Rougirel A. Unified Theory for Fractional and Entire Differential Operators 2024

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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.
Introduction
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
Nomenclature
Bibliography
Index