The numerical solution of linear time-invariant systems of fractional order is investigated. We construct a family of exponential integrators of Adams type possessing good convergence and stability properties. The methods are devised in order to keep at a suitable level, the computational effort necessary to solve problems of large size. Numerical experiments are provided to validate the theoretical results; the effectiveness of the proposed approach is tested and compared to some other classical methods.

In this paper we propose the generalization of the Grunwald–Letnikov scheme to fractional differential operators of Havriliak–Negami type; these operators have important applications in the description and simulation of polarization processes in anomalous dielectrics with hereditary properties. We discuss in details the technique used for generalizing the proposed scheme, we provide a recursive relationship for the evaluation of the corresponding weights and we study some of their main properties.

We consider an initial-boundary-value problem for a time-fractional diffusion equation with initial condition u0(x) and homogeneous Dirichlet boundary conditions in a bounded interval [0, L]. We study a semidiscrete approximation scheme based on the pseudo-spectral method on Chebyshev-Gauss-Lobatto nodes. In order to preserve the high accuracy of the spectral approximation we use an approach based on the evaluation of the Mittag-Leffler function on matrix arguments for the integration along the time variable. Some examples are presented and numerical experiments illustrate the effectiveness of the proposed approach

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.

Time–fractional partial differential equations can be numerically solved by first discretizing with respect to the spatial derivatives and hence applying a time–step integrator. An exponential integrator for fractional differential equations is proposed to overcome the stability issues due to the stiffness in the resulting semi–discrete system. Convergence properties and the main implementation issues are studied. The advantages of the proposed method are illustrated by means of some test problems.

This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized Mittag-Leffler function. Then, suitable quadrature rules are devised and order conditions of algebraic type are derived. Theoretical findings are validated by means of numerical experiments and the effectiveness of the proposed approach is illustrated by means of comparisons with other standard methods.

The Mittag-Leffler function plays a central role in fractional calculus; however its numerical evaluation still remains an expensive and challenging task. In this work we discuss the evaluation of this function for pure imaginary arguments by means of a numerical method performing the inversion of its Laplace transform on a suitably selected integral contour. By means of some numerical experiments we show that the proposed algorithm behaves in an efficient and fast way.

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. (C) 2011 Elsevier Ltd. All rights reserved.

In this work we investigate the application of some model order reduction techniques, based on Krylov subspace methods, to large linear time--invariant systems of fractional order. Theoretical and technical aspects are discussed. The effectiveness of the proposed approach is verified by numerical simulations on a test problem describing the heat conduction in electro--thermal processes involved in micro--electromechanical systems. An application in a parameter identification problem is also presented.

In this paper we discuss numerical methods for fractional order problems. Some nonstandard finite difference schemes are presented and investigated. The application in the simulation of a fractional order Brusselator system is hence presented. By means of some numerical experiments we show the effectiveness of the proposed approach.

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 reducing 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.

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. (C) 2010 Elsevier B.V. All rights reserved.

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 (lavoro effettuato nell'ambito di ricerche finanziate dall'INdAM).

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.

We discuss the numerical solution of differential equations of fractional order with discontinuous right-hand side. Problems of this kind arise, for instance, in sliding mode control. After applying a set-valued regularization, the behavior of some generalizations of the implicit Euler method is investigated. We show that the scheme in the family of fractional Adams methods possesses the same chattering-free property of the implicit Euler method in the integer case. A test problem is considered to discuss in details some implementation issues and numerical experiments are presented

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.

We consider Runge-Kutta methods for second-kind Volterra Integral Equations with weakly singular kernel. Order conditions, whose number and structure depend on the singularity of the equation, are derived in a recursive manner using an approach originally devised by P. Albrecht for Ordinary Differential Equations. Order conditions are hence generated, in an automatic way, by means of a symbolic algorithm and some numerical experiments are presented. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

This paper presents an extension of the trapezoidal integration rule, that in the present work is applied to devise a pseudo-recursive numerical algorithm for the numerical evaluation of fractional-order integrals. The main benefit of pseudo recursive implementation arises in terms of higher accuracy when the algorithm is run in the "short memory" version. The rule is suitably generalized in order to build a numerical solver for a class of fractional differential equations. The algorithm is also specialized to derive an efficient numerical algorithm for the on-line implementation of linear fractional order controllers. The accuracy of the method is theoretically analyzed and its effectiveness is illustrated by simulation examples.

The time-fractional Schrödinger 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 (lavoro effettuato nell'ambito di ricerche finanziate dall'INdAM).

The numerical approximation of linear multiterm fractional differential equations is investigated. Convolution quadratures based on Runge-Kutta methods together with formulas for the efficient inversion of the Laplace transform are considered to provide highly accurate numerical solutions. Implementation issues are discussed and good stability properties are shown. The effectiveness of the algorithm is analyzed by means of some numerical experiments.

In this paper, a chaos control algorithm for a class of piece-wise continuous chaotic systems of fractional order, in the Caputo sense, is proposed. With the aid of Filippov’s convex regularization and via differential inclusions, the underlying discontinuous initial value problem is first recast in terms of a set-valued problem and hence it is continuously approximated by using Cellina’s Theorem for differential inclusions. For chaos control, an active control technique is implemented so that the unstable equilibria become stable. As example, Shimizu–Morioka’s system is considered. Numerical simulations are obtained by means of the Adams–Bashforth–Moulton method for differential equations of fractional-order.

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 Grünwald–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.

The time-simulation of models described by the Havriliak-Negami response function is a challenging problem due to the absence of an explicit formulation for the corresponding differential operator in the time-domain. In this work, we discuss a convolution quadrature rule with convolution weights evaluated on the basis of the representation of the response function in the Laplace transform domain. We describe a general and straightforward technique for the computation of the weights and we present some numerical experiments to illustrate the effectiveness of the proposed approach.

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