Email this article Investigation of the Cauchy problem for one fractional order equation with the Riemann–Liouville operator
- Authors: Hasanov I.I.1, Akramova D.I.1, Rahmonov A.A.2
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Affiliations:
- Bukhara State University
- Institute of Mathematics named after V.I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan
- Issue: Vol 27, No 1 (2023)
- Pages: 64-80
- Section: Differential Equations and Mathematical Physics
- URL: https://ogarev-online.ru/1991-8615/article/view/145890
- DOI: https://doi.org/10.14498/vsgtu1952
- ID: 145890
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Abstract
The article is dedicated to solving the Cauchy problem for a differential equation with a Riemann–Liouville fractional derivative. The initial condition is formulated in a natural way and it is proven that the resulting solution is regular. Firstly, a fundamental solution is constructed and its properties are analyzed. Then, based on these properties, the solution to the homogeneous equation in the Cauchy problem is studied. Furthermore, unlike other problems of this type, the solution to the Cauchy problem presented for a nonhomogeneous equation is explicitly obtained in this work using the Duhamel’s principle and the three-parameter Mittag–Leffler function. By applying additional conditions to these problems, it is also demonstrated that this solution is classical.
About the authors
Ibrohim I. Hasanov
Bukhara State University
Email: ihasanov998@gmail.com
ORCID iD: 0000-0002-9634-5550
Teacher; Dept. of Differential Equation
Uzbekistan, 705018, Bukhara, st. Muhammad Ikbol, 11Dilshoda I. Akramova
Bukhara State University
Email: akramova.shoda@mail.ru
ORCID iD: 0000-0001-9596-9401
Teacher; Dept. of Mathematical Analysis
Uzbekistan, 705018, Bukhara, st. Muhammad Ikbol, 11Askar A. Rahmonov
Institute of Mathematics named after V.I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan
Author for correspondence.
Email: araxmonov@mail.ru
ORCID iD: 0000-0002-7641-9698
SPIN-code: 2647-3705
Scopus Author ID: 57202852322
Cand. Phys. & Math. Sci.; Senior Researcher
Uzbekistan, 100174, Tashkent, st. Universitetskaya, 46References
- Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Amsterdam, Elsevier, 2006, xx+523 pp. EDN: RLZKLZ. DOI: https://doi.org/10.1016/s0304-0208(06)x8001-5.
- Pskhu A. V. Fractional diffusion equation with discretely distributed differentiation operator, Sib. Èlektron. Mat. Izv., 2016, vol. 13, pp. 1078–1098. EDN: XRNEPH. DOI: https://doi.org/10.17377/semi.2016.13.086.
- Parovik R. I. Cauchy problem for the nonlocal equation diffusion-advection radon in fractal media, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2010, no. 1(20), pp. 127–132. EDN: NBOEJN. DOI: https://doi.org/10.14498/vsgtu742.
- Mamchuev M. O. Modified cauchy problem for a loaded second-order parabolic equation with constant coefficients, Diff. Equat., 2015, vol. 51, no. 9, pp. 1137–1144. EDN: VAHISF. DOI: https://doi.org/10.1134/S0012266115090037.
- Mamchuev M. O. Fundamental solution of a loaded second-order parabolic equation with constant coefficients, Diff. Equat., 2015, vol. 51, no. 5, pp. 620–629. EDN: UEWBJX. DOI: https://doi.org/10.1134/S0012266115050055.
- Durdiev D. K., Shishkina E. L., Sitnik S. M. The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space, Lobachevskii J. Math., 2021, vol. 42, no. 6, pp. 1264–1273. EDN: HGPFMT. DOI: https://doi.org/10.1134/S199508022106007X; arXiv: 2009.10594 [math.CA]. DOI: https://doi.org/10.48550/arXiv.2009.10594.
- Sultanov M. A., Durdiev D. K., Rahmonov A. A. Construction of an explicit solution of a time-fractional multidimensional differential equation, Mathematics, 2021, vol. 9, no. 17, 2052. EDN: HZEAME. DOI: https://doi.org/10.3390/math9172052.
- Gorenflo R., Kilbas A. A., Mainardi F., Rogosin S. V. Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Berlin, Heidelberg, Springer, 2014, xiv+443 pp. DOI: https://doi.org/10.1007/978-3-662-43930-2.
- Mathai A. M., Saxena R. K., Haubold H. J. The H-Function. Theory and Applications. Berlin, Heidelberg: Springer, 2010. xiv+268 pp. DOI: https://doi.org/10.1007/978-1-4419-0916-9.
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