On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative

Atangana, Abdon; Kılıçman, Adem

doi:10.1155/2014/542809

Cited by 20 publications

(10 citation statements)

References 35 publications

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“…Recent theoretical and experimental studies have shown that transport processes in complex media are often characterized by either hybrid or anomalous mechanisms. Further, the nature of the transport processes transitions across very different underlying physical phenomena such as transitions from sub-diffusive flow to diffusive flow, and from diffusive flow to superdiffusive flow [34,127,128,[160][161][162]. These complex transport processes have been observed experimentally in various fields, including fluid flow through porous media [5,[134][135][136]163,164], reaction-diffusion interactions between chemical substances leading to pattern formations in nature [165][166][167], diffusion of ions in human neurons [168], analysis of financial data [169], advection-diffusion of groundwater [134][135][136] and elastography [7].…”

Section: Application Of Vo-fc To the Modelling Of Transport Processesmentioning

confidence: 99%

Applications of variable-order fractional operators: a review

2020

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Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

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Section: Application Of Vo-fc To the Modelling Of Transport Processesmentioning

confidence: 99%

Applications of variable-order fractional operators: a review

2020

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“…This approach is very recent, and many work has to be done for a complete study of the subject (see, e.g., (Atangana and Kilicman, 2014;Coimbra, Soon and Kobayashi, 2005;Sheng et al, 2011;Valério et al, 2009) Expansion formulas for fractional derivatives…”

Section: Fractional Variational Problems Of Variable-ordermentioning

confidence: 99%

The Variable-Order Fractional Calculus of Variations

Almeida

Tavares

Torres

2019

SpringerBriefs in Applied Sciences and Technology

127

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PrefaceThis book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators.Fractional calculus is a recent field of mathematical analysis and it is a generalization of integer differential calculus, involving derivatives and integrals of real or complex order (Kilbas, Srivastava and Trujillo, 2006;Podlubny, 1999). The first note about this ideia of differentiation, for non-integer numbers, dates back to 1695, with a famous correspondence between Leibniz and L'Hôpital. In a letter, L'Hôpital asked Leibniz about the possibility of the order n in the notation d n y/dx n , for the nth derivative of the function y, to be a non-integer, n = 1/2. Since then, several mathematicians investigated this approach, like Lacroix, Fourier, Liouville, Riemann, Letnikov, Grünwald, Caputo, and contributed to the grown development of this field. Currently, this is one of the most intensively developing areas of mathematical analysis as a result of its numerous applications. The first book devoted to the fractional calculus was published by Oldham and Spanier in 1974, where the authors systematized the main ideas, methods and applications about this field (Mainardi, 2010).In the recent years, fractional calculus has attracted the attention of many mathematicians, but also some researchers in other areas like physics, chemistry and engineering. As it is well known, several physical phenomena are often better described by fractional derivatives (Herrmann, 2013;Odzijewicz, Malinowska and Torres, 2012a; Sheng, 2012). This is mainly due to the fact that fractional operators take into consideration the evolution of the system, by taking the global correlation, and not only local characteristics. Moreover, integer-order calculus sometimes contradict the experimental results and therefore derivatives of fractional order may be more suitable (Hilfer, 2000).In 1993, Samko and Ross devoted themselves to investigate operators when the order α is not a constant during the process, but variable on time: α(t) ). An interesting recent generalization of the theory of fractional calculus is developed to allow the fractional order of the derivative to be non-constant, depending on time (Chen, Liu and Burrage, VI 2014; Odzijewicz, Torres, 2012b, 2013a). With this approach of variable-order fractional calculus, the non-local properties are more evident and numerous applications have been found in physics, mechanics, control and signal processing (Coimbra, Soon and Kobayashi, 2005; Ingman and Suzdalnitsky, 2004;Odzijewicz, Malinowska and Torres, 2013b; Ostalczyk et al., 2015;Ramirez and Coimbra, 2011; Rapaić and Pisano, 2014; Valério and Costa, 2013).Although there are many definitions of fractional derivative, the most commonly used are the Riemann-Liouville, the Caputo, and the Grünwald-Letnikov derivatives. For more about the development of fractional calculus, we suggest (Samko, Kilbas and Marichev, One difficult issue that usually arises when dealing with such fractional operators, is the extreme di...

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“…Then we may seek what is the best function α(·) such that the variable order fractional differential equation D α(·) y(t) = f (t, y(t)) better describes the model. This approach is very recent, and many work has to be done for a complete study of the subject (see, e.g., [2,3,13,14,17]).…”

Section: Introductionmentioning

confidence: 99%

Constrained fractional variational problems of variable order

Tavares

Almeida

Torres

2017

IEEE/CAA J. Autom. Sinica

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Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral, as well the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.Comment: This is a preprint of a paper whose final and definite form will appear in the IEEE/CAA Journal of Automatica Sinica, ISSN 2329-9266. Submitted 05-Sept-2015; Revised 19-April-2016; Accepted 22-June-201

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On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative

Cited by 20 publications

References 35 publications

Applications of variable-order fractional operators: a review

Applications of variable-order fractional operators: a review

The Variable-Order Fractional Calculus of Variations

Constrained fractional variational problems of variable order

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