Roberto Garrappa

Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions

The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.

On linear stability of predictor-corrector algorithms for fractional differential equations

This paper deals with the numerical approximation of differential equations of fractional order by means of predictor–corrector algorithms. A linear stability analysis is performed and the stability regions of different methods are compared. Furthermore the effects on stability of multiple corrector iterations are verified.

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

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.

On the use of matrix functions for fractional partial differential equations

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.

On accurate product integration rules for linear fractional differential equations

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.

Trapezoidal methods for fractional differential equations

The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical experiments are provided to illustrate potential and limitations of the different methods under investigation.

Generalized exponential time differencing methods for fractional order problems

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.

Fractional Adams-Moulton methods

In the simulation of dynamical systems exhibiting an ultraslow decay, differential equations of fractional order have been successfully proposed. In this paper we consider the problem of numerically solving fractional differential equations by means of a generalization of k-step Adams-Moulton multistep methods. Our investigation is focused on stability properties and we determine intervals for the fractional order for which methods are at least A(@p/2)-stable. Moreover we prove the A-stable character of k-step methods for k=0 and k=1.

On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics

The three parameters Mittag-Leffler function (often referred to as the Prabhakar function) has important applications, mainly in physics of dielectrics, in describing anomalous relaxation of non-Debye type. This paper concerns with the investigation of the conditions, on the characteristic parameters, under which the function is locally integrable and completely monotonic; these properties are essential for the physical feasibility of the corresponding models. In particular the classical Havriliak-Negami model is extended to a wider range of the parameters. The problem of the numerical evaluation of the three parameters Mittag-Leffler function is also addressed and three different approaches are discussed and compared. Numerical simulations are hence used to validate the theoretical findings and present some graphs of the function under investigation.

Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching

In this paper, a simple parameter switching (PS) methodology is proposed for sustaining the stable dynamics of a fractional-order chaotic financial system. This is achieved by switching a controllable parameter of the system, within a chosen set of values and for relatively short periods of time. The effectiveness of the method is confirmed from a computer-aided approach, and its applications to chaos control and anti-control are demonstrated. In order to obtain a numerical solution of the fractional-order financial system, a variant of the Grunwald-Letnikov scheme is used. Extensive simulation results show that the resulting chaotic attractor well represents a numerical approximation of the underlying chaotic attractor, which is obtained by applying the average of the switched values. Moreover, it is illustrated that this approach is also applicable to the integer-order financial system.