Fractional Scale Calculus: Hadamard vs. Liouville
Manuel Duarte Ortigueira, Gary W. Bohannan
Abstract
A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed.
Topics & Concepts
Hadamard transformFractional calculusTime-scale calculusMathematicsScale (ratio)LogarithmDerivative (finance)Pure mathematicsHadamard three-lines theoremLogarithmic scaleInvariant (physics)Logarithmic derivativeDifferential calculusCalculus (dental)Scale invarianceSeries (stratigraphy)Applied mathematicsMathematical analysisMathematical physicsPhysicsMultivariable calculusStatisticsHadamard matrixQuantum mechanicsControl engineeringPaleontologyEconomicsFinancial economicsBiologyAcousticsEngineeringDentistryMedicineFractional Differential Equations SolutionsAdvanced Mathematical Theories and ApplicationsMathematical functions and polynomials