Journal ArticleDOI
A new collection of real world applications of fractional calculus in science and engineering
TLDR
This review article aims to present some short summaries written by distinguished researchers in the field of fractional calculus that will guide young researchers and help newcomers to see some of the main real-world applications and gain an understanding of this powerful mathematical tool.About:
This article is published in Communications in Nonlinear Science and Numerical Simulation.The article was published on 2018-11-01. It has received 922 citations till now.read more
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Circuit Implementation of Variable-Order Scaling Fractal-Ladder Fractor with High Resolution
TL;DR: In this article , a scaling fractal-ladder circuit for achieving high resolution variable-order fractor based on scaling expansion theory using a high-resolution multiplying digital-to-analog converter (HMDAC).
Journal ArticleDOI
On the existence of traveling fronts in the fractional-order Amari neural field model
TL;DR: In this paper , the authors establish the existence of traveling fronts in a fractional-order formulation of the Amari neural field model, and they find that lower fractional orders imply a slower power-law tendency towards the excited state.
Journal ArticleDOI
A numerical method based on quadrature rules for ψ-fractional differential equations
Aneela Sabir,Mujeeb ur Rehman +1 more
TL;DR: In this article , a numerical method for the solution of a class of ψ-fractional differential equations involving Caputo derivative with respect to a function is presented, where the numerical solutions to the original problem are recovered from a solution of an algebraic system.
Journal ArticleDOI
Robust MPC For Fractional MIMO systems
Khaled Hcheichi,Faouzi Bouani +1 more
TL;DR: The discretization of this type of system is discussed to finally adapt the Robust Model Predictive Control to apply it and show its efficiency and performance in these systems.
Journal ArticleDOI
Dynamical Behavior of Fractional-Order Delayed Feedback Control on the Mathieu Equation by Incremental Harmonic Balance Method
TL;DR: In this paper , the authors applied the incremental harmonic balance method (IHBM) to the fractional-order delayed feedback control system, and derived the general formulas of the first-order approximate periodic solution for the Mathieu equation.
References
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Book
Theory and Applications of Fractional Differential Equations
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Book
Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
BookDOI
Impedance spectroscopy : theory, experiment, and applications
Abstract: Preface. Preface to the First Edition. Contributors. Contributors to the First Edition. Chapter 1. Fundamentals of Impedance Spectroscopy (J.Ross Macdonald and William B. Johnson). 1.1. Background, Basic Definitions, and History. 1.1.1 The Importance of Interfaces. 1.1.2 The Basic Impedance Spectroscopy Experiment. 1.1.3 Response to a Small-Signal Stimulus in the Frequency Domain. 1.1.4 Impedance-Related Functions. 1.1.5 Early History. 1.2. Advantages and Limitations. 1.2.1 Differences Between Solid State and Aqueous Electrochemistry. 1.3. Elementary Analysis of Impedance Spectra. 1.3.1 Physical Models for Equivalent Circuit Elements. 1.3.2 Simple RC Circuits. 1.3.3 Analysis of Single Impedance Arcs. 1.4. Selected Applications of IS. Chapter 2. Theory (Ian D. Raistrick, Donald R. Franceschetti, and J. Ross Macdonald). 2.1. The Electrical Analogs of Physical and Chemical Processes. 2.1.1 Introduction. 2.1.2 The Electrical Properties of Bulk Homogeneous Phases. 2.1.2.1 Introduction. 2.1.2.2 Dielectric Relaxation in Materials with a Single Time Constant. 2.1.2.3 Distributions of Relaxation Times. 2.1.2.4 Conductivity and Diffusion in Electrolytes. 2.1.2.5 Conductivity and Diffusion-a Statistical Description. 2.1.2.6 Migration in the Absence of Concentration Gradients. 2.1.2.7 Transport in Disordered Media. 2.1.3 Mass and Charge Transport in the Presence of Concentration Gradients. 