On a Fractional Operator Combining Proportional and Classical Differintegrals

Dumitru Baleanu, Arran Fernandez and Ali Akgül
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Dumitru Baleanu: Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey
Arran Fernandez: Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, 99628 Famagusta, Northern Cyprus, via Mersin-10, Turkey
Ali Akgül: Department of Mathematics, Faculty of Arts and Sciences, Siirt University, TR-56100 Siirt, Turkey

Mathematics, 2020, vol. 8, issue 3, 1-13

Abstract: The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

Keywords: fractional integrals; Caputo fractional derivatives; fractional differential equations; bivariate Mittag-Leffler functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (8)

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