TY  - BOOK
AU  - Cattani,Carlo
AU  - Srivastava,Hari M.
AU  - Yang,Xiao-Jun
TI  - Fractional Dynamics
SN  - 9783110472097
AV  - QA314 .F733 2015 
U1  - 515.83 
PY  - 2016///
CY  - Warschau/Berlin
PB  - Walter de Gruyter GmbH
KW  - Fractional calculus
KW  - Electronic books
N1  - Intro -- Fractional Dynamics -- Local Fractional Calculus on Shannon Wavelet Basis -- 1 Introduction -- 2 Preliminary Remarks -- 2.1 Shannon Wavelets in the Fourier Domain -- 2.2 Properties of the Shannon Wavelet -- 3 Connection Coefficients -- 3.1 Properties of Connection Coefficients -- 4 Differential Properties of L2( R )-functions in Shannon Wavelet Basis -- 4.1 Taylor Series -- 4.2 Functional Equations -- 4.3 Error of the Approximation by Connection Coefficients -- 5 Fractional Derivatives of the Wavelet Basis -- 5.1 Complex Shannon Wavelets on Fractal Sets of Dimension -- 5.2 Local Fractional Derivatives of Complex Functions -- 5.3 Example: Fractional Derivative of a Gaussian on a Fractal Set -- Discretely and Continuously Distributed Dynamical Systems with Fractional Nonlocality -- 1 Introduction -- 2 Lattice with Long-range Properties -- 3 Lattice Fractional Nonlinear Equations -- 4 Continuum Fractional Derivatives of the Riesz Type -- 5 From Lattice Equations to Continuum Equations -- 6 Fractional Continuum Nonlinear Equations -- 7 Conclusion -- Temporal Patterns in Earthquake Data-series -- 1 Introduction -- 2 Dataset -- 3 Mathematical Tools -- 3.1 Hierarchichal Clustering -- 3.2 Multidimensional Scaling -- 4 Data Analysys and Pattern Visualization -- 4.1 Hierarchical Clustering Analysis and Comparison -- 4.2 MDS Analysis and Visualization -- 5 Conclusions -- An Integral Transform arising from Fractional Calculus -- 1 Integral Transform R -- 2 Dirac's -function and R -- 3 Space of Generalized Functions Spanned by a(n) -- 4 Extended Borel Transform -- 5 The Transform R and Extended Borel Transform -- 6 Application of R to Fractional Differential Equations -- Approximate Solutions to Time-fractional Models by Integral-balance Approach -- 1 Introduction -- 1.1 Subdiffusion -- 1.2 Time-Fractional Derivatives in Rheology; 1.3 Common Methods of Solutions Involving Time-Fractional Derivatives -- 2 Preliminaries Necessary Mathematical Background -- 2.1 Time-Fractional Integral and Derivatives -- 2.2 Integral-Balance Method -- 3 Introductory Examples -- 3.1 Fading Memory in the Diffusion Term -- 3.2 Example 1: Diffusion of Momentum with Elastic Effects Only -- 4 Examples Involving Time-fractional Derivatives -- 4.1 Example 2: Time-Fractional Subdiffusion Equation -- 4.2 Approximate Parabolic Profiles -- 4.3 Calibration of the Profile Exponent and Results Thereof -- 4.4 Example 3: Subdiffusion Equation: A Solution by a Weak Approximate Profile -- 5 Transient Flows of Viscoelastic Fluids -- 5.1 Example 4: Stokes' First Problem of a Second Grade Fractional (viscoelastic) Fluid -- 5.2 Example 5: Transient Flow of a Generalized Second Grade Fluid Due to a Constant Surface Shear Stress -- 6 Final Comments and Results Outlines -- A Study of Sequential Fractional q-integro-difference Equations with Perturbed Anti-periodic Boundary Conditions -- 1 Introduction -- 2 Preliminaries -- 3 Main Results -- 4 Example -- Fractional Diffusion Equation, Sorption and Reaction Processes on a Surface -- 1 Introduction -- 2 Diffusion and Reaction -- 3 Discussion and Conclusions -- Fractional Order Models for Electrochemical Devices -- 1 Introduction -- 2 Fractional Modeling of Supercapacitors -- 3 Fractional Modeling of Lead Acid Batteries with Application to State of Charge and State of Health Estimation -- 4 Fractional Modeling of Lithium-ion Batteries with Application to State of Charge -- 5 Conclusion -- Results for an Electrolytic Cell Containing Two Groups of Ions: PNP - Model and Fractional Approach -- 1 Introduction -- 2 Fractional Diffusion and Impedance -- 3 Conclusions -- Application of Fractional Calculus to Epidemiology -- 1 Introduction; 1.1 Modelling Epidemic of Whooping Cough with Concept of Fractional Order Derivative -- 2 Conclusion -- On Numerical Methods for Fractional Differential Equation on a Semi-infinite Interval -- 1 Introduction -- 2 Preliminaries and Notations -- 3 Generalized Laguerre Polynomials/Functions -- 3.1 Generalized Laguerre Polynomials -- 3.2 Fractional-order Generalized Laguerre Functions -- 3.3 Fractional-order Generalized Laguerre-Gauss-type Quadratures -- 4 Operational Matrices of Caputo Fractional Derivatives -- 4.1 GLOM of Fractional Derivatives -- 4.2 FGLOM of Fractional Derivatives -- 5 Operational Matrices of Riemann-Liouville Fractional Integrals -- 5.1 GLOM of Fractional Integration -- 5.2 FGLOM of Fractional Integration -- 6 Spectral Methods for FDEs -- 6.1 Generalized Laguerre Tau Operational Matrix Formulation Method -- 6.2 FGLFs Tau Operational Matrix Formulation Method -- 6.3 Tau Method Based on FGLOM of Fractional Integration -- 6.4 Collocation Method for Nonlinear FDEs -- 6.5 Collocation Method for System of FDEs -- 7 Applications and Numerical Results -- From Leibniz's Notation for Derivative to the Fractal Derivative, Fractional Derivative and Application in Mongolian Yurt -- 1 Introduction -- 2 Fractal Derivative -- 3 On Definitions of Fractional Derivatives -- 3.1 Variational Iteration Method -- 3.2 Definitions on Fractional Derivatives -- 4 Mongolian Yurt, Biomimic Design of Cocoon and its Evolution -- 4.1 Pupa-cocoon System -- 4.2 Fractal Hierarchy and Local Fractional Model -- 5 Conclusions -- Cantor-type spherical-coordinate Method for Differential Equations within Local Fractional Derivatives -- 1 Introduction -- 2 Mathematical Tools -- 3 Cantor-type Spherical-coordinate Method -- 4 Examples -- 5 Conclusions -- Approximate Methods for Local Fractional Differential Equations -- 1 Introduction; 2 The Theory of Local Fractional Calculus -- 3 Analysis of the Methods -- 3.1 The local fractional variational iteration method -- 3.2 The local fractional Adomian decomposition method -- 3.3 The local fractional series expansion method -- 4 Applications to Solve Partial Differential Equations Involving Local Fractional Derivatives -- 4.1 Solving the linear Boussinesq equation occurring in fractal long water waves with local fractional variational iteration method -- 4.2 Solving the equation of the fractal motion of a long string by the local fractional Adomian decomposition method -- 4.3 Solving partial differential equations arising from the fractal transverse vibration of a beam with local fractional series expansion method -- 5 Conclusions -- Numerical Solutions for ODEs with Local Fractional Derivative -- 1 Introduction -- 2 The Generalized Local Fractional Taylor Theorems -- 3 Extended DTM -- 4 Four Illustrative Examples -- 5 Conclusions -- Local Fractional Calculus Application to Differential Equations Arising in Fractal Heat Transfer -- 1 Introduction -- 2 Theory of Local Fractional Vector Calculus -- 3 The Local Fractional Heat Equations Arising in Fractal Heat Transfer -- 3.1 The Non-homogeneous Heat Problems Arising in Fractal Heat Flow -- 3.2 The Homogeneous Heat Problems Arising in Fractal Heat Flow -- 4 Local Fractional Poisson Problems Arising in Fractal Heat Flow -- 5 Local Fractional Laplace Problems Arising From Fractal Heat Flow -- 6 The 2D Partial Differential Equations of Fractal Heat Transfer in Cantor-type Circle Coordinate Systems -- 7 Conclusions -- Local Fractional Laplace Decomposition Method for Solving Linear Partial Differential Equations with Local Fractional Derivative -- 1 Introduction -- 2 Mathematical Fundamentals -- 3 Local Fractional Laplace Decomposition Method -- 4 Illustrative Examples -- 5 Conclusions; Calculus on Fractals -- 1 Introduction -- 2 Calculus on Fractal Subset of Real-Line -- 2.1 Staircase Functions -- 2.2 F-Limit and F-Continuity -- 2.3 F-Integration -- 2.4 F-Differentiation -- 2.5 First Fundamental Theorem of F-calculus -- 2.6 Second Fundamental Theorem of F-calculus -- 2.7 Taylor Series on Fractal Sets -- 2.8 Integration by Parts in F-calculus -- 3 Fractal F-differential Equation -- 4 Calculus on Fractal Curves -- 4.1 Staircase Function on Fractal Curves -- 4.2 F-Limit and F-Continuity on Fractal Curves -- 4.3 F-integration on Fractal Curves -- 4.4 F-Differentiation on Fractal Curves -- 4.5 First Fundamental Theorem on Fractal Curve -- 4.6 Second Fundamental Theorem on Fractal Curve -- 5 Gradient, Divergent, Curl and Laplacian on Fractal Curves -- 5.1 Gradient on Fractal Curves -- 5.2 Divergent on Fractal Curves -- 5.3 Laplacian on Fractal Curves -- 6 Function Spaces in F-calculus -- 6.1 Spaces of F-differentiable Functions -- 6.2 Spaces of F-Integrable Functions -- 7 Calculus on Fractal Subsets of R3 -- 7.1 Integral Staircase for Fractal Subsets of R3 -- 7.2 F-integration on Fractal Subset of R3 -- 7.3 F-differentiation on Fractal Subsets of R3 -- 8 F-differential Form -- 8.1 F-Fractional 1-forms -- 8.2 F- Fractional Exactness -- 8.3 F-Fractional 2-forms -- 9 Gauge Integral and F-calculus -- 10 Application of F-calculus -- 10.1 Lagrangian and Hamiltonian Mechanics on Fractals -- 11 Quantum Mechanics on Fractal Curve -- 11.1 Generalized Feynman Path Integral Method -- 12 Continuity Equation and Probability on Fractal -- 13 Newtonian Mechanics on Fractals -- 13.1 Kinematics of Motion -- 13.2 Dynamics of Motion -- 14 Work and Energy Theorem on Fractals -- 15 Langevin F-Equation on Fractals -- 16 Maxwell's Equation on Fractals; Solutions of Nonlinear Fractional Differential Equations Systems through an Implementation of the Variational Iteration Method
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