Third Boundary-Value Problem in the Half-Strip for the $B$-Parabolic Equation
Matematičeskie zametki, Tome 109 (2021) no. 2, pp. 290-301.

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The third boundary-value problem in the half-strip for a partial differential equation with Bessel operator is studied. Existence and uniqueness theorems are proved. The representation of the solution is found in terms of the Laplace convolution of the exponential function and Mittag-Leffler type function with power multipliers. Uniqueness is proved for the class of bounded functions.
Keywords: Bessel operator, Mittag-Leffler type function, third boundary-value problem.
Mots-clés : anomalous diffusion, $B$-parabolic equation
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F. G. Khushtova. Third Boundary-Value Problem in the Half-Strip for the $B$-Parabolic Equation. Matematičeskie zametki, Tome 109 (2021) no. 2, pp. 290-301. http://geodesic.mathdoc.fr/item/MZM_2021_109_2_a11/

[1] I. A. Kipriyanov, V. V. Katrakhov, V. M. Lyapin, “O kraevykh zadachakh v oblastyakh obschego vida dlya singulyarnykh parabolicheskikh sistem uravnenii”, Dokl. AN SSSR, 230:6 (1976), 1271–1274 | MR | Zbl

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

[3] A. M. Nakhushev, Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003 | Zbl

[4] A. M. Nakhushev, “O pravilnoi postanovke kraevykh zadach dlya parabolicheskikh uravnenii so znakoperemennoi kharakteristicheskoi formoi”, Differents. uravneniya, 9:1 (1973), 130–135 | MR | Zbl

[5] V. Alexiades, “Generalized axially symmetric heat potentials and singular parabolic initial boundary value problems”, Arch. Rational Mech. Anal., 79:4 (1982), 325–350 | DOI | MR

[6] D. Colton, “Cauchy's problem for a singular parabolic partial differential equation”, J. Differential Equations, 8:2 (1970), 250–257 | DOI | MR

[7] S. A. Tersenov, Parabolicheskie uravneniya s menyayuschimsya napravleniem vremeni, Nauka, Sibirsk. otdel., M., 1985 | MR | Zbl

[8] M. I. Matiichuk, Parabolichni singulyarni kraiovi zadachi, In-t matem. NAN Ukraïni, Kiïv, 1999

[9] M. Gevrey, “Sur les équations aux dérivées partielles du type parabolique”, Journ. de Math., 9:6 (1913), 305–476

[10] M. Gevrey, “Sur les équations aux dérivées partielles du type parabolique (suite)”, Journ. de Math., 10:6 (1914), 105–148 | Zbl

[11] O. Arena, “On a singular parabolic equation related to axially symmetric heat potentials”, Ann. Mat. Pura Appl. (4), 105 (1975), 347–393 | DOI | MR

[12] M. Giona, H. E. Roman, “Fractional diffusion equation on fractals: one-dimensional case and asymptotic behavior”, J. Phys. A, 25:8 (1992), 2093–2105 | DOI | MR

[13] C. D. Pagani, “On the parabolic equation $\operatorname{sgn}(x)x^pu_y-u_{xx}=0$ and a related one”, Ann. Mat. Pura Appl. (4), 99 (1974), 333–339 | DOI | MR

[14] O. Arena, “On a degenerate elliptic-parabolic equation”, Comm. Partial Differential Equations, 3:11 (1978), 1007–1040 | DOI | MR

[15] Yu. P. Gorkov, “O predstavlenii resheniya odnoi kraevoi zadachi dlya statsionarnogo uravneniya brounovskogo dvizheniya”, Vych. met. programmirovanie, 5:1 (2004), 118–123

[16] Yu. P. Gorkov, “Ob asimptotike resheniya zadachi brounovskogo dvizheniya”, Vych. met. programmirovanie, 4:1 (2003), 19–25

[17] Yu. P. Gorkov, “Postroenie fundamentalnogo resheniya parabolicheskogo uravneniya s vyrozhdeniem”, Vych. met. programmirovanie, 6:1 (2005), 66–70

[18] S. Kepinski, “Über die Differentialgleichung $\frac{\partial^2 z}{\partial x^2} +\frac{m+1}{x}\frac{\partial z}{\partial x} -\frac{n}{x}\frac{\partial z}{\partial t}=0$”, Math. Ann., 61:3 (1905), 397–405 | DOI | MR

[19] S. Kepinski, “Integration der Differentialgleichung $\frac{\partial^2j}{\partial\xi^2} -\frac{1}{\xi}\frac{\partial j}{\partial t}=0$”, Krakau Anz., 1905, 198–205 | Zbl

[20] W. Myller-Lebedeff, “Über die Anwendung der Integralgleichungen in einer parabolischen Randwertaufgabe”, Math. Ann., 66:3 (1908), 325–330 | DOI | MR

[21] B. O'Shaugnessy, I. Procaccia, “Analytical Solutions for Diffusion on Fractal Objects”, Phys. Rev. Lett., 54 (1985), 455–458 | DOI

[22] A. B. Muravnik, “Funktsionalno-differentsialnye parabolicheskie uravneniya: integralnye predstavleniya i kachestvennye svoistva reshenii zadachi Koshi”, Uravneniya v chastnykh proizvodnykh, SMFN, 52, RUDN, M., 2014, 3–141

[23] V. V. Katrakhov, S. M. Sitnik, “Metod operatorov preobrazovaniya i kraevye zadachi dlya singulyarnykh ellipticheskikh uravnenii”, Singulyarnye differentsialnye uravneniya, SMFN, 64, no. 2, RUDN, M., 2018, 211–426 | DOI | MR

[24] S. M. Sitnik, E. L. Shishkina, Metod operatorov preobrazovaniya dlya differentsialnykh uravnenii s operatorami Besselya, Fizmatlit, M., 2018

[25] A. A. Samarskii, Vvedenie v teoriyu raznostnykh skhem, Nauka, M., 1971 | MR

[26] A. N. Tikhonov, A. A. Samarskii, Uravneniya matematicheskoi fiziki, Nauka, M., 1966 | MR | Zbl

[27] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1965 | MR | Zbl

[28] N. N. Lebedev, Spetsialnye funktsii i ikh prilozheniya, Fizmatlit, M.–L., 1963 | MR | Zbl

[29] M. M. Dzhrbashyan, Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966 | MR | Zbl

[30] A.Yu. Popov, A.M. Sedletskii, “Raspredelenie kornei funktsii Mittag-Lefflera”, Teoriya funktsii, SMFN, 40, RUDN, M., 2011, 3–171 | MR | Zbl

[31] S. A. Tersenov, Vvedenie v teoriyu uravnenii parabolicheskogo tipa s menyayuschimsya napravleniem vremeni, Institut matematiki SO AN SSSR, Novosibirsk, 1982 | MR