Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals

Area of Science:

• Fractional Calculus
• Mathematical Analysis

Background:

• Builds upon previous work on Riemann (ABR) and Caputo (ABC) type fractional operators with Mittag-Leffler kernels.
• Introduces fractional integrals of arbitrary order using the infinite binomial theorem.

Purpose of the Study:

• To derive and analyze new fractional integrals and derivatives.
• To prove existence and uniqueness theorems for ABC-fractional initial value problems.
• To explore semigroup properties and alternative representations of fractional operators.

Main Methods:

• Utilizing the infinite binomial theorem for deriving fractional integrals.
• Investigating semigroup properties of the derived operators.
• Applying fractional operators to ABC-type fractional derivatives.
• Calculating Laplace transforms for AB fractional integrals and iterated differ-integrals.

Main Results:

• Derived fractional integrals of arbitrary order with demonstrated semigroup properties.
• Established existence and uniqueness for linear ABC-type initial value problems with nontrivial solutions.
• Introduced iterated fractional differ-integrals with a fourth parameter and derived their semigroup property.
• Provided an alternative representation for ABR-derivatives and compared it with iterated AB differ-integrals.

Conclusions:

• The developed fractional operators generalize and improve existing results in fractional calculus.
• The study offers a deeper understanding of Atangana-Baleanu fractional calculus properties.
• New theoretical tools are provided for solving fractional differential equations.