Fractional Dynamics.

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The book is devoted to recent developments in the theory of fractional calculus and its applications. Particular attention is paid to the applicability of this currently popular research field in various branches of pure and applied mathematics. In particular, the book focuses on the more recent res...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Cattani, Carlo (Autor), Srivastava, Hari M. (Autor), Yang, Xiao-Jun (Autor)
Formato: Electrónico Software eBook
Idioma:Inglés
Publicado: De Gruyter 2015.
Temas:
Mathematics.
Physics.
Mathématiques.
Physique.
mathematics.
applied mathematics.
physics.
Mathematics
Physics
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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