Unified Theory for Fractional and Entire Differential Operators:An Approach via Differential Quadruplets and Boundary Restriction Operators (Frontiers in Mathematics)

by: Aaud Rougirel (Author)

Publisher: Birkhäuser

Edition: 2024th

Publication Date: 2024/6/28

Language: English

Print Length: 508 pages

ISBN-10: 3031583558

ISBN-13: 9783031583551

Book Description

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 conces differential triplets and differential quadruplets; the second conces 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 keel.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.

About the Author

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 conces differential triplets and differential quadruplets; the second conces 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 keel.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.

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