On a Fractional Operator Combining Proportional and Classical Differintegrals

Băleanu, Dumitru; Fernandez, Arran; Akgül, Ali

doi:10.3390/math8030360

Cited by 221 publications

(137 citation statements)

References 20 publications

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“…We undertook the task of properly defining these functions and operators, doing the mathematics which may then be used in analysing those experimental results. The same bivariate Mittag-Leffler functions also arise naturally from some elementary systems of fractional differential equations (Huseynov et al), and from other types of fractional-calculus operators (Baleanu et al 2020). Therefore, we are confident that the functions and operators defined herein will have a rapid impact in several different fields of study.…”

Section: Introductionmentioning

confidence: 62%

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

2020

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We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly, their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly, the fact that they emerge naturally from certain applications in bioengineering. Keywords Mittag-Leffler functions • Fractional integrals • Fractional derivatives • Fractional differential equations • Bivariate Mittag-Leffler functions ρ α,β (x) and E α,β,γ (x), the "two-parameter" and "three-parameter" Mittag-Leffler functions, being defined by power series similar to the Communicated by Roberto Garrappa.

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Section: Introductionmentioning

confidence: 62%

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

2020

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“…Where (y, t) is the constant proportional-Caputo hybrid derivative and is defined as [33]. Taking Laplace transform of eqs.…”

Section: S346mentioning

confidence: 99%

“…In 2020 a new kind of fractional operator with power law recently suggested by Baleanu et al [33] and called it a hybrid fractional derivatives because it is linear combination of two fractional operators known as constant proportional and Caputo type fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

New analytical solutions of heat transfer flow of clay-water base nanoparticles with the application of novel hybrid fractional derivative

et al. 2020

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Clay nanoparticles are hanging in three different based fluids (water, kerosene, and engine oil). The exact terminologies of Maxwell-Garnett and Brinkman for the current thermophysical properties of clay nanofluids are used, while the flow occurrence is directed by a set linear PDE with physical initial and boundary conditions. The classical governing equations are extended to non-integer order hybrid fractional derivative which is introduced in [33]. Analytical solutions for temperature and ve?locity fields are attained via Laplace transform technique. Some limiting solutions are also obtained from the existing literature and compared for different values of fractional parameter. To vision the impact of several flow parameters on the temperature and velocity some graphs are drawn using Mathcad software and designed in different figures. As a result, we found that hybrid fractional model is better in describing the decay behavior of temperature and velocity in comparison of classical derivatives. In comparison of nanofluid with different base fluids, it is concluded that water-based nanofluid has higher velocity than others.

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“…Baleanu et al [27] gave the new direction in the field of fractional calculus and defined some new definitions and successfully applied in fractional partial differential equations and expressed the obtained results in terms of Mittage-Leffler function. The advantage of this new operator is that it consists of two fractional operators' namely constant proportional derivative and Caputo type with power law kernel.…”

Section: Introductionmentioning

confidence: 99%

Thermal analysis of magnetohydrodynamic viscous fluid with innovative fractional derivative

2020

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In this study, an attempt is made to investigate a fractional model of unsteady and an incompressible MHD viscous fluid with heat transfer. The fluid is lying over a vertical and moving plate in its own plane. The problem is modeled by using the constant proportional Caputo fractional derivatives with suitable boundary conditions. The non-dimensional governing equations of problem have been solved analytically with the help of Laplace transform techniques and explicit expressions for respective field variable are obtained. The transformed solutions for energy and momentum balances are appeared in terms of series form. The analytical results regarding velocity and temperature are plotted graphically by MATHCAD software to see the influence of physical parameters. Some graphic comparisons are also mad among present results with hybrid fractional derivatives and the published results that have been obtained by Caputo. It is found that the velocity and temperature with constant proportional Capu?to fractional derivative are portrait better decay than velocities and temperatures that obtained with Caputo and Caputo-Fabrizio derivative. Further, rate of heat transfer and skin friction can be enhanced with smaller values of fractional parameter.

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On a Fractional Operator Combining Proportional and Classical Differintegrals

Cited by 221 publications

References 20 publications

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

New analytical solutions of heat transfer flow of clay-water base nanoparticles with the application of novel hybrid fractional derivative

Thermal analysis of magnetohydrodynamic viscous fluid with innovative fractional derivative

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