1,721,030 research outputs found
On some explicit Adams multistep methods for fractional differential equations
No full textIn this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order. The proposed methods are obtained by modifying, in a suitable way, Fractional-Adams-Moulton methods and they represent a way for extending classical Adams-Bashforth multistep methods to the fractional case. The attention is hence focused on the investigation of stability properties. Intervals of stability for k-step methods, k = 1,...,5, are computed and plots of stability regions in the complex plane are presented. (C) 2008 Elsevier B.V. All rights reserved
Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models
No full textSeveral classes of differential and integral operators of non integer order have been proposed in the past to model systems exhibiting anomalous and hereditary properties. A wide range of complex and heterogeneous systems are described in terms of laws of Havriliak-Negami type involving a special fractional relaxation whose behavior in the time-domain can not be represented by any of the existing operators. In this work we introduce new integral and differential operators for the description of Havriliak-Negami models in the time-domain. In particular we propose a formulation of Grünwald-Letnikov type which turns out to be effective not only to provide a theoretical characterization of the operators associated to Havriliak-Negami systems but also for computational purposes. We study some properties of the new operators and, by means of some numerical experiments, we present their use in practical computation and we show the superiority with respect to the few other approaches previously proposed in literature
Order conditions for Volterra Runge-Kutta methods
No full textWe 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
On some generalizations of the implicit Euler method for discontinuous fractional differential equations
No full textWe 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 presente
On linear stability of predictor-corrector algorithms for fractional differential equations
No full textThis 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
STABILITY-PRESERVING HIGH-ORDER METHODS FOR MULTITERM FRACTIONAL DIFFERENTIAL EQUATIONS
No full textThe 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
Trapezoidal methods for fractional differential equations: theoretical and computational aspects
No full textThe 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 investigatio
An analysis of convergence for two-stage waveform relaxation methods
No full textThis paper consider a two-stage (or inner/outer) strategy for waveform relaxation (WR) iterations, applied to initial value problems for linear systems of ordinary differential equations (ODEs) in the form y(t) + Qy(t) = f (t). Outer WR iterations are defined by y (k+1)(t) + Dyk+1(t) = N1 y(k) (t) + f (t), where Q = D - N-1, and each iteration y(k+1) (t) is computed using an inner iterative process, based on an other splitting D = M - N-2. Each ODE is then discretized by means of Theta method. For an M-matrix Q we prove that the method converges under the assumption that the whole splitting Q = M - N-1 - N-2 is an M-splitting, independently of the number of inner iterations. Moreover, some comparison results are given in order to relate the ratio of convergence of the whole inner/outer process both to the number of inner iterations actually done and to discretization parameters h and theta. Finally numerical experiments are presented. (C) 2004 Elsevier B.V. All rights reserved
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