POISSON FORMULA FOR SOLVING THE RADIAL CAUCHY PROBLEM FOR A SINGULAR ULTRAHYPERBOLIC EQUATION

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The singular ultrahyperbolic equation (Δ𝐵𝛽)𝑦𝑢= (Δ𝐵𝛾)𝑥𝑢 is considered under the assumption that the I.A. Kipriyanov condition is satisfied, where fractional dimensions of the Δ𝐵𝛾 -operators included in the equation are equal to the same positive number 𝜎. Three types of solutions to the radial Cauchy problem are studied, one of them is based on the T-pseudoshift operator, generalized T-shift and S.A. Tersenov’s method for determining solutions to equations that degenerate on the boundary. Poisson formulas for solving the Cauchy problem for the Euler–Poisson–Darboux equation are given for various values of the parameters in this equation.

About the authors

L. N Lyakhov

Voronezh State University; Lipetsk State Pedagogical University named after P.P. Semenov-Tyan-Shansky; Yelets State University named after I.A. Bunin

Email: levnlya@mail.ru

Yu. N Bulatov

Yelets State University named after I.A. Bunin

Email: y.bulatov@bk.ru

References

  1. Lyakhov, L.N. and Sanina, E.L., Differential and integral operations in hidden spherical symmetry and the dimension of the Koch curve, Math. Notes, 2023, vol. 113, no. 4, pp. 502–511.
  2. Lyakhov, L.N. and Sanina, E.L., Kipriyanov–Beltrami operator with negative dimension of the Bessel operators and the singular Dirichlet problem for the 𝐵-harmonic equation, Differ. Equat., 2020, vol. 56, no. 12, pp. 1564–1574.
  3. Kipriyanov, I.A., Singulyarnye ellipticheskie kraevye zadachi (Singular Elliptic Boundary Value Problems), Moscow: Nauka, 1997.
  4. Mandelbrot, B., The Fractal Geometry of Nature, United States: Henry Holt and Co., 1983.
  5. Metzler, R. Fractional model equation for anomalous diffusion / R. Metzler, W.G. Gl¨ockle, Th.F. Nonnenmacher // Phys. A: Stat. Mech. Appl. — 1994. — V. 211, № 1. — P. 13–24.
  6. Levitan, B.M., Expansion into Fourier series and integrals in Bessel functions, Usp. Mat. Nauk, 1951, vol. 6, no. 2 (42), pp. 102–143.
  7. Lyakhov, L.N., Polovinkin, I.P., and Shishkina, E.L., On a Kipriyanov problem for a singular ultrahyperbolic equation, Differ. Equat., 2014, vol. 50, no. 4, pp. 516–528.
  8. Lyakhov, L.N., Bulatov, Yu.N., Roshchupkin, S.A., and Sanina, E.L., Pseudoshift and the fundamental solution of the Kipriyanov Δ𝐵-operator, Differ. Equat., 2022, vol. 58, no. 12, pp. 1639–1650.
  9. Sabitov, K.B. and Zaitseva, N.V., The second initial-boundary value problem for a 𝐵-hyperbolic equation, Russ. Math., 2019, vol. 63, no. 10, pp. 66–76.
  10. Sabitov, K.B. and Zaitseva, N.V., Initial value problem for 𝐵-hyperbolic equation with integral condition of the second kind, Differ. Equat., 2018, vol. 54, no. 1, pp. 121–133.
  11. Sabitov, K.B., On the uniform convergence of the expansion of a function in the Fourier–Bessel range, Russ. Math., 2022, vol. 66, no. 11, pp. 79–85.
  12. Lyakhov, L.N., Sanina, E.L., Roshchupkin, S.A., and Bulatov, Yu.N., Fundamental solution of a singular Bessel differential operator with a negative parameter, Russ. Math., 2023, vol. 67, no. 7, pp. 43–54.
  13. Levitan, B.M., Teoriya operatorov obobshchennogo sdviga (Theory of Generalized Shift Operators), Moscow: Nauka, 1973.
  14. Levitan, B.M., Application of generalized shift operators to second-order linear differential equations, Usp. Mat. Nauk, 1949, vol. 4, no. 1 (29), pp. 3–112.
  15. Kipriyanov, I.A. and Klyuchantsev, M.I., Singular integrals that are generated by a generalized shift operator, Siberian Math. J., 1970, vol. 11, no. 5, pp. 787–804.
  16. Shishkina, E.L., Uniqueness of the solution of the Cauchy problem for the general Euler–Poisson–Darboux equation, Differ. Equat., 2022, vol. 58, no. 12, pp. 1673–1679.
  17. Tersenov, S.A., Vvedeniye v teoriyu uravneniy, vyrozhdayushchikhsya na granitse (Introduction to the Theory of Equations that Degenerate at the Boundary), Novosibirsk: NSU, 1973.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Russian Academy of Sciences

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).