Browsing by Author "Fernandez, Arran"

Now showing 1 - 17 of 17

Citation - WoS: 10
Citation - Scopus: 9
Balance Equations With Generalised Memory and the Emerging Fractional Kernels
(Springer, 2021) Baleanu, Dumitru; Fernandez, Arran; Nigmatullin, Raoul
In this paper, we consider the mechanism of a memory effect based on linear or nonlinear systems of balance equations. By considering a chain of balance equations, connecting each particle to the next by means of a memory kernel, it becomes possible to derive generalised expressions for the overall memory kernel that connects the initial particle to the last particle. We consider several different cases and types of systems, both linear and nonlinear. By assuming a general type of fractional integral operator to describe each balance equation, we derive an expression for the generalised memory which yields a more general type of fractional integral operator based on multivariate series. Some cases of this, such as multivariate Mittag-Leffler-type functions, are already known in mathematics, but they have never discovered real applications until now.
Citation - WoS: 2
Citation - Scopus: 3
Brownian Motion on Cantor Sets
(Walter de Gruyter Gmbh, 2020) Ashrafi, Saleh; Baleanu, Dumitru; Fernandez, Arran; Golmankhaneh, Ali Khalili; Khalili Golmankhaneh, Ali
In this paper, we have investigated the Langevin and Brownian equations on fractal time sets using F-alpha-calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the classical method of deriving the Fokker Planck equation in order to obtain the Fokker-Planck equation on fractal time sets.
Citation - WoS: 52
Citation - Scopus: 52
Classes of Operators in Fractional Calculus: a Case Study
(Wiley, 2021) Baleanu, Dumitru; Fernandez, Arran
The notion of general classes of operators has recently been proposed as an approach to fractional calculus that respects pure and applied viewpoints equally. Here we demonstrate this approach as it applies to the operators with three-parameter Mittag-Leffler kernels defined by Prabhakar in 1971. By considering the general such operator as a class, we are able to better understand its fundamental nature and the different special cases that emerge. In particular, we show that many other named models of fractional calculus can fit within the class of operators defined by Prabhakar and that this class contains both singular and nonsingular operators together. We characterise completely the cases in which these operators are singular or nonsingular and the cases in which they can be written as finite or infinite sums of Riemann-Liouville differintegrals, to obtain finally a catalogue of subclasses with different types of properties.
Corrigendum to Series Representations for Fractional-Calculus Operators Involving Generalised Mittag-Leffler Functions
(Elsevier B.V., 2020) Fernandez, Arran; Baleanu, Dumitru; Srivastava, H. M.
This corrigendum corrects two equations presented in the paper “Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions” [Commun. Nonlinear Sci. Numer. Simulat. 67 (2019) 517–527]. One error is inconsequential, while the other leads to a missing factor in the statement of one theorem.
Citation - WoS: 45
Citation - Scopus: 47
Diffusion on Middle- Cantor Sets
(Mdpi, 2018) Fernandez, Arran; Golmankhaneh, Ali Khalili; Baleanu, Dumitru; Golmankhaneh, Alireza Khalili
In this paper, we study C-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the C-calculus on the generalized Cantor sets known as middle- Cantor sets. We have suggested a calculus on the middle- Cantor sets for different values of with 0<<1. Differential equations on the middle- Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.
Citation - WoS: 13
Citation - Scopus: 14
A Generalisation of the Malgrange-Ehrenpreis Theorem To Find Fundamental Solutions To Fractional Pdes
(Univ Szeged, Bolyai institute, 2017) Fernandez, Arran; Baleanu, Dumitru
We present and prove a new generalisation of the Malgrange-Ehrenpreis theorem to fractional partial differential equations, which can be used to construct fundamental solutions to all partial differential operators of rational order and many of arbitrary real order. We demonstrate with some examples and mention a few possible applications.
Citation - WoS: 40
Citation - Scopus: 46
The Mean Value Theorem and Taylor's Theorem for Fractional Derivatives With Mittag-Leffler Kernel
(Pushpa Publishing House, 2018) Baleanu, Dumitru; Fernandez, Arran
We establish analogues of the mean value theorem and Taylor's theorem for fractional differential operators defined using a Mittag-Leffler kernel. We formulate a new model for the fractional Boussinesq equation by using this new Taylor series expansion.
