Fractional operators defined by generalized special functions and their applications

Most of the fractional integral and derivative operators in the literature are defined either from classical derivative operator or from integrals with the help of various special functions. In this study, two different types of fractional integral and derivative operators which have more general structure than the mentioned operators are defined. The first type was obtained by using integrals containing power and confluent hypergeometric function in their kernels, and the second type was obtained by using the M-series in the definition of classical derivative operator. Besides, the properties of all the operators defined here were examined. Finally, examples of differential equations including both types of fractional derivative operators separately, were given and analytical solutions of these equations were also obtained.