2.1.3.1 Diffusion. 2.1.3.2 Mixed Electronic-Ionic Conductors. 2.1.3.3 Concentration Polarization. 2.1.4 Interfaces and Boundary Conditions. 2.1.4.1 Reversible and Irreversible Interfaces. 2.1.4.2 Polarizable Electrodes. 2.1.4.3 Adsorption at the Electrode-Electrolyte Interface. 2.1.4.4 Charge Transfer at the Electrode-Electrolyte Interface. 2.1.5 Grain Boundary Effects. 2.1.6 Current Distribution, Porous and Rough Electrodes- the Effect of Geometry. 2.1.6.1 Current Distribution Problems. 2.1.6.2 Rough and Porous Electrodes. 2.2. Physical and Electrochemical Models. 2.2.1 The Modeling of Electrochemical Systems. 2.2.2 Equivalent Circuits. 2.2.2.1 Unification of Immitance Responses. 2.2.2.2 Distributed Circuit Elements. 2.2.2.3 Ambiguous Circuits. 2.2.3 Modeling Results. 2.2.3.1 Introduction. 2.2.3.2 Supported Situations. 2.2.3.3 Unsupported Situations: Theoretical Models. 2.2.3.4 Unsupported Situations: Equivalent Network Models. 2.2.3.5 Unsupported Situations: Empirical and Semiempirical Models. Chapter 3. Measuring Techniques and Data Analysis. 3.1. Impedance Measurement Techniques (Michael C. H. McKubre and Digby D. Macdonald). 3.1.1 Introduction. 3.1.2 Frequency Domain Methods. 3.1.2.1 Audio Frequency Bridges. 3.1.2.2 Transformer Ratio Arm Bridges. 3.1.2.3 Berberian-Cole Bridge. 3.1.2.4 Considerations of Potentiostatic Control. 3.1.2.5 Oscilloscopic Methods for Direct Measurement. 3.1.2.6 Phase-Sensitive Detection for Direct Measurement. 3.1.2.7 Automated Frequency Response Analysis. 3.1.2.8 Automated Impedance Analyzers. 3.1.2.9 The Use of Kramers-Kronig Transforms. 3.1.2.10 Spectrum Analyzers. 3.1.3 Time Domain Methods. 3.1.3.1 Introduction. 3.1.3.2 Analog-to-Digital (A/D) Conversion. 3.1.3.3 Computer Interfacing. 3.1.3.4 Digital Signal Processing. 3.1.4 Conclusions. 3.2. Commercially Available Impedance Measurement Systems (Brian Sayers). 3.2.1 Electrochemical Impedance Measurement Systems. 3.2.1.1 System Configuration. 3.2.1.2 Why Use a Potentiostat? 3.2.1.3 Measurements Using 2, 3 or 4-Terminal Techniques. 3.2.1.4 Measurement Resolution and Accuracy. 3.2.1.5 Single Sine and FFT Measurement Techniques. 3.2.1.6 Multielectrode Techniques. 3.2.1.7 Effects of Connections and Input Impedance. 3.2.1.8 Verification of Measurement Performance. 3.2.1.9 Floating Measurement Techniques. 3.2.1.10 Multichannel Techniques. 3.2.2 Materials Impedance Measurement Systems. 3.2.2.1 System Configuration. 3.2.2.2 Measurement of Low Impedance Materials. 3.2.2.3 Measurement of High Impedance Materials. 3.2.2.4 Reference Techniques. 3.2.2.5 Normalization Techniques. 3.2.2.6 High Voltage Measurement Techniques. 3.2.2.7 Temperature Control. 3.2.2.8 Sample Holder Considerations. 3.3. Data Analysis (J. Ross Macdonald). 3.3.1 Data Presentation and Adjustment. 3.3.1.1 Previous Approaches. 3.3.1.2 Three-Dimensional Perspective Plotting. 3.3.1.3 Treatment of Anomalies. 3.3.2 Data Analysis Methods. 3.3.2.1 Simple Methods. 3.3.2.2 Complex Nonlinear Least Squares. 3.3.2.3 Weighting. 3.3.2.4 Which Impedance-Related Function to Fit? 3.3.2.5 The Question of "What to Fit" Revisited. 3.3.2.6 Deconvolution Approaches. 3.3.2.7 Examples of CNLS Fitting. 3.3.2.8 Summary and Simple Characterization Example. Chapter 4. Applications of Impedance Spectroscopy. 4.1. Characterization of Materials (N. Bonanos, B. C. H. Steele, and E. P. Butler). 4.1.1 Microstructural Models for Impedance Spectra of Materials. 4.1.1.1 Introduction. 4.1.1.2 Layer Models. 4.1.1.3 Effective Medium Models. 4.1.1.4 Modeling of Composite Electrodes. 4.1.2 Experimental Techniques. 4.1.2.1 Introduction. 4.1.2.2 Measurement Systems. 4.1.2.3 Sample Preparation-Electrodes. 4.1.2.4 Problems Associated With the Measurement of Electrode Properties. 4.1.3 Interpretation of the Impedance Spectra of Ionic Conductors and Interfaces. 4.1.3.1 Introduction. 4.1.3.2 Characterization of Grain Boundaries by IS. 4.1.3.3 Characterization of Two-Phase Dispersions by IS. 4.1.3.4 Impedance Spectra of Unusual Two-phase Systems. 4.1.3.5 Impedance Spectra of Composite Electrodes. 4.1.3.6 Closing Remarks. 