Abstract
We study the second boundary-value problem in a half-strip for a differential equation with Bessel operator and the Riemann–Liouville partial derivative. In the case of a zero initial condition, a representation of the solution is obtained in terms of the Fox H-function. The uniqueness of the solution is proved for the class of functions satisfying an analog of the Tikhonov condition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. V. Pskhu, Partial Differential Equations of Fractional Order (Nauka, Moscow, 2005) [in Russian].
Yu. P. Gor’kov, “Construction of the fundamental solution to a parabolic equation with degeneracy,” Vychisl. Metody Programm. 6 (1), 66–70 (2005).
S. A. Tersenov, Parabolic Equations with Changing TimeDirection (Nauka, Sibirsk.Otd., Moscow, 1985) [in Russian].
S. Kh. Gekkieva, “A boundary-value problem for the generalized transport equation with fractional derivative in a semi-infinite domain,” Izv. Kabardino-Balkarsk. Nauchn. Tsentra RAN 1 (8), 6–8 (2002).
A. A. Voroshilov and A. A. Kilbas, “A Cauchy-type problem for the diffusion-wave equation with Riemann–Liouville partial derivative,” Dokl. Akad. Nauk 406 (1), 12–16 (2006) [Dokl. Math. 73 (1), 6–10 (2006)].
A. N. Kochubei, “The Cauchy problem for evolution equations of fractional order,” Differ. Uravn. 25 (8), 1359–1368 (1989) [Differ. Equations 25 (8), 967–974 (1989)].
A. N. Kochubei, “Diffusion of fractional order,” Differ. Uravn. 26 (4), 660–670 (1990) [Differ. Equations 26 (4), 485–492 (1990)].
M. Giona and H. E. Roman, “Fractional diffusion equation for fractals: the one-dimensional case and asymptotic behavior,” J. Phys. A 25 (8), 2093–2105 (1992).
R. Metzler, W. G. Glöckle, and T. F. Nonnenmacher, “Fractional model equation for anomalous diffusion,” Phys. A 211 (1), 13–24 (1994).
R. Metzler and J. Klafter, “The randomwalk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339 (1), 1–77 (2000).
R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description anomalous transport by fractional dynamics,” J. Phys. A 37, R161–R208 (2004).
F. G. Khushtova, “First Boundary-Value Problemin theHalf-Strip for a Parabolic-TypeEquation with Bessel Operator and Riemann–Liouville Derivative,” Mat. Zametki 99 (6), 921–928 (2016) [Math. Notes 99 (6), 916–923 (2016)].
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 3: Special Functions. Additional Chapters (Fizmatlit, Moscow, 2003) [in Russian].
A. A. Kilbas and M. Saigo, H-Transform. Theory and Applications, in Vol. 9: Anal.Methods Spec. Funct. (Chapman & Hall/CRC, Boca Raton, FL, 2004).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1966) [in Russian].
D. S. Kuznetsov, Special Functions (Vyssh. Shkola, Moscow, 1965) [in Russian].
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2: Bessel Functions, Parabolic Cylinder Functions, Orthogonal Polynomials (McGraw–Hill,New York–Toronto–London, 1953; Nauka, Moscow, 1974).
R. Gorenflo, Y. Luchko, and F. Mainardi, “Analytical properties and applications of the Wright function,” Fract. Calc. Appl. Anal. 2 (4), 383–414 (1999).
F. G. Khushtova, “Fundamental solution of the model equation of anomalous diffusion of fractional order,” Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki 19 (4), 722–735 (2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © F. G. Khushtova, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 3, pp. 460–470.
Rights and permissions
About this article
Cite this article
Khushtova, F.G. The Second Boundary-Value Problem in a Half-Strip for a Parabolic-Type Equation with Bessel Operator and Riemann–Liouville Partial Derivative. Math Notes 103, 474–482 (2018). https://doi.org/10.1134/S0001434618030136
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434618030136