Fundamental solution of a singular Bessel differential operator with a negative parameter

L. N. Lyakhov ; Yu. N. Bulatov ; S. A. Roshchupkin ; E. L. Sanina

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2023), pp. 52-65.

Résumé

The singular differential Bessel operator $B_{-\gamma}$ with negative parameter $-\gamma0$ is considered. Solutions of the singular differential Bessel equation $B_{-\gamma} u+\lambda^2u=0$ are represented by linearly independent functions $\mathbb{J}_\mu$ and $\mathbb{J}_{-\mu},~{\mu}=\dfrac{\gamma+1}{2}$. Studied some properties of the functions $\mathbb{J}_\mu$, which are expressed in terms of the properties of the Bessel–Levitan j-function. Direct and inverse Bessel $\mathbb J_\mu$-transforms are introduced. Based on the $\mathbb T$-pseudo-shift operator introduced earlier, a a generalized $\mathbb T$-shift operator belonging to the Levitan class of generalized shifts, commuting with the Bessel operator $B_{-\gamma}$. A fundamental solution is found for the singular differential operator $B_{-\gamma}$ with a singularity at an arbitrary point on the semiaxis $[0,\infty).$

Détail

Keywords: spherical symmetry, singular Bessel differential operator, Bessel transforms, generalized Levitan shift, fundamental solution.

Bibliographie

[1] Lyakhov L.N., Sanina E.L., “Operator Kipriyanova-Beltrami s otritsatelnoi razmernostyu operatorov Besselya i singulyarnaya zadacha Dirikhle dlya V-garmonicheskogo uravneniya”, Differents. uravneniya, 56:12 (2020), 1610–1620 | DOI | Zbl

[2] Lyakhov L.N., Bulatov Yu.N., Roschupkin S.A., Sanina E.L., “Psevdosdvig i fundamentalnoe reshenie $\Delta_B$-operatora Kipriyanova”, Differents. uravneniya, 58:12 (2022), 1654–1665 | DOI | MR | Zbl

[3] Sabitov K.B., Zaitseva N.V., “Vtoraya nachalno-granichnaya zadacha dlya V-giperbolicheskogo uravneniya”, Izv. vuzov. Matem., 2019, no. 10, 75–86 | Zbl

[4] Sabitov K.B., Zaitseva N.V., “Nachalnaya zadacha dlya B-giperbolicheskogo uravneniya s integralnym usloviem vtorogo roda”, Differents. uravneniya, 54:1 (2018), 123–135 | DOI | MR | Zbl

[5] Lyakhov L.N., Sanina E.L., “Differentsialnye i integralnye operatsii v skrytoi sfericheskoi simmetrii i razmernost krivoi Kokha”, Matem. zametki, 113:4 (2023), 517–528

[6] Metzler R., Glöckle W.G., Nonnenmacker T.F., “Fractional model equation for anomalous diffusion”, Physica A: Stat. Mech. Appl., 211:1 (1994), 13–24 | DOI

[7] Vilenkin N.Ya., “Besselevy funktsii i predstavleniya gruppy evklidovykh dvizhenii”, UMN, 11:3(69) (1956), 69–112 | MR | Zbl

[8] Kakichev V.A., “O svertkakh dlya integralnykh preobrazovanii”, Izv. AN BSSR. Ser. fiz-matem. nauk, 1967, no. 2, 48–57 | Zbl

[9] Britvina L.E., “Polisvertki preobrazovaniya Khankelya i differentsialnye operatory”, Dokl. RAN, 382:3 (2002), 298–300 | MR | Zbl

[10] Britvina L.E., “O nekotorykh polisvertkakh, porozhdennykh preobrazovaniem Khankelya”, Matem. zametki, 76:1 (2004), 20—26 | DOI | MR | Zbl

[11] Levitan B.M., “Razlozhenie po funktsiyam Besselya v ryady i integraly Fure”, UMN, 6:2 (42) (1951), 102–143 | MR | Zbl

[12] Kipriyanov I.A., Singulyarnye ellipticheskie kraevye zadachi, Nauka, M., 1997 | MR

[13] Vatson G.N., Teoriya Besselevykh funktsii, In. lit., M., 1949, 798 pp.

[14] Tolstov G.P., Ryady Fure, Nauka, M., 1980, 381 pp. | MR

[15] Lyakhov L.N., “Fundamentalnye resheniya singulyarnykh differentsialnykh uravnenii s $D_B$-operatorom Besselya”, Tr. MIAN, 278, 2012, 148–160 | Zbl