FDA Express

New exact solutions for the conformable space-time fractional KdV, CDG, (2+1)-dimensional CBS and (2+1)-dimensional AKNS equations
By: Yaslan, H. C.; Girgin, A.
JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE Volume: 13 Issue: 1 Pages: 1-8 Published: DEC 11 2019

Optimal control problem for coupled time-fractional diffusion systems with final observations
By: Bahaa, G. M.; Hamiaz, A.
JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE Volume: 13 Issue: 1 Pages: 124-135 Published: DEC 11 2019

NEW EXACT SOLUTIONS FOR SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS VIA IMPROVED SUB-EQUATION METHOD
By: Karaagac, Berat
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S Volume: 12 Issue: 3 Pages: 447-454 Published: JUN 2019

THE FIRST INTEGRAL METHOD FOR TWO FRACTIONAL NON-LINEAR BIOLOGICAL MODELS
By: Kolebaje, Olusola; Bonyah, Ebenezer; Mustapha, Lateef
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S Volume: 12 Issue: 3 Pages: 487-502 Published: JUN 2019

A UNIFIED FINITE DIFFERENCE CHEBYSHEV WAVELET METHOD FOR NUMERICALLY SOLVING TIME FRACTIONAL BURGERS' EQUATION
By: Oruc, Omer; Esen, Alaattin; Bulut, Fatih
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S Volume: 12 Issue: 3 Pages: 533-542 Published: JUN 2019

NUMERICAL ANALYSIS AND PATTERN FORMATION PROCESS FOR SPACE-FRACTIONAL SUPERDIFFUSIVE SYSTEMS
By: Owolabi, Kolade M.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S Volume: 12 Issue: 3 Pages: 543-566 Published: JUN 2019

HIGH-ORDER SOLVERS FOR SPACE-FRACTIONAL DIFFERENTIAL EQUATIONS WITH RIESZ DERIVATIVE
By: Owolabi, Kolade M.; Atangana, Abdon
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S Volume: 12 Issue: 3 Pages: 567-590 Published: JUN 2019

Analytical and numerical solutions of time and space fractional advection-diffusion-reaction equation
By: Jannelli, Alessandra; Ruggieri, Marianna; Speciale, Maria Paola
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 70 Pages: 89-101 Published: MAY 2019

Fractional and integer derivatives with continuously distributed lag
By: Tarasov, Vasily E.; Tarasova, Svetlana S.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 70 Pages: 125-169 Published: MAY 2019

Electrical transport properties and fractional dynamics of twist-bend nematic liquid crystal phase
By: Ribeiro de Almeida, R. R.; Evangelista, L. R.; Lenzi, K.; etc.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 70 Pages: 248-256 Published: MAY 2019

On power law scaling dynamics for time-fractional phase field models during coarsening
By: Zhao, Jia; Chen, Lizhen; Wang, Hong
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 70 Pages: 257-270 Published: MAY 2019

Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations
By: Garrappa, Roberto
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 70 Pages: 302-306 Published: MAY 2019

Fractional-order modeling of a diode
By: Tenreiro Machado, J. A.; Lopes, Antonio M.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 70 Pages: 343-353 Published: MAY 2019

Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains
By: Feng, Libo; Liu, Fawang; Turner, Ian
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 70 Pages: 354-371 Published: MAY 2019

Fractional calculus via Laplace transform and its application in relaxation processes
By: Capelas de Oliveira, E.; Jarosz, S.; Vaz, J., Jr.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 69 Pages: 58-72 Published: APR 2019

On an accurate discretization of a variable-order fractional reaction-diffusion equation
By: Hajipour, Mojtaba; Jajarmi, Amin; Baleanu, Dumitru; etc.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 69 Pages: 119-133 Published: APR 2019

Emergence of death islands in fractional-order oscillators via delayed coupling
By: Xiao, Rui; Sun, Zhongkui; Yang, Xiaoli; etc.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 69 Pages: 168-175 Published: APR 2019

On the series representation of nabla discrete fractional calculus
By: Wei, Yiheng; Gao, Qing; Liu, Da-Yan; etc.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 69 Pages: 198-218 Published: APR 2019

Numerical solution of Caputo fractional differential equations with infinity memory effect at initial condition
By: Mendes, Eduardo M. A. M.; Salgado, Gustavo H. O.; Aguirre, Luis A.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION Volume: 69 Pages: 237-247 Published: APR 2019

Stable evaluations of fractional derivative of the Muntz-Legendre polynomials and application to fractional differential equations
By: Erfani, S.; Babolian, E.; Javadi, S.; etc.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS Volume: 348 Pages: 70-88 Published: MAR 1 2019

