Fast methods for the computation of the Mittag-Leffler function
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Roberto Garrappa
Abstract
In this chapter we discuss the problem of the fast and reliable computation of the Mittag-Leffler function. We start from the series representation, by which the function is commonly defined, to illustrate how it turns out to be not suitable for computation in most of the cases; then we present an alternative approach based on the numerical inversion of the Laplace transform. In particular, we describe a technique, known as the optimal parabolic contour, in which the inversion of the Laplace transform is performed by applying the trapezoidal rule along a parabolic contour in the complex plane; the contour and the integration parameters are chosen, on the basis of the error analysis and the location of the singularities of the Laplace transform, with the aim of achieving a target accuracy (which can be very close to machine precision) with a substantially low computational effort. Applications to the evaluation of derivatives of the Mittag-Leffler function and to matrix arguments are also discussed.
Abstract
In this chapter we discuss the problem of the fast and reliable computation of the Mittag-Leffler function. We start from the series representation, by which the function is commonly defined, to illustrate how it turns out to be not suitable for computation in most of the cases; then we present an alternative approach based on the numerical inversion of the Laplace transform. In particular, we describe a technique, known as the optimal parabolic contour, in which the inversion of the Laplace transform is performed by applying the trapezoidal rule along a parabolic contour in the complex plane; the contour and the integration parameters are chosen, on the basis of the error analysis and the location of the singularities of the Laplace transform, with the aim of achieving a target accuracy (which can be very close to machine precision) with a substantially low computational effort. Applications to the evaluation of derivatives of the Mittag-Leffler function and to matrix arguments are also discussed.
Kapitel in diesem Buch
- 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
Kapitel in diesem Buch
- 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
