Numerical solution of time fractional diffusion systems

Burrage, Kevin; Cardone, Angelamaria; D’Ambrosio, Raffaele; Paternoster, Beatrice

doi:10.1016/j.apnum.2017.02.004

Cited by 49 publications

(27 citation statements)

References 35 publications

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“…We must mention that several other approaches have been however discussed in the literature: see, for instance, the generalized Adams methods [10], extensions of the Runge-Kutta methods [11], generalized exponential integrators [12,13], spectral methods [14,15], spectral collocation methods [16], methods based on matrix functions [17][18][19][20], and so on. In this paper, for brevity, we focus only on PI rules and FLMMs, and we refer the reader to the existing literature for alternative approaches.…”

Section: Multi-step Methods For Fdesmentioning

confidence: 99%

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

2018

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Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines.

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Section: Multi-step Methods For Fdesmentioning

confidence: 99%

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

2018

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“…This idea may be exploited to obtain high order methods for solving other types of equations as well. For example, recently two-step collocation methods have been proposed for fractional differential equations [44], and further developments may be achieved for other fractional models, as time fractional differential equations [45]. Further issues of this research will focus on oscillatory problems [46,47] and in particular on the application of multistep collocation methods to periodic integral equations [48,49].…”

Section: Discussionmentioning

confidence: 99%

“…The discretized collocation methods are a special class of the Runge-Kutta extended methods and preserve the order of convergence and superconvergence of exact collocation methods, if the quadrature Formulae (44) and (45) are sufficiently accurate [3].…”

Section: Discretized One-step Collocation Methodsmentioning

confidence: 99%

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

Cardone

Conte

D’Ambrosio

et al. 2018

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We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper.

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“…Recently, for the simulation of complex or smooth physical phenomena, spectral methods have emerged as a very powerful and efficient numerical technique, which are a well-known class of numerical methods for the solutions of various differential equations due to the spectral rate of convergence [10][11][12]. These methods for the numerical solutions of stochastic fractional differential equations were most recently used by Raffaele D' Ambrosio et al [13][14][15], while Jacobi polynomials for the numerical solution of time fractional diffusion systems was used in [16,17]. The aim of this research work is to develop the spectral collocation method using Legendre-Gauss-Lobatto points for the approximate solution of SVIDEs.…”

Section: Introductionmentioning

confidence: 99%

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

Khan

Ali

2019

Adv Differ Equ

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Volterra integro-differential equations arise in the modeling of natural systems where the past influence the present and future, for example pollution, population growth, mechanical systems and financial market. Furthermore, as many real-world phenomena are subject to perturbations or random noise, it is natural to move from deterministic models to stochastic models. Generally exact solutions of such models are not available and numerical methods are used to obtain the approximate solutions. Therefore the efficiency and long-term behavior of approximate solutions for these systems is an important area of investigation. This paper presents a new numerical approach for the approximate solution of stochastic Volterra integro-differential (SVID) equations based on the Legendre-spectral collocation method. In order to fully use the properties of orthogonal polynomials, we use some function and a variable transformation to change the given SVID equation into a new equation, which is defined on the standard interval [-1, 1]. For the evaluation of the integral term efficiently a Legendre-Gauss quadrature formula will be used. A rigorous error analysis of the proposed scheme will be provided under the assumption that the solution of the given SVID is sufficiently smooth. For the illustration of our theoretical results a number of numerical experiments will be performed.

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Numerical solution of time fractional diffusion systems

Cited by 49 publications

References 35 publications

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

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