Generalization of Fractional Calculus Operators with Applications

Abstract

Generalized forms of fractional calculus operators (integrals and derivatives) are introduced. Caputo -fractional derivative and Hadamard Caputo type -fractional derivative are discussed and their results with some applications are presented. Extensions of Weyl -fractional integral and Hadamard -fractional integral are also introduced. Boundedness of the extended Hadamard -fractional integral in spaces is determined. The generalized -fractional derivative and generalized Caputo type -fractional derivative are introduced and their properties and results are found. Finaly, the generalized type -fractional integral (unifying eleven existing fractional integrals) is introduced and its boundedness in spaces is also determined. Further, integral transforms of - fractional and extended -fractional operators are found. Proofs of properties including semigroup, commutative and some other results for -Weyl fractional integral are given. Moreover, some inequalities for -Weyl fractional integral are discussed and examples are also given to illustrate the results. Relationship between these new generalized forms of fractional calculus operators with the existing fractional operators are discussed by substituting the different values of involved parameters. Integral transforms of new fractional operators and their applications are also given.