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THE Q-ANALOGUES OF NONSINGULAR FRACTIONAL OPERATORS WITH MITTAG-LEFFLER AND EXPONENTIAL KERNELS
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This paper is devoted to introducing a new [Formula: see text]-fractional calculus in the framework of Atangana–Baleanu ([Formula: see text]) and Caputo–Fabrizio ([Formula: see text]) operators. First, an appropriate [Formula: see text]-Mittag-Leffler function is defined, and then [Formula: see text]-analogues of fractional derivatives of Atangana–Baleanu–Riemann ([Formula: see text]) and Atangana–Baleanu–Caputo ([Formula: see text]) are derived. Next, the [Formula: see text]-analogues of proper fractional integrals in the [Formula: see text] sense are proved. Several important properties of these definitions are investigated by using the [Formula: see text]-Laplace transform. Additionally, a suitable [Formula: see text]-exponential function is defined, and the [Formula: see text]-analogues of [Formula: see text] fractional derivatives with their inverse operators are introduced. The higher-order extension of the [Formula: see text]-analogues of [Formula: see text] and [Formula: see text] fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of [Formula: see text]-[Formula: see text] calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.
World Scientific Pub Co Pte Ltd
Title: THE Q-ANALOGUES OF NONSINGULAR FRACTIONAL OPERATORS WITH MITTAG-LEFFLER AND EXPONENTIAL KERNELS
Description:
This paper is devoted to introducing a new [Formula: see text]-fractional calculus in the framework of Atangana–Baleanu ([Formula: see text]) and Caputo–Fabrizio ([Formula: see text]) operators.
First, an appropriate [Formula: see text]-Mittag-Leffler function is defined, and then [Formula: see text]-analogues of fractional derivatives of Atangana–Baleanu–Riemann ([Formula: see text]) and Atangana–Baleanu–Caputo ([Formula: see text]) are derived.
Next, the [Formula: see text]-analogues of proper fractional integrals in the [Formula: see text] sense are proved.
Several important properties of these definitions are investigated by using the [Formula: see text]-Laplace transform.
Additionally, a suitable [Formula: see text]-exponential function is defined, and the [Formula: see text]-analogues of [Formula: see text] fractional derivatives with their inverse operators are introduced.
The higher-order extension of the [Formula: see text]-analogues of [Formula: see text] and [Formula: see text] fractional operators is discussed.
Finally, a demonstrative example is enhanced to check the effectiveness of [Formula: see text]-[Formula: see text] calculus.
We believe that these outcomes will be the care of many researchers in the field of fractional calculus.
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