Marina Popolizio

Acceleration Techniques for Approximating the Matrix Exponential Operator

In this paper we investigate some well-established and more recent methods that aim at approximating the vector

Evaluation of generalized Mittag---Leffler functions on the real line

\exp(A)v

On the use of matrix functions for fractional partial differential equations

when

On accurate product integration rules for linear fractional differential equations

A

Generalized exponential time differencing methods for fractional order problems

is a large symmetric negative semidefinite matrix, by efficiently combining subspace projections and spectral transformations. We show that some recently developed acceleration procedures may be restated as preconditioning techniques for the partial fraction expansion form of an approximating rational function. These new results allow us to devise a priori strategies to select the associated acceleration parameters; theoretical and numerical results are shown to justify these choices. Moreover, we provide a performance evaluation among several numerical approaches to approximate the action of the exponential of large matrices. Our numerical experiments provide a new, and in some cases, unexpected picture of the actual behavior of the discussed methods.

Solving the time-fractional Schrödinger equation by Krylov projection methods

This paper addresses the problem of the numerical computation of generalized Mittag–Leffler functions with two parameters, with applications in fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are investigated and combined with quadrature rules for the numerical integration. An in-depth error analysis is carried out to select suitable contour’s parameters, depending on the parameters of the Mittag–Leffler function, in order to achieve any fixed accuracy. We present numerical experiments to validate theoretical results and some computational issues are discussed.

The restarted shift-and-invert Krylov method for matrix functions

Abstract: The main focus of this paper is the solution of some partial differential equations of fractional order. Promising methods based on matrix functions are taken in consideration. The features of different approaches are discussed and compared with results provided by classical convolution quadrature rules. By means of numerical experiments accuracy and performance are examined.

Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus

This paper addresses the numerical solution of linear fractional differential equations with a forcing term. Competitive and highly accurate Product Integration rules are derived by starting from an equivalent formulation in terms of a Volterra integral equation with a generalized Mittag-Leffler function in the kernel. The error analysis is reported and aspects related to the computational complexity are treated. Numerical tests confirming the theoretical findings are presented.

On the time-fractional Schrödinger equation: Theoretical analysis and numerical solution by matrix Mittag-Leffler functions

The main aim of this paper is to discuss the generalization of exponential integrators to differential equations of non-integer orders. Two methods of this kind are devised and the accuracy and stability are investigated. Some numerical experiments are presented to validate the theoretical findings.

Exponential Quadrature Rules for Linear Fractional Differential Equations

The time-fractional Schrodinger equation is a fundamental topic in physics and its numerical solution is still an open problem. Here we start from the possibility to express its solution by means of the Mittag-Leffler function; then we analyze some approaches based on the Krylov projection methods to approximate this function; their convergence properties are discussed, together with related issues. Numerical tests are presented to confirm the strength of the approach under investigation.