SOLVABILITY OF A MIXED PROBLEM FOR A DEGENERATE SUB-DIFFUSION EQUATION IN A RECTANGULAR DOMAIN

Ikramova Nargis Saidakbarovna; Yunusalieva Mokhinur Tulqinjon qizi

doi:10.56292/SJFSU/vol31_iss6/a201

Authors

Ikramova Nargis Saidakbarovna

Fergana State University

https://orcid.org/0000-0002-1597-644X
Yunusalieva Mokhinur Tulqinjon qizi

Fergana State University

https://orcid.org/0009-0001-0204-6403

DOI:

https://doi.org/10.56292/SJFSU/vol31_iss6/a201

Keywords:

fractional calculus, degenerate differential equation, mixed problem, Caputo fractional differential operator, spectral problem.

Abstract

In the present paper, a mixed boundary value problem for a degenerate sub-diffusion equation in a rectangular domain is studied. By applying the method of separation of variables, a spectral problem for a space-dependent ordinary differential equation is derived. The existence of eigenvalues and eigenfunctions of the resulting spectral problem is established using the theory of positive differential operators. The solution to the problem is constructed in the form of a Fourier series. The convergence of the obtained series is proved, and the uniqueness of the solution is established by exploiting the completeness of the system of eigenfunctions.

Author Biographies

Ikramova Nargis Saidakbarovna, Fergana State University

Senior Lecturer, Fergana State University
Yunusalieva Mokhinur Tulqinjon qizi, Fergana State University

Master’s Student, Fergana State University

References

1. V.V. Uchaikin, Fractional derivatives for Physicists and Engineers. Vol. 1, Background and Theory. Vol. 2, Application, Springer. 2013.

2. R. Hilfer, aditor: Applications of fractional calculas in physics. Singapore, World Scientific. 2000.

3. S. Umarov, M. Hahn, K. Kobayashi, Beyond the triangle: Browian motion, Ito calculas, and Fokker-Plank equation-fractional generalizations. World Scientific. 2017.

4. Ji Sh., Huang R. On the Budyko-Sellers climate model with mushy region // Journal of Mathematical Analysis and Applications. 2016. Vol. 434. Issue 1, pp. 581-598.https://doi.org/10.1016/j.jmaa.2015.09.028

5. Camasta A., Fragnelli G. Boundary controllability for a degenerate beam equation // Mathematical Methods in the Applied Sciences. 2024. Vol. 47. Issue 2, pp. 907-927. https://doi.org/10.1002/mma.9692

6. Camasta A., Fragnelli G. A stability result for a degenerate beam equation // SIAM Journal on Control and Optimization. 2024. Vol. 62. Issue 1, pp. 630-649. https://doi.org/10.1137/23M1565668

7. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam etc. (2006).

8. Podlubny I. Fractional Differential Equations. United States, Academic Press. -1999.

9. S. G. Mikhlin. Variational Methods in Mathematical Physics, Pergamon Press, New York, 1964.

10. V.I. Kondrashov, On the theory of boundary-value problems with boundary conditions containing parameters, Dokl. Akad. Nauk SSSR, 142(6) (1962), pp. 1243–1246.