Generalized fractional derivatives generated by a class of local proportional derivatives

• Published in: The European physical journal. ST, Special topics Vol. 226; no. 16-18; pp. 3457 - 3471
• Main Authors: Jarad, Fahd, Abdeljawad, Thabet, Alzabut, Jehad
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-12-2017; Springer Nature B.V
• Subjects: Atomic; Classical and Continuum Physics; Condensed Matter Physics; Exponential functions; Fractional Dynamical Systems - Recent Trends in Theory and Applications; Integrals; Laplace transforms; Materials Science; Mathematical analysis; Measurement Science and Instrumentation; Molecular; Optical and Plasma Physics; Physics; Physics and Astronomy; Proportional derivative; Regular Article
• ISSN: 1951-6355; 1951-6401
• DOI: 10.1140/epjst/e2018-00021-7

Recently, Anderson and Ulness [Adv. Dyn. Syst. Appl.
10
, 109 (2015)] utilized the concept of the proportional derivative controller to modify the conformable derivatives. In parallel to Anderson’s work, Caputo and Fabrizio [Progr. Fract. Differ. Appl.
1
, 73 (2015)] introduced a fractional derivative with exponential kernel whose corresponding fractional integral does not have a semi-group property. Inspired by the above works and based on a special case of the proportional-derivative, we generate Caputo and Riemann-Liouville generalized proportional fractional derivatives involving exponential functions in their kernels. The advantage of the newly defined derivatives which makes them distinctive is that their corresponding proportional fractional integrals possess a semi-group property and they provide undeviating generalization to the existing Caputo and Riemann-Liouville fractional derivatives and integrals. The Laplace transform of the generalized proportional fractional derivatives and integrals are calculated and used to solve Cauchy linear fractional type problems.