Analytical solution of fractional variable order differential equations
By: Malesza, W.; Macias, M.; Sierociuk, D.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS Volume: 348 Pages: 214-236 Published: MAR 1 2019

Weakly singular Gronwall inequalities and applications to fractional differential equations
By: Webb, J. R. L.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS Volume: 471 Issue: 1-2 Pages: 692-711 Published: MAR 2019

[Back]

==========================================================================

Call for Papers

－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－

Special Issue "Applications of Statistical Thermodynamics"

Entropy (IF:2.305)

Special Issue Information

Dear Colleagues,

Statistical thermodynamics span the bridge between the visible macroscopic world and the invisible atomistic world to evaluate values of atomistic interaction parameters with unambiguous physical significance from measured values of state parameters, such as temperature, pressure and chemical composition under equilibrium state. Unlike conventional thermodynamics, in which entropy, enthalpy, and free energy are defined mathematically in terms of state parameters and thus applicable universally to any system, even without knowing exactly the nature of compound under consideration, statistical thermodynamic analysis must be started from unambiguous a priori modeling of compounds under consideration. When an unrealistic model is chosen at the onset of the statistical thermodynamic approach, the evaluated parameters are without valid physical significance. This inherent nature of the statistical thermodynamic approach might make use of this unique analysis tool somewhat difficult for experimentalists to use casually. However, there also lies a merit of this unique analysis tool to a provide feedback channel to check the validity of the a priori model with reference to the compatibility of the evaluated atomistic interaction parameter values with the macroscopic state parameter values.

The Guest Editor wishes this Special Issue will attract the attention of authors who have been working on entropy and enthalpy aspects of materials science, as well as physicists and chemists using statistical thermodynamics as an analysis tool.

Prof. Dr. Nobumitsu Shohoji
Guest Editor

Special Issue Editor

Guest Editor
Prof. Dr. Nobumitsu Shohoji
LNEG - Laboratório Nacional de Energia e Geologia, LEN - Laboratório de Energia Estrada do Paço do Lumiar, 22 1649-038 Lisboa, Portugal
Website | E-Mail
Phone: +351 21 092 9600 (ext. 4234)
Interests: 1. Statistical thermodynamic analysis of non-stoichiometric interstitial compounds; 2. Synthesis of carbide, nitride and carbo-nitride (using concentrated solar beam as the heat source as well as using conventional electric furnace); 3. Formation and characterization of non-equilibrium solid phases

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1500 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Deadline for manuscript submissions: 30 June 2019

Keywords
-Statistical thermodynamics
-Entropy (configurational, electronic)
-Enthalpy
-Free Energy
-Saddle point approach
-Non-stoichiometry
-Interstitial
-Substitutional

Further information, see https://www.mdpi.com/si/entropy/Statistical_Thermodynamics

[Back]

===========================================================================
Books
－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－
Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales

( Svetlin G. Georgiev)

Details: https://link.springer.com/book/10.1007/978-3-319-73954-0#toc

Introduction

Pedagogically organized, this monograph introduces fractional calculus and fractional dynamic equations on time scales in relation to mathematical physics applications and problems. Beginning with the definitions of forward and backward jump operators, the book builds from Stefan Hilger' s basic theories on time scales and examines recent developments within the field of fractional calculus and fractional equations. Useful tools are provided for solving differential and integral equations as well as various problems involving special functions of mathematical physics and their extensions and generalizations in one and more variables. Much discussion is devoted to Riemann-Liouville fractional dynamic equations and Caputo fractional dynamic equations.

Intended for use in the field and designed for students without an extensive mathematical background, this book is suitable for graduate courses and researchers looking for an introduction to fractional dynamic calculus and equations on time scales.

Chapters

-Elements of Time Scale Calculus

-The Laplace Transform on Time Scales

-Convolution on Time Scales

-The Riemann–Liouville Fractional Δ-Integral and the Riemann–Liouville Fractional Δ-Derivative on Time Scales

-Cauchy-Type Problems with the Riemann–Liouville Fractional Δ-Derivative

-Riemann–Liouville Fractional Dynamic Equations with Constant Coefficients

-The Caputo Fractional Δ-Derivative on Time Scales

-Cauchy-Type Problems with the Caputo Fractional Δ-Derivative

-Caputo Fractional Dynamic Equations with Constant Coefficients

-Appendix: The Gamma Function

-Appendix: The Beta Function

The Contents is available at: https://link.springer.com/book/10.1007/978-3-319-73954-0#toc

[Back]

========================================================================
Journals
－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－

Journal of Computational Physics

(Selected)

A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation

Ruilian Du, Yubin Yan, Zongqi Liang

Fractional magneto-hydrodynamics: Algorithms and applications

Fangying Song, George Em Karniadakis

Boundary conditions for two-sided fractional diffusion

James F. Kelly, Harish Sankaranarayanan, Mark M. Meerschaert

Fractional Hermite interpolation using RBFs in high dimensions over irregular domains with application

M. Esmaeilbeigi, O. Chatrabgoun, M. Cheraghi

A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem

Siwei Duo, Hans Werner van Wyk, Yanzhi Zhang

Parallel algorithms for nonlinear time–space fractional parabolic PDEs

T. A. Biala, A. Q. M. Khaliq

What is a fractional derivative?

