: Hermann, 1951.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yuriy Fedkovych Chernivtsi National University, Chernivtsi, 58012, Ukraine

    V. A. Litovchenko

Authors
  1. V. A. Litovchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Litovchenko.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Litovchenko, V.A. Pseudodifferential Local Interaction Equation for Moving Objects. Diff Equat 58, 44–52 (2022). https://doi.org/10.1134/S0012266122010062

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266122010062

Search

Navigation