Space-time fractional diffusion equation associated with Jacobi expansions

Abstract

We consider the operator ${\mathrm{\Delta }}^{(\alpha ,\beta )}-I$, where ${\mathrm{\Delta }}^{(\alpha ,\beta )}$ is the three term recurrence relation for the Jacobi polynomial $R_n^{(\alpha ,\beta )}(x)$ normalized at the point x = 1 and I is the identity operator acting on $\mathbf{N}$. Knowing that for $\alpha \ge \beta \ge -\frac{1}{2}$, these polynomials satisfy a product formula which provides a hypergroup structure on $\mathbf{N}$, we use the tools of harmonic analysis on this hypergroup to solve the space-time fractional diffusion equation $\{\begin{array}{l}{\mathbb{D}}_t^{\sigma }u(n,t)=-(-{\mathrm{\Delta }}^{(\alpha ,\beta )}{)}^{s}u(n,t),t>0,n\in \mathbf{N},\\ u(n,0)=f(n).\end{array}$Here, $0<s,\sigma \le 1$, ${\mathbb{D}}_t^{\sigma }$ is the Caputo time fractional derivative of order σ. The solution is given by $f\ast G_{\sigma ,s}^{(\alpha ,\beta )}(\cdot ,t)(n)$, $G_{\sigma ,s}^{(\alpha ,\beta )}(n,t)$ being the fundamental solution expressed in terms of Mittag-Leffler function and $\ast$ is the convolution on the hypergroup $\mathbf{N}$. We prove that, for $0<s<\sigma \le 1$, $G_{\sigma ,s}^{(\alpha ,\beta )}(n,t)$ is nonnegative and we give some of its properties. Next, we study the one parameter operators associated with this fundamental solution.

Keywords:

• Jacobi polynomial
• heat kernel
• space-time fractional diffusion
• Mittag-Leffler function
• one parameter semigroup

2010 Mathematics Subject Classifications:

• 26A33
• 33C45
• 33E12
• 35R11
• 43A62
• 47D06

Acknowledgments

The author is grateful to the referee for his valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).