Boundary value problem for a degenerate equation with a Riemann–Liouville operator
https://doi.org/10.17586/2220-8054-2023-14-5-511-517
Abstract
In the article, the uniqueness and solvability of one boundary value problem for a high-order equation with two lines of degeneracy with a fractional Riemann–Liouville derivative in a rectangular domain is studied by the Fourier method. Sufficient conditions for the well-posedness of the problem posed are obtained.
About the Author
B. Yu. Irgashev
Namangan Engineering Construction Institute; Institute of Mathematics named after V. I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan
Uzbekistan
Uzbekistan
Bakhrom Yu. Irgashev
Namangan
References
1. Bogolyubov A.N., Koblikov A.A., Smirnova D.D. and Shapkina N.E. Mathematical modelling of media with time dispersion using fractional derivatives. Matem. Mod., 2013, 25(12), P. 50–64.
2. Nakhushev A.M. Fractional Calculus and Its Application. Fizmatlit, Moscow, 2003 [in Russian].
3. Artyushin A.N. Fractional integral inequalities and their applications to degenerate differential equations with the Caputo fractional derivative. Siberian Mathematical Journal, 2020, 62(2), P. 208–221.
4. Kilbas A.A., Saigo Megumi. Solution in closed form of a class of linear differential equations of fractional order. Different. Equat., 1997, 33(2), P. 194–204.
5. Samko S.G., Kilbas A.A., Marichev O.I. Fractional integrals and derivatives and some of their applications. Minsk, Science and technology, 1987, p. 688 [in Russian].
6. Oldham, K.B., Spanier, J. The Fractional Calculus. Academic Press, New York, London, 1974.
7. Wiener K. On solutions of a differential equation of nonintegral order that occurs in the theory of polarography, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reiche, 1983, 32(1), P. 41–46 (in German).
8. Karimov E., Ruzhansky M., Toshtemirov B. Solvability of the boundary-value problem for a mixed equation involving hyper-Bessel fractional differential operator and bi-ordinal Hilfer fractional derivative. Mathematical Methods in the Applied Sciences, 2023, 41(1), P.54–77.
9. Ruzhansky M., Torebek B. T., Turmetov B. Well-posedness of Tricomi-Gellerstedt-Keldysh-type fractional elliptic problems. Journal of Integral Equations and Applications. 2022, 34(3), P. 373–387.
10. Smadiyeva A.G. Degenerate time-fractional diffusion equation with initial and initial-boundary conditions. Georgian Mathematical Journal, 2023, 30(3), P.435–443.
11. Smadiyeva A.G. Well-posedness of the initial-boundary value problems for the time-fractional degenerate diffusion equations. Bulletin of the Karaganda University. Mathematics series, 2022, 107(3), P.145–151.
12. Urinov A.K., Usmonov D.A. Non-local initial-boundary value problem for a degenerate fourth-order equation with a fractional Gerasimov-Caputo derivative. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42(1), P. 123–139.
13. Irgashev B.Yu. A boundary value problem with conjugation conditions for a degenerate equation with the Caputo fractional derivative. Russian Mathematics, 2022, 66(4), P. 24–31.
14. Kilbas Anatoly A., Srivastava Hari M., Trujillo Juan J. Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
15. Irgashev B.Y. Initial-boundary problem for degenerate high order equation with fractional derivative. Indian J. Pure Appl. Math., 2022, 53, P. 170– 180.
16. Hilfer R. (Ed.). Applications of fractional calculus in physics / World Scientific Publishing Co, Singapore. 1999.
17. Baleanu D., Guvenc Z.B., Machado J.A.T. (Eds.). New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Berlin, 2010.
18. Baltaeva U., Alikulov Y., Baltaeva I.I., Ashirova A. Analog of the Darboux problem for a loaded integro-differential equation involving the Caputo fractional derivative. Nanosystems: Phys. Chem. Math. 2021, 12(4), P. 418–424.
19. Lotfi Boudabsa, Thomas Simon. Some Properties of the Kilbas-Saigo Function. Mathematics, 217(9), 2021, P. 1–24.
20. Zvonarev S.V., Kortov V.S., Shtang T.V. Modeling the structure and properties of nanosystems. Educational methodical manual. Yekaterinburg, Publishing House of the Ural University, 2014, p. 120 (in Russian).
21. Sibatov R.T., Uchaikin V.V. Fractional differential approach to dispersive transport in semiconductors. Physics-Uspekhi, 2009, 52(10), P. 1019– 1043.
22. Sibatov R.T., Svetukhin V.V. Subdiffusion kinetics of nanoprecipitate growth and destruction in solid solutions. Theor. Math. Phys., 2015, 183, P. 846–859.
23. Uchaikin V.V., Sibatov R.T. Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems. CRC Press, Boca Raton, FL, 2013.
Review
For citations:
Irgashev B.Yu. Boundary value problem for a degenerate equation with a Riemann–Liouville operator. Nanosystems: Physics, Chemistry, Mathematics. 2023;14(5):511-517. https://doi.org/10.17586/2220-8054-2023-14-5-511-517
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