4.2. Characterization of the Electrical Response of High Resistivity Ionic and Dielectric Solid Materials by Immittance Spectroscopy (J. Ross Macdonald). 4.2.1 Introduction. 4.2.2 Types of Dispersive Response Models: Strengths and Weaknesses. 4.2.2.1 Overview. 4.2.2.2 Variable-slope Models. 4.2.2.3 Composite Models. 4.2.3 Illustration of Typical Data Fitting Results for an Ionic Conductor. 4.3. Solid State Devices (William B. Johnson and Wayne L. Worrell). 4.3.1 Electrolyte-Insulator-Semiconductor (EIS) Sensors. 4.3.2 Solid Electrolyte Chemical Sensors. 4.3.3 Photoelectrochemical Solar Cells. 4.3.4 Impedance Response of Electrochromic Materials and Devices (Gunnar A. Niklasson, Anna Karin Johsson, and Maria Stromme). 4.3.4.1 Introduction. 4.3.4.2 Materials. 4.3.4.3 Experimental Techniques. 4.3.4.4 Experimental Results on Single Materials. 4.3.4.5 Experimental Results on Electrochromic Devices. 4.3.4.6 Conclusions and Outlook. 4.3.5 Time-Resolved Photocurrent Generation (Albert Goossens). 4.3.5.1 Introduction-Semiconductors. 4.3.5.2 Steady-State Photocurrents. 4.3.5.3 Time-of-Flight. 4.3.5.4 Intensity-Modulated Photocurrent Spectroscopy. 4.3.5.5 Final Remarks. 4.4. Corrosion of Materials (Digby D. Macdonald and Michael C. H. McKubre). 4.4.1 Introduction. 4.4.2 Fundamentals. 4.4.3 Measurement of Corrosion Rate. 4.4.4 Harmonic Analysis. 4.4.5 Kramer-Kronig Transforms. 4.4.6 Corrosion Mechanisms. 4.4.6.1 Active Dissolution. 4.4.6.2 Active-Passive Transition. 4.4.6.3 The Passive State. 4.4.7 Point Defect Model of the Passive State (Digby D. Macdonald). 4.4.7.1 Introduction. 4.4.7.2 Point Defect Model. 4.4.7.3 Electrochemical Impedance Spectroscopy. 4.4.7.4 Bilayer Passive Films. 4.4.8 Equivalent Circuit Analysis (Digby D. Macdonald and Michael C. H. McKubre). 4.4.8.1 Coatings. 4.4.9 Other Impedance Techniques. 4.4.9.1 Electrochemical Hydrodynamic Impedance (EHI). 4.4.9.2 Fracture Transfer Function (FTF). 4.4.9.3 Electrochemical Mechanical Impedance. 4.5. Electrochemical Power Sources. 4.5.1 Special Aspects of Impedance Modeling of Power Sources (Evgenij Barsoukov). 4.5.1.1 Intrinsic Relation Between Impedance Properties and Power Sources Performance. 4.5.1.2 Linear Time-Domain Modeling Based on Impedance Models, Laplace Transform. 4.5.1.3 Expressing Model Parameters in Electrical Terms, Limiting Resistances and Capacitances of Distributed Elements. 4.5.1.4 Discretization of Distributed Elements, Augmenting Equivalent Circuits. 4.5.1.5 Nonlinear Time-Domain Modeling of Power Sources Based on Impedance Models. 4.5.1.6 Special Kinds of Impedance Measurement Possible with Power Sources-Passive Load Excitation and Load Interrupt. 4.5.2 Batteries (Evgenij Barsoukov). 4.5.2.1 Generic Approach to Battery Impedance Modeling. 4.5.2.2 Lead Acid Batteries. 4.5.2.3 Nickel Cadmium Batteries. 4.5.2.4 Nickel Metal-hydride Batteries. 4.5.2.5 Li-ion Batteries. 4.5.3 Impedance Behavior of Electrochemical Supercapacitors and Porous Electrodes (Brian E. Conway). 4.5.3.1 Introduction. 4.5.3.2 The Time Factor in Capacitance Charge or Discharge. 4.5.3.3 Nyquist (or Argand) Complex-Plane Plots for Representation of Impedance Behavior. 4.5.3.4 Bode Plots of Impedance Parameters for Capacitors. 4.5.3.5 Hierarchy of Equivalent Circuits and Representation of Electrochemical Capacitor Behavior. 4.5.3.6 Impedance and Voltammetry Behavior of Brush Electrode Models of Porous Electrodes. 4.5.3.7 Impedance Behavior of Supercapacitors Based on Pseudocapacitance. 4.5.3.8 Deviations of Double-layer Capacitance from Ideal Behavior: Representation by a Constant-phase Element (CPE). 4.5.4 Fuel Cells (Norbert Wagner). 4.5.4.1 Introduction. 4.5.4.2 Alkaline Fuel Cells (AFC). 4.5.4.3 Polymer Electrolyte Fuel Cells (PEFC). 4.5.4.4 Solid Oxide Fuel Cells (SOFC). Appendix. Abbreviations and Definitions of Models. References. Index.
Book
Applications Of Fractional Calculus In Physics
TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.
Journal ArticleDOI
An Extension Problem Related to the Fractional Laplacian
TL;DR: In this article, the square root of the Laplacian (−△) 1/2 operator was obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition.