Regularity of the solution to Riesz-type fractional differential equation

In this paper, the Riesz-type fractional differential equation is studied. For the equation defined on , its analytical solution is obtained. The existence and uniqueness of the solution are proved when the right-hand side term belongs to Lebesgue space. Furthermore, it is continuous and differentiable provided that the right-hand side term is continuously differentiable and in Sobolev space or the weighted one. For the equation constrained on a bounded domain, we focus on the case with . Mapping property of Riesz derivative operator on bounded domain indicates end-point singularities in the non-weighted space. Based on the observation of null space of the Riesz derivative operator, the well-posed definite problems of the Riesz-type fractional differential equation are proposed. Through a spectral-type method, solutions to those definite problems are obtained along with corresponding regularity analyses in the weighted space.