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

Abstract

The generalized (or coupled) Abel equations exist in many BVPs of fractional-order differential equations and play a key role during the process of converting weak solutions to the true solutions. Motivated by the analysis of double-sided fractional diffusion-advection-reaction equations, this article develops the mapping properties of generalized Abel operators ${\alpha }_a{D}_x^{-s}+{\beta }_x{D}_b^{-s}$ in fractional Sobolev spaces, where , $\alpha +\beta =1$, and ${}_a{D}_x^{-s}$, ${}_x{D}_b^{-s}$ are fractional Riemann–Liouville integrals. It is mainly concerned with the regularity property of $\left({\alpha }_a{D}_x^{-s}+{\beta }_x{D}_b^{-s}\right)u=f$ by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of $u\left(x\right)$ while letting $f\left(x\right)$ become smoother and imposing homogeneous boundary restrictions $u\left(a\right)=u\left(b\right)=0$.