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A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 October 2016) | Viewed by 40087

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Special Issue Editor

Dr. Rui A. C. Ferreira

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Guest Editor

Grupo Física-Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Interests: fractional calculus; differential equations; difference equations

Special Issue Information

Dear Colleagues,

The Special Issue, “Fractional Differential and Difference Equations”, invites papers that focus on recent and novel developments in the theory of fractional differential and difference equations, especially on analytical and numerical results for fractional ordinary and partial differential equations.

Papers presenting original ideas and dealing with applications of fractional operators are especially welcome.

Dr. Rui A. C. Ferreira
Guest Editor

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Keywords

Fractional derivatives
differential and difference equations
numerical results

Published Papers (7 papers)

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6132 KiB

Open AccessArticle

Fractional Fokker-Planck Equation

by Gerd Baumann and Frank Stenger

Mathematics 2017, 5(1), 12; https://doi.org/10.3390/math5010012 - 11 Feb 2017

Cited by 10 | Viewed by 6017

Abstract

We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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277 KiB

Open AccessArticle

Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay

by Selvaraj Suganya and Mani Mallika Arjunan

Mathematics 2017, 5(1), 9; https://doi.org/10.3390/math5010009 - 25 Jan 2017

Cited by 9 | Viewed by 3663

Abstract

In this manuscript, we implement Bohnenblust–Karlin’s fixed point theorem to demonstrate the existence of mild solutions for a class of impulsive fractional integro-differential inclusions (IFIDI) with state-dependent delay (SDD) in Banach spaces. An example is provided to illustrate the obtained abstract results. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

283 KiB

Open AccessArticle

From the Underdamped Generalized Elastic Model to the Single Particle Langevin Description

by Alessandro Taloni

Mathematics 2017, 5(1), 3; https://doi.org/10.3390/math5010003 - 06 Jan 2017

Cited by 1 | Viewed by 3794

Abstract

The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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1710 KiB

Open AccessFeature PaperArticle

Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus

by Abel Garcia-Bernabé, S. I. Hernández, L. F. Del Castillo and David Jou

Mathematics 2016, 4(4), 67; https://doi.org/10.3390/math4040067 - 09 Dec 2016

Cited by 2 | Viewed by 4117

Abstract

The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have derived a space-time generalized telegrapher’s equation with a fractional nested hierarchy which can be used in separate developments for the mass transport, for the heat conduction and for the flux of ions. We have obtained a new formalism which includes the contribution of fast of higher-order fluxes in the mesoscopic and inhomogeneous media. The results take the form of continued fraction expansions. The balance equations are used in a scheme of continued fractions, and they appear as a closure condition. In this way the transport equation and its corresponding wave number-frequency relation are obtained, both of them in the mathematical structure of the continued fraction scheme. Numerical examples are included to show the dispersive nature of the solutions, and the generalized fractional transport equation in the same mathematical form, which can be applied to the mass transport, the heat conduction and the flux of ions. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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271 KiB

Open AccessArticle

Effective Potential from the Generalized Time-Dependent Schrödinger Equation

by Trifce Sandev, Irina Petreska and Ervin K. Lenzi

Mathematics 2016, 4(4), 59; https://doi.org/10.3390/math4040059 - 28 Sep 2016

Cited by 12 | Viewed by 4420

Abstract

We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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920 KiB

Open AccessArticle

Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

by Yanmei Liu, Monzorul Khan and Yubin Yan

Mathematics 2016, 4(3), 45; https://doi.org/10.3390/math4030045 - 01 Jul 2016

Cited by 6 | Viewed by 6379

Abstract

Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the

L^{2}

-norm are obtained, and numerical examples are given. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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415 KiB

Open AccessFeature PaperArticle

Fractional Schrödinger Equation in the Presence of the Linear Potential

by André Liemert and Alwin Kienle

Mathematics 2016, 4(2), 31; https://doi.org/10.3390/math4020031 - 04 May 2016

Cited by 50 | Viewed by 10248

Abstract

In this paper, we consider the time-dependent Schrödinger equation: [...] Read more.

In this paper, we consider the time-dependent Schrödinger equation:

i \frac{\partial ψ (x, t)}{\partial t} = \frac{1}{2} {(- Δ)}^{\frac{α}{2}} ψ (x, t) + V (x) ψ (x, t),

x \in R, t > 0

with the Riesz space-fractional derivative of order

0 < α \leq 2

in the presence of the linear potential

V (x) = β x

. The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of

α = 1

, which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential

V (ρ) = F \cdot ρ

including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach. Full article

(This article belongs to the Special Issue Fractional Differential and Difference Equations)

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