Sparse representations of approximation to identity via time–space fractional heat equations

Abstract

In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: $\{\begin{array}{rl}& {\mathrm{\partial }}_t^{\beta }u(t,x)=-\nu (-\mathrm{\Delta }{)}^{\alpha /2}u(t,x),(t,x)\in {\mathbb{R}}_{+}^{1+n};\\ & u(0,x)=f\left(x\right),x\in {\mathbb{R}}^n.\end{array}$Due to the time-fractional derivative, the semigroup property is invalid for the solutions $u(\cdot ,\cdot )$ to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of $u(\cdot ,\cdot )$, which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time–space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations.

KEYWORDS:

• Sparse representations
• fractional time–space heat equations
• adaptive Fourier decomposition
• stochastic fractional time–space heat equations

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M80
• 41A30
• 65N80
• 35K08
• 35K05

Disclosure statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Funding

This work was supported by Research grant of Macau University of Science and Technology (Grant Number FRG-22-075-MCMS), Macao Government Research Funding (Grant Number FDCT0128/2022/A), the National Natural Science Foundation of China (Grant Number 1207127,11871293), the Shandong Natural Science Foundation of China (Grant Number ZR2020MA004) and Zhejiang Provincial Natural Science Foundation of China (Grant Number LQ23A010014).