The Craft of Fractional Modelling in Science and Engineering: II and III

Hristov, Jordan

doi:10.3390/fractalfract5040281

Cited by 4 publications

(3 citation statements)

References 11 publications

(9 reference statements)

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“…where E ν denotes the modified Mittag-Leffler function defined by (28) and H denotes the Hankel contour defined by (4)- (6).…”

Section: Ab Derivativesmentioning

confidence: 99%

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A complex analysis approach to Atangana–Baleanu fractional calculus

Fernandez¹

2019

Math Methods in App Sciences

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The standard definition for the Atangana-Baleanu fractional derivative involves an integral transform with a Mittag-Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana-Baleanu fractional integral and for iterated Atangana-Baleanu fractional differintegrals.

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“…where E ν denotes the modified Mittag-Leffler function defined by (28) and H denotes the Hankel contour defined by (4)- (6).…”

Section: Ab Derivativesmentioning

confidence: 99%

“…Some of the newer models of fractional calculus are justified by means of their real-life applications, while others are purely mathematical extensions of the idea of fractional differintegration. See for example [20,23,28,37,39] for some discussion and examples of these newer models and their interrelationships.…”

Section: Introductionmentioning

confidence: 99%

A complex analysis approach to Atangana–Baleanu fractional calculus

Fernandez¹

2019

Math Methods in App Sciences

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“…Partly this is due to the discovery of practical applications in various areas including fluid dynamics, chaos theory, bioengineering, etc. [7,9,12,13,21]. Partly also the expansion is due to the realisation that the classical definitions of fractional differintegrals are only the tip of the iceberg: dozens of other models can be proposed and analysed, each with their own properties and applications [1,19,11,16,3,10].…”

Section: Introductionmentioning

confidence: 99%

Analytical Development of Incomplete Riemann-Liouville Fractional Calculus

Fernandez¹,

Ustaoğlu²,

Özarslan³

2019

Preprint

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The theory of fractional calculus has developed in a number of directions over the years, including:• the formulation of multiple different definitions of fractional differintegration,• the extension of various properties of standard calculus into the fractional scenario,• the application of fractional differintegrals to assorted special functions. Recently, a new variant of fractional calculus has arisen, namely incomplete fractional calculus. In two very recent papers, incomplete versions of the Riemann-Liouville and Caputo fractional differintegrals have been formulated and applied to several important special functions. However, this recent development in the field still requires further analysis.In the current work, we develop the theory of incomplete fractional calculus in more depth than has been done before, investigating the further properties of the incomplete Riemann-Liouville fractional differintegrals and answering some fundamental questions about how these operators work.By considering appropriate function spaces, we formulate rigorously the definitions of incomplete Riemann-Liouville fractional integration, and justify how this model may be used to analyse a wider class of functions than classical fractional calculus can consider. By using the idea of analytic continuation from complex analysis, we formulate definitions for incomplete Riemann-Liouville fractional differentiation, hence extending the incomplete integrals to a fully-fledged model of fractional calculus.We also investigate and analyse these operators further, in order to prove new properties of the incomplete Riemann-Liouville fractional calculus. These include a Leibniz rule for incomplete differintegrals of products, and composition properties of incomplete differintegrals with classical calculus operations. These are natural and expected issues to investigate in any new model of fractional calculus, and in the incomplete Riemann-Liouville model the results emerge naturally from the definition previously proposed.

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