Generalized fractional derivatives generated by a class of local proportional derivatives

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