Fractional Differential Equations: Modeling, Discretization, and Numerical Solvers (Springer INdAM Series, 50)

Preface
Organization
Contents
About the Editors
A New Diffusive Representation for Fractional Derivatives, Part I: Construction, Implementation and Numerical Examples
	1 Introduction and Statement of the Problem
		1.1 Classical Discretizations in Fractional Calculus
		1.2 Diffusive Representations in Discretized Fractional Calculus
	2 The New Diffusive Representation and Its Properties
	3 The Complete Numerical Method
	4 Experimental Results and Conclusion
	References
Exact Solutions for the Fractional Nonlinear Boussinesq Equation
	1 Introduction
	2 Physical Motivation
	3 The Steady Solution
	4 The Unsteady Space-Fractional Case
	5 The Time-Fractional Case
	6 Conclusions
	References
A Numerical Procedure for Fractional-Time-Space Differential Equations with the Spectral Fractional Laplacian
	1 Introduction
	2 The Spectral Fractional Laplacian: A Brief Introduction
		2.1 Eigendecomposition of the Laplacian
		2.2 Spectral Fractional Laplacian
	3 Fractional-Time-Space Differential Equation
	4 Generalized Exponential Time-Differencing Methods
	5 Error Analysis
	6 Numerical Experiments
		6.1 Homogeneous Dirichlet Boundary Conditions in a 1D Domain
		6.2 Homogeneous Neumann Boundary Conditions in a 1D Domain
		6.3 Homogeneous Dirichlet Boundary Conditions in a 2D Domain
	7 Concluding Remarks
	References
Spectral Analysis of Matrices in B-Spline Galerkin Methods for Riesz Fractional Equations
	1 Introduction
	2 Preliminaries
		2.1 Fractional Derivatives
		2.2 Spectral Tools
		2.3 B-Splines and Cardinal B-Splines
	3 B-Spline Galerkin Discretization of the Fractional Riesz Operator
	4 Spectral Symbol of {n1-αAnp,α}n and Its Properties
	5 Numerical Results
	6 Conclusions
	References
Do the Mittag–Leffler Functions Preserve the Properties of Their Matrix Arguments?
	1 Introduction
	2 What Is Not Preserved
	3 Nonnegativity Preservation
	4 Centrosymmetric Matrices
	5 Circulant Matrices
	6 Quasi-Toeplitz Matrices
	7 The ML Function with Time-Dependent Matrix Arguments
	8 Conclusions
	References
On the Solutions of the Fractional GeneralizedGierer–Meinhardt Model
	1 Introduction
	2 Lie Transformation and FODEs
	3 Analytical Solutions of the Generalized Depletion Model
	4 Numerical Method and Solutions
	5 Concluding Remarks
	References
A Convolution-Based Method for an Integro-Differential Equation in Mechanics
	1 Introduction
	2 Fourier Semi-Discretization of the Problem
	3 Volume Penalization Technique
	4 The Fully Discrete Problem
		4.1 Störmer–Verlet Scheme
		4.2 Newmark-β Method
	5 Numerical Simulations
		5.1 Simulations on a 2D Lamina
		5.2 Simulations on a 1D Bar
	6 Conclusions
	References
A MATLAB Code for Fractional Differential Equations Based on Two-Step Spline Collocation Methods
	1 Introduction
	2 The Two-Step Spline Collocation Method
	3 Computation of Fractional Integrals
	4 Starting Procedure
	5 Convergence and Optimal Parameters Setting
	6 Matrix Formulation of the Method
		6.1 Matrix Formulation of Nonlinear System (13)
		6.2 Vector Formulation of the Numerical Solution yN
	7 The MATLAB Algorithm
	8 Input and Output Parameters
		8.1 Input Parameters
		8.2 Output Parameters
	9 Example of Usage
	10 Numerical Experiments
	References