Jarad, FahdJarad, FahdAbdeljawad, ThabetAbdeljawad, ThabetAlzabut, JehadAlzabut, Jehad2020-03-262020-03-262017Jarad, Fahd; Abdeljawad, Thabet; Alzabut, Jehad "Generalized fractional derivatives generated by a class of local proportional derivatives", European Physical Journal-Special Topics, Vol. 226, No. 16-18, pp. 3457-3471, (2017).1951-6355https://hdl.handle.net/20.500.12416/2752Recently, 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.eninfo:eu-repo/semantics/closedAccessOperatorsCalculusKernelArticle22616-183457347110.1140/epjst/e2018-00021-7