Manuel D. Ortigueira, J. A. Tenreiro Machado

Recovering an unknown source in a fractional diffusion problem

William Rundell, Zhidong Zhang

A fourth-order scheme for space fractional diffusion equations

Xu Guo, Yutian Li, Hong Wang

The finite difference/finite volume method for solving the fractional diffusion equation

Tie Zhang, Qingxin Guo

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Yujian Jiao, Li-Lian Wang, Can Huang

Spectral iterative method and convergence analysis for solving nonlinear fractional differential equation

M. Yarmohammadi, S. Javadi, E. Babolian

Parareal algorithms with local time-integrators for time fractional differential equations

Shu-Lin Wu, Tao Zhou

Tempered fractional calculus

Farzad Sabzikar, Mark M. Meerschaert, Jinghua Chen

An integrated fractional partial differential equation and molecular dynamics model of anomalously diffusive transport in heterogeneous nano-pore structures

Meng Zhao, Shuai He, Hong Wang, Guan Qin

[Back]

Computer Methods in Applied Mechanics and Engineering

(Selected)

High-order central difference scheme for Caputo fractional derivative

Yuping Ying, Yanping Lian, Shaoqiang Tang, Wing Kam Liu

A Petrov–Galerkin finite element method for the fractional advection–diffusion equation

Yanping Lian, Yuping Ying, Shaoqiang Tang, Stephen Lin, Wing Kam Liu

Fractional-order uniaxial visco-elasto-plastic models for structural analysis

J. L. Suzuki, M. Zayernouri, M. L. Bittencourt, G. E. Karniadakis

Adaptive finite element method for fractional differential equations using hierarchical matrices

Xuan Zhao, Xiaozhe Hu, Wei Cai, George Em Karniadakis

Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview

Bangti Jin, Raytcho Lazarov, Zhi Zhou

A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations

Fangying Song, Chuanju Xu, George Em Karniadakis

Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

Stig Larsson, Milena Racheva, Fardin Saedpanah

Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives

Akbar Mohebbi, Mostafa Abbaszadeh, Mehdi Dehghan

[Back]

========================================================================
Paper Highlight

fPINNs: Fractional Physics-Informed Neural Networks

Guofei Pang, Lu Lu, George Em Karniadakis

Publication information: https://arxiv.org/abs/1811.08967

Abstract

Physics-informed neural networks (PINNs) are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE-residuals and the mean-squared error in initial/boundary conditions is minimized with respect to the NN parameters. We extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we demonstrate their accuracy and effectiveness in solving multi-dimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. We consider 1D time-dependent fractional ADEs and compare white-box (WB) and black-box (BB) forcing. We observe that for the BB forcing fPINNs outperform FDM. Subsequently, we consider multi-dimensional time-, space-, and space-time-fractional ADEs using the directional fractional Laplacian and we observe relative errors of 10⁻⁴. Finally, we solve several inverse problems in 1D, 2D, and 3D to identify the fractional orders, diffusion coefficients, and transport velocities and obtain accurate results even in the presence of significant noise.

[Back]

－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－

A scale-dependent finite difference approximation for time fractional differential equation

XiaoTing Liu, Hong Guang Sun, Yong Zhang, Zhuojia Fu

Publication information: Computational Mechanics Pages: 1-14 Published: 2018

https://link.springer.com/article/10.1007/s00466-018-1601-x#citeas

Abstract

This study proposes a scale-dependent finite difference method (S-FDM) to approximate the time fractional differential equations (FDEs), using Hausdroff metric to conveniently link the order of the time fractional derivative (α) and the non-uniform time intervals. The S-FDM is unconditional stable and exhibits a convergence rate on the order of 2-α. Numerical tests show that the S-FDM is superior to the standard methods with either uniform or non-uniform time steps in computing time or cost, accuracy, and convergence rate, especially for a large time range. Hence, although many numerical schemes have been developed in the last decades for various FDEs, the unique S-FDM proposed in this study fits the requirement of calculating anomalous transport in natural systems involving a large spatiotemporal scale, which might be the future direction to extend the application of FDEs especially in Earth sciences, the ideal testbed for FDEs.

[Back]

==========================================================================
The End of This Issue
∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽∽