Fractional Calculus

Băleanu, Dumitru; Diethelm, Kai; Scalas, Enrico; Trujillo, Juan J.

doi:10.1142/9789814355216

Cited by 382 publications

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“…The additional perturbation error is not large compared to the approximation errors for the other terms in (11), and hence (12), which is called as ADIM or ADI-Euler method, is consistent with order O(τ + + ). Computationally, the ADI-Euler method defined by (12) can now be solved by the following iterative scheme. At time :…”

Section: An Alternating Direction Methods For the 2d-rsfde-nrtmentioning

confidence: 99%

“…In this section, we discuss the stability and convergence of the implicit numerical method (12). We need to rewrite (13), (14) and (12) in matrix form [24].…”

Section: Stability and Convergence Of The Adimmentioning

confidence: 99%

“…According to the results from discussing the consistency of the implicit method defined by (12) in Section 2, we obtain the following error equation…”

Section: Lemmamentioning

confidence: 99%

“…Rabei et al [10] presented fractional Hamiltonï£¡CJacobi formulations for systems containing Riesz fractional derivatives Jarad et al [11] studied sequential Riesz-Caputo fractional variational problems with and without the presence of delay in the state variables and their derivatives. Baleanu et al [12] introduced the fractional diffusion problem. Ochoa-Tapia et al [13] proposed a fractional Darcy's law to simulate shear stress phenomena in heterogeneous porous media.…”

Section: Introductionmentioning

confidence: 99%

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Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Liu¹,

Chen²,

Turner³

et al. 2013

Open Physics

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Abstract:Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.PACS (2008)

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Section: An Alternating Direction Methods For the 2d-rsfde-nrtmentioning

confidence: 99%

“…In this section, we discuss the stability and convergence of the implicit numerical method (12). We need to rewrite (13), (14) and (12) in matrix form [24].…”

Section: Stability and Convergence Of The Adimmentioning

confidence: 99%

“…According to the results from discussing the consistency of the implicit method defined by (12) in Section 2, we obtain the following error equation…”

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Liu¹,

Chen²,

Turner³

et al. 2013

Open Physics

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show abstract

“…As crucial realizations, those, which reveal such important dynamic properties as stability, controllability, and so forth, are considered. The first-order differential equations can be generalized to the fractional-order ones [3][4][5][6][7][8][9][10]. Hence, we get the fractional-order integrator, which can be used in a modelling of fractional-order dynamic systems.…”

Section: Introductionmentioning

confidence: 99%

Generalized Fractional-Order Discrete-Time Integrator

Mozyrska

Ostalczyk

2017

Complexity

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Delay Approximation of Fractional Integrals

Machado¹

2012

Asian Journal of Control

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This paper explores the calculation of fractional integrals by means of the time delay operator. The study starts by reviewing the memory properties of fractional operators and their relationship with time delay. Based on the time response of the Mittag‐Leffler function an approximation of fractional integrals consisting of time delayed samples is proposed. The tuning of the approximation is optimized by means of a genetic algorithm. The results demonstrate the feasibility of the new perspective and the limits of their application.

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Fractional Calculus

Cited by 382 publications

References 0 publications

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Generalized Fractional-Order Discrete-Time Integrator

Delay Approximation of Fractional Integrals

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