Citation - WoS: 234
Citation - Scopus: 302
On a Fractional Operator Combining Proportional and Classical Differintegrals
(Mdpi, 2020) Fernandez, Arran; Akgul, Ali; Baleanu, Dumitru
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.
Citation - WoS: 16
Citation - Scopus: 15
On a New Definition of Fractional Differintegrals With Mittag-Leffler Kernel
(Univ Nis, Fac Sci Math, 2019) Baleanu, Dumitru; Fernandez, Arran
We introduce a new family of fractional differential and integral operators which emerge from a fractional iteration process applied to some existing fractional operators with Mittag-Leffler kernels. We analyse the new operators and prove various facts about them, including a semigroup property. We also solve some ODEs in this new model by using Laplace transforms, and discuss applications of our results.
Citation - WoS: 180
Citation - Scopus: 192
On Fractional Calculus with General Analytic Kernels
(Elsevier Science Inc, 2019) Fernandez, Arran; Ozarslan, Mehmet Ali; Baleanu, Dumitru
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators. (C) 2019 Elsevier Inc. All rights reserved.
Citation - WoS: 205
Citation - Scopus: 225
On Fractional Operators and Their Classifications
(Mdpi, 2019) Fernandez, Arran; Baleanu, Dumitru
Fractional calculus dates its inception to a correspondence between Leibniz and L'Hopital in 1695, when Leibniz described "paradoxes" and predicted that "one day useful consequences will be drawn" from them. In today's world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the "useful consequences" predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a "fractional derivative" can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators.
Citation - WoS: 297
Citation - Scopus: 322
On Some New Properties of Fractional Derivatives With Mittag-Leffler Kernel
(Elsevier Science Bv, 2018) Fernandez, Arran; Baleanu, Dumitru
We establish a new formula for the fractional derivative with Mittag-Leffler kernel, in the form of a series of Riemann-Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes. We also prove existence and uniqueness results for certain families of linear and nonlinear fractional ODEs defined using this fractional derivative. We consider the possibility of a semigroup property for these derivatives, and establish extensions of the product rule and chain rule, with an application to fractional mechanics. (C) 2017 Elsevier B.V. All rights reserved.
Citation - WoS: 24
Citation - Scopus: 28
Relations Between Fractional Models With Three-Parameter Mittag-Leffler Kernels
(Springer, 2020) Baleanu, Dumitru; Fernandez, Arran; Abdeljawad, Thabet
We consider two models of fractional calculus which are defined using three-parameter Mittag-Leffler functions: the Prabhakar definition and a recently defined extension of the Atangana-Baleanu definition. By examining the relationships between the two, we are able to find some new properties of both, as well as of the original Atangana-Baleanu model and its iterated form.
Citation - WoS: 129
Citation - Scopus: 134
Series Representations for Fractional-Calculus Operators Involving Generalised Mittag-Leffler Functions
(Elsevier, 2019) Baleanu, Dumitru; Srivastava, H. M.; Fernandez, Arran
We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann-Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties. (C) 2018 Elsevier B.V. All rights reserved.
Citation - WoS: 3
Citation - Scopus: 2
Series Representations for Fractional-Calculus Operators Involving Generalised Mittag-Leffler Functions (Vol 67, Pg 517, 2019)
(Elsevier, 2020) Baleanu, Dumitru; Srivastava, H. M.; Fernandez, Arran
Citation - WoS: 50
Citation - Scopus: 52
Solving Pdes of Fractional Order Using the Unified Transform Method
(Elsevier Science inc, 2018) Baleanu, Dumitru; Fokas, Athanassios S.; Fernandez, Arran
We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class of partial differential equations of fractional order. We demonstrate the applicability of the method by implementing it to solve a model fractional problem. (C) 2018 Elsevier Inc. All rights reserved.
Citation - WoS: 44
Citation - Scopus: 53
Some New Fractional-Calculus Connections Between Mittag-Leffler Functions
(Mdpi, 2019) Fernandez, Arran; Baleanu, Dumitru; Srivastava, Hari M.
We consider the well-known Mittag-Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag-Leffler function as a fractional derivative of the two-parameter Mittag-Leffler function, which is in turn a fractional integral of the one-parameter Mittag-Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag-Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse.