[Paper Review] Characterization of Image Spaces of Riemann-Liouville Fractional Integral Operators on Sobolev Spaces $W^{m,p}(\Omega)$

Lijing Zhao, Weihua Deng+1 more|arXiv (Cornell University)|Mar 21, 2016

Fractional Differential Equations Solutions32 references4 citations

TL;DR

This paper characterizes the image spaces of Riemann-Liouville fractional integral operators on Sobolev spaces $W^{m,p}(Ω)$, providing equivalent characterizations of these spaces and their intersections. It reveals that functions in these image spaces exhibit specific regularity at boundary points and asymptotic decay near boundaries, with explicit representations using Jacobi and Legendre polynomials. The key contribution is establishing that tempered fractional operators are reciprocal to Riemann-Liouville operators within these image spaces, offering theoretical support for numerical methods in fractional PDEs.

ABSTRACT

Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity result, or the relationship between the left-side and right-side fractional operators are seldom mentioned. In stead of considering the fractional derivative spaces, this paper starts from discussing the image spaces of Riemann-Liouville fractional integrals of $L_p(\Omega)$ functions, since the fractional derivative operators that often used are all pseudo-differential. Then high regularity situation---the image spaces of Riemann-Liouville fractional integral operators on $W^{m,p}(\Omega)$ space are considered. Equivalent characterizations of the defined spaces, as well as of the intersection of the left-side and right-side spaces are given. The behavior of the functions in the defined spaces at both the nearby boundary point/ponits and the points in the domain are demonstrated in a clear way. Besides, tempered fractional operators show to be reciprocal to the corresponding Riemann-Liouville fractional operators, which is expected to make some efforts on theoretical support for relevant numerical methods. Last, we also provide some instructions on how to take advantage of the introduced spaces when numerically solving fractional equations.

Motivation & Objective

• To address the lack of clear theoretical regularity analysis for fractional operators in mathematical models.
• To characterize the image spaces of Riemann-Liouville fractional integral operators on $W^{m,p}(Ω)$, focusing on boundary behavior and regularity.
• To clarify the relationship between left- and right-sided fractional operators and their impact on solution regularity.
• To provide a theoretical foundation for numerical methods by identifying function spaces that capture the true regularity of fractional solutions.
• To offer practical guidance on using these image spaces in the numerical solution of fractional differential equations.

Proposed method

• The paper extends the image space framework from $L^p(\Omega)$ to $W^{m,p}(\Omega)$, analyzing the structure of functions in the image of Riemann-Liouville fractional integrals.
• It provides equivalent characterizations of the image spaces and their intersections using polynomial expansions and asymptotic behavior near boundaries.
• Functions in the image spaces are decomposed into a smooth part in the interior and a boundary-layer part with specific singular behavior, represented via Jacobi and Legendre polynomials.
• The method employs weighted Sobolev spaces and asymptotic expansions to describe the regularity at boundary points, particularly at $x = -1$.
• It establishes reciprocity between tempered fractional operators and Riemann-Liouville operators within the image spaces of $L^p(\Omega)$.
• Numerical implementation is guided by constructing approximations using orthogonal polynomial bases with boundary conditions enforced via $\phi_n^{(k)}(-1) = 0$.

Experimental results

Research questions

• RQ1 What is the precise regularity structure of functions in the image space of Riemann-Liouville fractional integrals on $W^{m,p}(\Omega)$?
• RQ2 How do the singularities at the boundary points influence the global behavior of solutions to fractional PDEs?
• RQ3 What is the relationship between left-sided and right-sided fractional operators in terms of their image spaces?
• RQ4 Can the image spaces of fractional integrals be used to improve the convergence and stability of numerical methods for fractional equations?
• RQ5 How can the asymptotic behavior near boundaries be systematically represented and exploited in spectral approximations?

Key findings

• The image space of the Riemann-Liouville fractional integral on $W^{m,p}(\Omega)$ consists of functions that are smooth in the interior but exhibit specific regularity and asymptotic decay near the boundary, particularly at $x = -1$.
• Functions in the image space can be decomposed into a polynomial part $v_1(x) \in (x+1)^{\gamma_1 - \gamma_2} P_{m-1}(x)$ and a remainder $v_2(x) = o((x+1)^{m + \gamma_1 - \gamma_2 - 1/2})$ as $x \to -1$.
• The intersection of left- and right-sided image spaces admits an equivalent characterization via orthogonal polynomial expansions with controlled boundary behavior.
• Tempered fractional operators are shown to be reciprocal to Riemann-Liouville operators within the image spaces of $L^p(\Omega)$, providing theoretical justification for numerical schemes.
• For $u(x) \in I^{\gamma_1}_{(-1)+}[H^m(\Omega)]$, the function $h(x)$ in the variational formulation lies in $H^m(\Omega)$ with $h(x) = \hat{h}_1(x, \gamma_1, \gamma_2) + \hat{h}_2(x, \gamma_1, \gamma_2)$, where $\hat{h}_1 \in (x+1)^{\gamma_1 - \gamma_2} P_{m-1}(x)$ and $\hat{h}_2 = o((x+1)^{m + \gamma_1 - \gamma_2 - 1/2})$.
• When $\gamma_1 \in (1, 3/2)$, the approximation $\tilde{u}_N(x)$ is constructed using $\phi_n(x)$ satisfying $\phi_n^{(k)}(-1) = 0$ for $k = 0, \dots, m$, ensuring boundary compatibility.