# Cauchy problems for the time-fractional degenerate diffusion equations

## DOI:

https://doi.org/10.26577/JMMCS.2023.v117.i1.02## Keywords:

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

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*Journal of Mathematics, Mechanics and Computer Science*,

*117*(1). https://doi.org/10.26577/JMMCS.2023.v117.i1.02