Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions

TL;DR

This paper studies the regularity properties of generalized Abel operators in fractional Sobolev spaces, focusing on boundary conditions, which is important for solving fractional differential equations numerically.

Contribution

It develops the mapping properties of generalized Abel operators in fractional Sobolev spaces with homogeneous boundary conditions, advancing understanding of their regularity behavior.

Findings

Established regularity results for solutions under boundary conditions.

Analyzed the effect of smoother right-hand side functions on solution regularity.

Provided theoretical foundation for numerical methods in fractional differential equations.

Abstract

The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$\ddot{\text{o}}$lderian spaces that admit integrable singularities at the endpoints and relatively inadequate in other functional spaces. In recent years, such operators have appeared in BVPs of fractional-order differential equations such as fractional diffusion equations that are usually studied in the frame of fractional Sobolev spaces for weak solution and numerical approximation; and their analysis plays the key role during the process of converting weak solutions to the true solutions. This article develops the mapping properties of generalized Abel operators $\alpha {_aD_x^{-s}}+\beta {_xD_b^{-s}}$ in fractional Sobolev spaces, where $0<\alpha,\beta$, $\alpha+\beta=1$, $ 0<s<1$ and $ {_aD_x^{-s}}$, $ {_xD_b^{-s}}$ are fractional Riemann-Liouville integrals. It is mainly concerned with the regularity property of $(\alpha {_aD_x^{-s}}+\beta {_xD_b^{-s}})u=f$ by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of $u(x)$ while letting $f(x)$ become smoother and imposing homogeneous boundary restrictions $u(a)=u(b)=0$.