Unified Theory for Fractional and Entire Differential Operators - An Approach via Differential Quadruplets and Boundary Restriction Operators

This monograph proposes a unified theory of the calculus of fractional and standard derivatives by means of an abstract operator-theoretic approach. By highlighting the axiomatic properties shared by standard derivatives, Riemann-Liouville and Caputo derivatives, the author introduces two new classes of objects. The first class concerns differential triplets and differential quadruplets; the second concerns boundary restriction operators. Instances of boundary restriction operators can be generalized fractional differential operators supplemented with homogeneous boundary conditions. The analysis of these operators comprises: The computation of adjoint operators;The definition of abstract boundary values;The solvability of equations supplemented with inhomogeneous abstract linear boundary conditions;The analysis of fractional inhomogeneous Dirichlet Problems. As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators are closely related to their kernel. Unified Theory for Fractional and Entire Differential Operators will appeal to researchers in analysis and those who work with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and fractional calculus.Table of ContentIntroduction.- Background on Functional Analysis.- Background on Fractional Calculus.- Differential Triplets on Hilbert Spaces.- Differential Quadruplets on Banach Spaces.- Fractional Differential Triplets and Quadruplets on Lebesgue Spaces.- Endogenous Boundary Value Problems.- Abstract and Fractional Laplace Operators.This monograph proposes a unified theory of the calculus of fractional and standard derivatives by means of an abstract operator-theoretic approach. By highlighting the axiomatic properties shared by standard derivatives, Riemann-Liouville and Caputo derivatives, the author introduces two new classes of objects. The first class concerns differential triplets and differential quadruplets; the second concerns boundary restriction operators. Instances of boundary restriction operators can be generalized fractional differential operators supplemented with homogeneous boundary conditions. The analysis of these operators comprises: The computation of adjoint operators;The definition of abstract boundary values;The solvability of equations supplemented with inhomogeneous abstract linear boundary conditions;The analysis of fractional inhomogeneous Dirichlet Problems. As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators are closely related to their kernel. Unified Theory for Fractional and Entire Differential Operators will appeal to researchers in analysis and those who work with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and fractional calculus.Proposes a unified theory of fractional and entire derivatives Focuses on the solvability of linear boundary value problems based on usual and fractional differential operators Establishes the theory utilizing a simple framework that makes it more accessible to researchers