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

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