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Journal of Mathematical Sciences

, Volume 130, Issue 5, pp 4911–4940 | Cite as

Linear Differential Relations Between Solutions of the Class of Euler-Poisson-Darboux Equations

  • A. V. Aksenov
Article

Abstract

All the linear first-order relations of the form
$$u^{(\beta )} = A(r,z)\frac{{\partial u^{(\alpha )} }}{{\partial r}} + B(r,z)\frac{{\partial u^{(\alpha )} }}{{\partial z}} + C(r,z)u^{(\alpha )}$$
between solutions u = u(α) and u = u(β) of the class of Euler-Poisson-Darboux (EPD) equations are obtained. We consider applications of the obtained relations for obtaining identities between the EPD operators, recursive relations for the Bessel function, and general solutions of the EPD equation in special cases in application to the gas dynamics of a polytropic gas.

Keywords

General Solution Bessel Function Recursive Relation Differential Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    A. V. Aksenov, “Periodic invariant solutions of equations of absolutely unstable media,” Izv. AN. Mekh. Tverd. Tela, No. 2, 14–20 (1997).Google Scholar
  2. 2.
    A. V. Aksenov, “Symmetries and the relations between the solutions of the class of Euler-Poisson-Darboux equations,” Dokl. Ross. Akad. Nauk, 381, No.2, 176–179 (2001).Google Scholar
  3. 3.
    E. Beltrami, “Sulla teoria delle funzioni potenziali simmetriche,” Mem. R. Accad. sci. Bologna (1880), 2, pp. 461–505 (Opere mat., 3, 349–382 (1911)).Google Scholar
  4. 4.
    S. A. Chaplygin, “On gas jets,” Collection of Works. Vol. II [in Russian], Gos. Izd. Tekhn. Teor. Lit., Moscow-Leningrad (1948), pp. 19–137.Google Scholar
  5. 5.
    R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves [Russian translation], IL, Moscow (1950).Google Scholar
  6. 6.
    G. Darboux, Lecons sur la Theorie Generale des Surfaces et les Applications Geometriques du Calcul Infinitesimal. Vol. II, 2 ed. Paris (1915) (1 ed., 1888).Google Scholar
  7. 7.
    G. V. Dzhaiani, Solution of Certain Problems for One Degenerating Elliptic Equation and Their Application to the Prismatic Shells [in Russian], Izd. Tbilisskogo Gos. Univ., Tbilisi (1982).Google Scholar
  8. 8.
    G. V. Dzhaiani, The Euler-Poisson-Darboux Equation [in Russian], Izd. Tbilisskogo Gos. Univ., Tbilisi (1984).Google Scholar
  9. 9.
    L. Euler, Institutiones calculi integralis, Vol III. Petropoli. 1770. Part. II. Ch. III, IV, V (Opera Omnia. Ser. 1, 13 Leipzig, Berlin (1914), pp. 212–230).Google Scholar
  10. 10.
    N. Kh. Ibragimov, Groups of Transformations in Mathematical Physics [in Russian], Nauka, Moscow (1983).Google Scholar
  11. 11.
    E. Kamke, Reference Book on Ordinary Differential Equation [in Russian], Nauka, Moscow (1976).Google Scholar
  12. 12.
    S. Lie, “Uber die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen,” Arch. Math., 6, No.3, 328–368 (1881).Google Scholar
  13. 13.
    W. Miller (Jr.), “Symmetries of differential equations. The hypergeometric and Euler-Darboux equations,” SIAM J. Math. Anal., 4, No.2, 314–328 (1973).CrossRefGoogle Scholar
  14. 14.
    R. von Mises, Mathematical Theory of Compressible Fluid Flow [Russian translation], IL, Moscow (1961).Google Scholar
  15. 15.
    M. N. Olevskii, “The solution of the Dirichlet problem related to the equation \(\Delta u + \frac{p}{{x_n }}\frac{{\partial u}}{{\partial x_n }} = p\) for the semispheric domain,” Dokl. AN SSSR, 64, No.6, 767–770 (1949).Google Scholar
  16. 16.
    L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).Google Scholar
  17. 17.
    S. D. Poisson, “Memoire sur l'integration des equations lineaires aux differences partielles,” J. L'Ecole Polytech., Ser. 1, 19, 215–248 (1823).Google Scholar
  18. 18.
    M. Abramovits and I. Stigan (Eds.), Reference Book on Special Functions [in Russian], Nauka, Moscow (1979).Google Scholar
  19. 19.
    B. Riemann, On the Propagation of Flat Waves of Finite Amplitude, Ouvres, OGIZ, Moscow-Leningrad (1948), pp. 376–395.Google Scholar
  20. 20.
    K. V. Solyanik-Krassa, Torsion of Shafts of Variable Cross Section [in Russian], Gos. Izd. Tekhn. Teor. Lit., Moscow-Leningrad (1949).Google Scholar
  21. 21.
    K. V. Solyanik-Krassa, Axially Symmetric Problems of Elasticity Theory [in Russian], Stroiizdat, Moscow-Leningrad (1987).Google Scholar
  22. 22.
    K. P. Stanyukovich, Unsteady Motions of Continuum [in Russian], Nauka, Moscow (1971).Google Scholar
  23. 23.
    O. Tsaldastani, “One-dimensional isentropic flow of fluid,” In: Problems of Mechanics. Collection of Papers. R. von Mises and T. Karman (Eds.) [Russian translation] (1955), pp. 519–552.Google Scholar
  24. 24.
    I. N. Vekua, New Methods of Solving of Elliptic Equations [in Russian], OGIZ, Gostekhizdat, Moscow-Leningrad (1948).Google Scholar
  25. 25.
    A. Weinstein, “Discontinuous integrals and generalized potential theory,” Trans. Amer. Math. Soc., 63, No.2, 342–354 (1948).Google Scholar
  26. 26.
    A. Weinstein, “The singular solutions and the Cauchy problem for generalized Tricomi equations,” Commun. Pure Appl. Math., 7, No.1, 105–116 (1954).Google Scholar
  27. 27.
    A. Weinstein, “Some applications of generalized axially symmetric potential theory to continuum mechanics,” Applications of the Theory of Functions in Continuum Mechanics. Papers of Intern. Symp. Vol. 2 (Mechanics of Fluid and Gas, Mathematical Methods) [in Russian], Nauka, Moscow (1965), pp. 440–453.Google Scholar
  28. 28.
    V. F. Zaitsev and A. D. Polyanin, Reference Book on Ordinary Differential Equations [in Russian], Fizmatlit, Moscow (2001).Google Scholar
  29. 29.
    V. K. Zhdanov and B. A. Trubnikov, Quasigas Unstable Media [in Russian], Nauka, Moscow (1991).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. V. Aksenov
    • 1
  1. 1.Moscow State UniversityMoscowRussia

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