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4 pages, 164 KiB

Open AccessEditorial

Fractional Calculus: Theory and Applications

by Francesco Mainardi

Mathematics 2018, 6(9), 145; https://doi.org/10.3390/math6090145 - 21 Aug 2018

Cited by 36 | Viewed by 5462

Abstract

Fractional calculus is allowing integrals and derivatives of any positive order (the term fractional is kept only for historical reasons).[...] Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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16 pages, 990 KiB

Open AccessArticle

Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches

by Guoxing Lin

Mathematics 2018, 6(2), 17; https://doi.org/10.3390/math6020017 - 29 Jan 2018

Cited by 5 | Viewed by 3911

Abstract

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may [...] Read more.

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected, such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase-shift diffusion equation (EPSDE) method and a method based on observing the signal intensity at the origin. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including the FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function-type PFG signal attenuation. The methods here provide a clear view of spin evolution under a field gradient, and their results will help the analysis of PFG anomalous diffusion. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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23 pages, 647 KiB

Open AccessFeature PaperArticle

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

by Roberto Garrappa

Mathematics 2018, 6(2), 16; https://doi.org/10.3390/math6020016 - 23 Jan 2018

Cited by 346 | Viewed by 25938

Abstract

Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In [...] Read more.

Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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9 pages, 875 KiB

Open AccessArticle

Storage and Dissipation of Energy in Prabhakar Viscoelasticity

by Ivano Colombaro, Andrea Giusti and Silvia Vitali

Mathematics 2018, 6(2), 15; https://doi.org/10.3390/math6020015 - 23 Jan 2018

Cited by 25 | Viewed by 4019

Abstract

In this paper, after a brief review of the physical notion of quality factor in viscoelasticity, we present a complete discussion of the attenuation processes emerging in the Maxwell–Prabhakar model, recently developed by Giusti and Colombaro. Then, taking profit of some illuminating plots, [...] Read more.

In this paper, after a brief review of the physical notion of quality factor in viscoelasticity, we present a complete discussion of the attenuation processes emerging in the Maxwell–Prabhakar model, recently developed by Giusti and Colombaro. Then, taking profit of some illuminating plots, we discuss some potential connections between the presented model and the modern mathematical modelling of seismic processes. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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12 pages, 731 KiB

Open AccessArticle

Best Approximation of the Fractional Semi-Derivative Operator by Exponential Series

by Vladimir D. Zakharchenko and Ilya G. Kovalenko

Mathematics 2018, 6(1), 12; https://doi.org/10.3390/math6010012 - 16 Jan 2018

Cited by 8 | Viewed by 4141

Abstract

A significant reduction in the time required to obtain an estimate of the mean frequency of the spectrum of Doppler signals when seeking to measure the instantaneous velocity of dangerous near-Earth cosmic objects (NEO) is an important task being developed to counter the [...] Read more.

A significant reduction in the time required to obtain an estimate of the mean frequency of the spectrum of Doppler signals when seeking to measure the instantaneous velocity of dangerous near-Earth cosmic objects (NEO) is an important task being developed to counter the threat from asteroids. Spectral analysis methods have shown that the coordinate of the centroid of the Doppler signal spectrum can be found by using operations in the time domain without spectral processing. At the same time, an increase in the speed of resolving the algorithm for estimating the mean frequency of the spectrum is achieved by using fractional differentiation without spectral processing. Thus, an accurate estimate of location of the centroid for the spectrum of Doppler signals can be obtained in the time domain as the signal arrives. This paper considers the implementation of a fractional-differentiating filter of the order of ½ by a set of automation astatic transfer elements, which greatly simplifies practical implementation. Real technical devices have the ultimate time delay, albeit small in comparison with the duration of the signal. As a result, the real filter will process the signal with some error. In accordance with this, this paper introduces and uses the concept of a “pre-derivative” of ½ of magnitude. An optimal algorithm for realizing the structure of the filter is proposed based on the criterion of minimum mean square error. Relations are obtained for the quadrature coefficients that determine the structure of the filter. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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5 pages, 280 KiB

Open AccessArticle

A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients

by Francesco Mainardi

Mathematics 2018, 6(1), 8; https://doi.org/10.3390/math6010008 - 9 Jan 2018

Cited by 15 | Viewed by 3920

Abstract

In this note, we show how an initial value problem for a relaxation process governed by a differential equation of a non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying [...] Read more.

In this note, we show how an initial value problem for a relaxation process governed by a differential equation of a non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying coefficient. This equivalence is shown for the simple fractional relaxation equation that points out the relevance of the Mittag–Leffler function in fractional calculus. This simple argument may lead to the equivalence of more general processes governed by evolution equations of fractional order with constant coefficients to processes governed by differential equations of integer order but with varying coefficients. Our main motivation is to solicit the researchers to extend this approach to other areas of applied science in order to have a deeper knowledge of certain phenomena, both deterministic and stochastic ones, investigated nowadays with the techniques of the fractional calculus. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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13 pages, 369 KiB

Open AccessFeature PaperArticle

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

by Marina Popolizio

Mathematics 2018, 6(1), 7; https://doi.org/10.3390/math6010007 - 9 Jan 2018

Cited by 29 | Viewed by 6678

Abstract

Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods [...] Read more.

Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs) to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag–Leffler (ML) functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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25 pages, 304 KiB

Open AccessArticle

Weyl and Marchaud Derivatives: A Forgotten History

by Fausto Ferrari

Mathematics 2018, 6(1), 6; https://doi.org/10.3390/math6010006 - 3 Jan 2018

Cited by 62 | Viewed by 7432

Abstract

In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different [...] Read more.

