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Fractional Differential Equations: Modeling, Discretization, and Numerical Solvers

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The content of the book collects some contributions related to the talks presented during the INdAM Workshop "Fractional Differential Equations: Modelling, Discretization, and Numerical Solvers", held in Rome, Italy, on July 12–14, 2021. All contributions are original and not published elsewhere.

The main topic of the book is fractional calculus, a topic that addresses the study and application of integrals and derivatives of noninteger order. These operators, unlike the classic operators of integer order, are nonlocal operators and are better suited to describe phenomena with memory (with respect to time and/or space). Although the basic ideas of fractional calculus go back over three centuries, only in recent decades there has been a rapid increase in interest in this field of research due not only to the increasing use of fractional calculus in applications in biology, physics, engineering, probability, etc., but also thanks to the availability of new and more powerful numerical tools that allow for an efficient solution of problems that until a few years ago appeared unsolvable. The analytical solution of fractional differential equations (FDEs) appears even more difficult than in the integer case. Hence, numerical analysis plays a decisive role since practically every type of application of fractional calculus requires adequate numerical tools.

The aim of this book is therefore to collect and spread ideas mainly coming from the two communities of numerical analysts operating in this field - the one working on methods for the solution of differential problems and the one working on the numerical linear algebra side - to share knowledge and create synergies. At the same time, the book intends to realize a direct bridge between researchers working on applications and numerical analysts. Indeed, the book collects papers on applications, numerical methods for differential problems of fractional order, and related aspects in numerical linear algebra.

The target audience of the book is scholars interested in recent advancements in fractional calculus.


Author(s): Angelamaria Cardone, Marco Donatelli, Fabio Durastante, Roberto Garrappa, Mariarosa Mazza, Marina Popolizio
Series: Springer INdAM Series, 50
Publisher: Springer-INdAM
Year: 2023

Language: English
Pages: 151
City: Salerno

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