In this paper, we use the generalized notions of Riemann-Liouville (fractional calculus with respect to a regular function σ ) to extend the definitions of fractional integration and derivative from the functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators.

We study the $\rho $-generalized fractional version of the classical Sturm–Liouville operator. We establish the existence and uniqueness of a boundary value problem for this operator with homogeneous boundary conditions. Using this problem, we present a s

This paper investigates the existence and uniqueness of solutions for a class of $\psi-$ generalized fractional differential equations subject to nonhomogeneous boundary value conditions within the framework of fractional Sobolev spaces. We introduce a novel fractional Sobolev space tailored to the $\psi-$ generalized fractional operator and establish its fundamental properties. A variational formulation for the proposed boundary value problem is developed, and the existence and uniqueness of a weak solution are proved by applying Stampacchia's theorem.