# Cauchy problems for the time-fractional degenerate diffusion equations

### Abstract

This paper is devoted to the Cauchy problems for the one-dimensional linear time-fractional diffusion equations with $\partial^{\alpha}_{t}$ the Caputo fractional derivative of order $\alpha\in(0,1)$ in the variable t and time-degenerate diffusive coefficients $t^{\beta}$ with $\beta >1-\alpha$. The solutions of Cauchy problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative $\partial^{\alpha}_{t}$ of order $\alpha\in(0,1)$ in the variable $t$, are shown. In the "Problem statement and main results" section of the paper, the solution of the time-fractional degenerate diffusion equation in a variable coefficient with two different initial conditions are considered. In this work, a solution is found by using the Kilbas-Saigo function $E_{\alpha,m,l}(z)$ and applying the Fourier transform $F$ and inverse Fourier transform $\mathcal{F}^{-1}$. Convergence of solution of problem 1 and problem 2 are proven using Plancherel theorem. The existence and uniqueness of the solution of the problem are confirmed.### References

[1] N. Al-Salti, E. Karimov, "Inverse source problems for degenerate time-fractional PDE Progr. Fract. Differ. Appl. 8: 1 (2022), 39–52.

[2] L. Boudabsa, T. Simon, "Some properties of the Kilbas-Saigo function Mathematics, 9: 3 (2021).

[3] A. A. Kilbas, M. Saigo, "On solution of integral equations of Abel-Volterra type". Differ. Integral Equ. 8: 5 (1995), 993–1011

[4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, "Theory and applications of fractional differential equations North-Holland Mathematics Studies, (2006).

[5] I. Podlubny, "Fractional Differential Equations Academic Press, San Diego, (1999).

[6] A. Carpinteri, F. Mainardi, "Fractals and Fractional Calculus in Continuum Mechanics Springer, Berlin, (1997).

[7] R. Hilfer, "Applications of Fractional Calculus in Physics World Sci. Publishing, River Edge, NJ, (2000).

[8] R. Metzler, J. Klafter, "The random walk’s guide to anomalous diffusion: a fractional dinamics approach Physics Reports, 339 (2000), 1–77.

[9] A. M. Nakhushev, "Fractional calculus and its applications". Fizmatlit, Moskva (2003), 272. (in Russian).

[10] S. G. Samko, A. A. Kilbas, O. I. Marichev, "Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, Amsterdam, (1993).

[11] D. Idczak, R. Kamocki, "On the existence and uniqueness and formula for the solution of R−L fractional Cauchy problem in Rn". Fract. Calc. Appl. Anal. 14: 4 (2011), 538–553.

[12] B. J. Kadirkulov, B. Kh. Turmetov, "On a generalization of heat equations Uzbek Math. Journal. 3 (2006), 40–45.

[13] Yu. Luchko, M. Yamamoto, "General time-fractional diffusion equation: Some uniqueness and existence results for the initial-boundary-value problems Fract. Calc. and Appl. Anal. 19: 3 (2018), 676–695.

[14] A. G. Smadiyeva, "Initial-boundary value problem for the time-fractional degenerate diffusion equation JMMCS. 113: 1 (2022), 32–41.

[15] T. K. Yuldashev, B. J. Kadirkulov, R. A. Bandaliyev, "On a mixed problem for Hilfer type fractional differential equation with degeneration Lobachevskii Journal of Mathematics, 43: 1 (2022), 263–274.

[16] M. Caputo, "Linear models of dissipation whose Q is almost frequency independent–II Geophysical Journal of the Royal Astronomical Society. 13: 5 (1967), 529–539.

[2] L. Boudabsa, T. Simon, "Some properties of the Kilbas-Saigo function Mathematics, 9: 3 (2021).

[3] A. A. Kilbas, M. Saigo, "On solution of integral equations of Abel-Volterra type". Differ. Integral Equ. 8: 5 (1995), 993–1011

[4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, "Theory and applications of fractional differential equations North-Holland Mathematics Studies, (2006).

[5] I. Podlubny, "Fractional Differential Equations Academic Press, San Diego, (1999).

[6] A. Carpinteri, F. Mainardi, "Fractals and Fractional Calculus in Continuum Mechanics Springer, Berlin, (1997).

[7] R. Hilfer, "Applications of Fractional Calculus in Physics World Sci. Publishing, River Edge, NJ, (2000).

[8] R. Metzler, J. Klafter, "The random walk’s guide to anomalous diffusion: a fractional dinamics approach Physics Reports, 339 (2000), 1–77.

[9] A. M. Nakhushev, "Fractional calculus and its applications". Fizmatlit, Moskva (2003), 272. (in Russian).

[10] S. G. Samko, A. A. Kilbas, O. I. Marichev, "Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, Amsterdam, (1993).

[11] D. Idczak, R. Kamocki, "On the existence and uniqueness and formula for the solution of R−L fractional Cauchy problem in Rn". Fract. Calc. Appl. Anal. 14: 4 (2011), 538–553.

[12] B. J. Kadirkulov, B. Kh. Turmetov, "On a generalization of heat equations Uzbek Math. Journal. 3 (2006), 40–45.

[13] Yu. Luchko, M. Yamamoto, "General time-fractional diffusion equation: Some uniqueness and existence results for the initial-boundary-value problems Fract. Calc. and Appl. Anal. 19: 3 (2018), 676–695.

[14] A. G. Smadiyeva, "Initial-boundary value problem for the time-fractional degenerate diffusion equation JMMCS. 113: 1 (2022), 32–41.

[15] T. K. Yuldashev, B. J. Kadirkulov, R. A. Bandaliyev, "On a mixed problem for Hilfer type fractional differential equation with degeneration Lobachevskii Journal of Mathematics, 43: 1 (2022), 263–274.

[16] M. Caputo, "Linear models of dissipation whose Q is almost frequency independent–II Geophysical Journal of the Royal Astronomical Society. 13: 5 (1967), 529–539.

How to Cite

BORIKHANOV, M.; SMADIYEVA, A..
Cauchy problems for the time-fractional degenerate diffusion equations.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 117, n. 1, apr. 2023. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1045>. Date accessed: 08 june 2023. doi: https://doi.org/10.26577/JMMCS.2023.v117.i1.02.
Section

Mathematics

Keywords
Time-fractional diffusion equation, Fourier transform, the Kilbas-Saigo function