Special Issue "Fractional Differential and Difference Equations"

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

Deadline for manuscript submissions: closed (31 October 2016)

Special Issue Editor

Guest Editor
Dr. Rui A. C. Ferreira

Grupo Física-Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Website | E-Mail
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

Manuscript Submission Information

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Keywords

  • Fractional derivatives
  • differential and difference equations
  • numerical results

Published Papers (7 papers)

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Research

Open AccessArticle
Fractional Fokker-Planck Equation
Mathematics 2017, 5(1), 12; https://doi.org/10.3390/math5010012
Received: 27 October 2016 / Revised: 13 January 2017 / Accepted: 3 February 2017 / Published: 11 February 2017
Cited by 2 | PDF Full-text (6132 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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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Open AccessArticle
Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay
Mathematics 2017, 5(1), 9; https://doi.org/10.3390/math5010009
Received: 15 October 2016 / Revised: 4 January 2017 / Accepted: 17 January 2017 / Published: 25 January 2017
Cited by 1 | PDF Full-text (277 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle
From the Underdamped Generalized Elastic Model to the Single Particle Langevin Description
Mathematics 2017, 5(1), 3; https://doi.org/10.3390/math5010003
Received: 8 November 2016 / Revised: 12 December 2016 / Accepted: 19 December 2016 / Published: 6 January 2017
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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 [...] Read more.
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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Open AccessFeature PaperArticle
Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus
Mathematics 2016, 4(4), 67; https://doi.org/10.3390/math4040067
Received: 1 November 2016 / Revised: 29 November 2016 / Accepted: 2 December 2016 / Published: 9 December 2016
Cited by 1 | PDF Full-text (1710 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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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Open AccessArticle
Effective Potential from the Generalized Time-Dependent Schrödinger Equation
Mathematics 2016, 4(4), 59; https://doi.org/10.3390/math4040059
Received: 5 August 2016 / Revised: 20 September 2016 / Accepted: 21 September 2016 / Published: 28 September 2016
Cited by 8 | PDF Full-text (271 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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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Open AccessArticle
Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations
Mathematics 2016, 4(3), 45; https://doi.org/10.3390/math4030045
Received: 13 March 2016 / Revised: 13 June 2016 / Accepted: 15 June 2016 / Published: 1 July 2016
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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 [...] Read more.
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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Open AccessFeature PaperArticle
Fractional Schrödinger Equation in the Presence of the Linear Potential
Mathematics 2016, 4(2), 31; https://doi.org/10.3390/math4020031
Received: 24 March 2016 / Revised: 18 April 2016 / Accepted: 21 April 2016 / Published: 4 May 2016
Cited by 20 | PDF Full-text (415 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we consider the time-dependent Schrödinger equation: iψ(x,t)t=12(Δ)α2ψ(x,t)+V(x)ψ(x, [...] Read more.
In this paper, we consider the time-dependent Schrödinger equation: i ψ ( x , t ) t = 1 2 ( Δ ) α 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , x R , t > 0 with the Riesz space-fractional derivative of order 0 < α 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 · 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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