Spectral methods for some kinds of fractional differential equations
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Chuanju Xu
Abstract
In this chapter, we design and analyze two types of spectral methods for fractional differential equations. The first spectral method makes use of the traditional polynomials and follows the standard Galerkin framework, while the second method is based on the Müntz polynomials and weighted Galerkin approach. For the first method we will first introduce suitable functional spaces and develop a theoretical framework for the weak solution of the STFDE. This allows to apply the existing theory for elliptic problems to prove the existence and uniqueness of the weak solution. Then we construct a Galerkin spectral method for efficiently solving the spacetime fractional diffusion problem. Optimal error estimates are derived under certain regularity conditions on the exact solution. For the second method, we first introduce a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials, and investigate their approximation properties. Then we propose an efficient scheme using GFJPs for the time-fractional diffusion equation. Our theoretical or numerical investigation shows that the proposed scheme is exponentially convergent for general right hand side functions, even though the exact solution has very limited regularity. Implementation details are also provided, along with a series of numerical examples to show the efficiency of the proposed methods.
Abstract
In this chapter, we design and analyze two types of spectral methods for fractional differential equations. The first spectral method makes use of the traditional polynomials and follows the standard Galerkin framework, while the second method is based on the Müntz polynomials and weighted Galerkin approach. For the first method we will first introduce suitable functional spaces and develop a theoretical framework for the weak solution of the STFDE. This allows to apply the existing theory for elliptic problems to prove the existence and uniqueness of the weak solution. Then we construct a Galerkin spectral method for efficiently solving the spacetime fractional diffusion problem. Optimal error estimates are derived under certain regularity conditions on the exact solution. For the second method, we first introduce a new class of generalized fractional Jacobi polynomials (GFJPs), also known as Müntz polynomials, and investigate their approximation properties. Then we propose an efficient scheme using GFJPs for the time-fractional diffusion equation. Our theoretical or numerical investigation shows that the proposed scheme is exponentially convergent for general right hand side functions, even though the exact solution has very limited regularity. Implementation details are also provided, along with a series of numerical examples to show the efficiency of the proposed methods.
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