In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

16 pages, 1569 KiB

Open AccessArticle

Application of Tempered-Stable Time Fractional-Derivative Model to Upscale Subdiffusion for Pollutant Transport in Field-Scale Discrete Fracture Networks

by Bingqing Lu, Yong Zhang, Donald M. Reeves, HongGuang Sun and Chunmiao Zheng

Mathematics 2018, 6(1), 5; https://doi.org/10.3390/math6010005 - 3 Jan 2018

Cited by 20 | Viewed by 5035

Abstract

Fractional calculus provides efficient physical models to quantify non-Fickian dynamics broadly observed within the Earth system. The potential advantages of using fractional partial differential equations (fPDEs) for real-world problems are often limited by the current lack of understanding of how earth system properties [...] Read more.

Fractional calculus provides efficient physical models to quantify non-Fickian dynamics broadly observed within the Earth system. The potential advantages of using fractional partial differential equations (fPDEs) for real-world problems are often limited by the current lack of understanding of how earth system properties influence observed non-Fickian dynamics. This study explores non-Fickian dynamics for pollutant transport in field-scale discrete fracture networks (DFNs), by investigating how fracture and rock matrix properties influence the leading and tailing edges of pollutant breakthrough curves (BTCs). Fractured reservoirs exhibit erratic internal structures and multi-scale heterogeneity, resulting in complex non-Fickian dynamics. A Monte Carlo approach is used to simulate pollutant transport through DFNs with a systematic variation of system properties, and the resultant non-Fickian transport is upscaled using a tempered-stable fractional in time advection–dispersion equation. Numerical results serve as a basis for determining both qualitative and quantitative relationships between BTC characteristics and model parameters, in addition to the impacts of fracture density, orientation, and rock matrix permeability on non-Fickian dynamics. The observed impacts of medium heterogeneity on tracer transport at late times tend to enhance the applicability of fPDEs that may be parameterized using measurable fracture–matrix characteristics. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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10 pages, 730 KiB

Open AccessArticle

A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability

by Roberto Garra, Enzo Orsingher and Federico Polito

Mathematics 2018, 6(1), 4; https://doi.org/10.3390/math6010004 - 1 Jan 2018

Cited by 44 | Viewed by 4322

Abstract

In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a [...] Read more.

In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

258 KiB

Open AccessFeature PaperArticle

Letnikov vs. Marchaud: A Survey on Two Prominent Constructions of Fractional Derivatives

by Sergei Rogosin and Maryna Dubatovskaya

Mathematics 2018, 6(1), 3; https://doi.org/10.3390/math6010003 - 25 Dec 2017

Cited by 25 | Viewed by 4607

Abstract

In this survey paper, we analyze two constructions of fractional derivatives proposed by Aleksey Letnikov (1837–1888) and by André Marchaud (1887–1973), respectively. These derivatives play very important roles in Fractional Calculus and its applications. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

255 KiB

Open AccessArticle

An Iterative Method for Solving a Class of Fractional Functional Differential Equations with “Maxima”

by Khadidja Nisse and Lamine Nisse

Mathematics 2018, 6(1), 2; https://doi.org/10.3390/math6010002 - 22 Dec 2017

Cited by 6 | Viewed by 3685

Abstract

In the present work, we deal with nonlinear fractional differential equations with “maxima” and deviating arguments. The nonlinear part of the problem under consideration depends on the maximum values of the unknown function taken in time-dependent intervals. Proceeding by an iterative approach, we [...] Read more.

In the present work, we deal with nonlinear fractional differential equations with “maxima” and deviating arguments. The nonlinear part of the problem under consideration depends on the maximum values of the unknown function taken in time-dependent intervals. Proceeding by an iterative approach, we obtain the existence and uniqueness of the solution, in a context that does not fit within the framework of fixed point theory methods for the self-mappings, frequently used in the study of such problems. An example illustrating our main result is also given. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

275 KiB

Open AccessArticle

On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

by Yuri Luchko

Mathematics 2017, 5(4), 76; https://doi.org/10.3390/math5040076 - 8 Dec 2017

Cited by 27 | Viewed by 3994

Abstract

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral [...] Read more.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

1128 KiB

Open AccessFeature PaperArticle

Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation

by Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi and Simonetta Pagnutti

Mathematics 2017, 5(4), 73; https://doi.org/10.3390/math5040073 - 4 Dec 2017

Cited by 8 | Viewed by 3311

Abstract

The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is [...] Read more.

The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is shown to provide straightforward analytical solutions for cases hardly solvable with conventional means. The possibility of extending the method by the use of expansion using different polynomials (two variable Legendre like) expansion is also discussed. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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341 KiB

Open AccessArticle

Generalized Langevin Equation and the Prabhakar Derivative

by Trifce Sandev

Mathematics 2017, 5(4), 66; https://doi.org/10.3390/math5040066 - 20 Nov 2017

Cited by 62 | Viewed by 6218

Abstract

We consider a generalized Langevin equation with regularized Prabhakar derivative operator. We analyze the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. We further introduce the so-called tempered regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term [...] Read more.

We consider a generalized Langevin equation with regularized Prabhakar derivative operator. We analyze the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. We further introduce the so-called tempered regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term represented through the tempered derivative. Various diffusive behaviors are observed. We show the importance of the three parameter Mittag-Leffler function in the description of anomalous diffusion in complex media. We also give analytical results related to the generalized Langevin equation for a harmonic oscillator with generalized friction. The normalized displacement correlation function shows different behaviors, such as monotonic and non-monotonic decay without zero-crossings, oscillation-like behavior without zero-crossings, critical behavior, and oscillation-like behavior with zero-crossings. These various behaviors appear due to the friction of the complex environment represented by the Mittag-Leffler and tempered Mittag-Leffler memory kernels. Depending on the values of the friction parameters in the system, either diffusion or oscillations dominate. Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

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