Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

Liu, Fawang; Zhuang, Pinghui; Anh, Vo; Turner, Ian; Burrage, Kevin

doi:10.1016/j.amc.2006.08.162

Cited by 390 publications

(222 citation statements)

References 5 publications

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“…The following definitions and Lemmas for the next consideration are easily based on the results of Ref. [12,13,14,20]. Definition 2.1.…”

Section: Problem Formulationmentioning

confidence: 99%

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

Zhang¹,

Liu²

2016

J. Nonlinear Sci. Appl.

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The full state hybrid projective synchronization (FSHPS) definition for variable-order fractional chaotic/hyperchaotic systems with nonlinear external disturbances and unknown parameters is firstly presented. Then by introducing a compensator and a nonlinear controller, the FSHPS scheme is generated to eliminate the influence of nonlinear external disturbances effectively. Moreover, the parameters are estimated validly. Based on these control methods, appropriate parameters and controller to achieve FSHPS for the variable-order fractional chaotic/hyperchaotic systems are chosen impactfully. Simulations of variable-order fractional Chen and Lü system and fractional order hyperchaotic Lorenz system in the sense of FSHPS are performed and results show the effectiveness of our method.

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“…The following definitions and Lemmas for the next consideration are easily based on the results of Ref. [12,13,14,20]. Definition 2.1.…”

Section: Problem Formulationmentioning

confidence: 99%

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

Zhang¹,

Liu²

2016

J. Nonlinear Sci. Appl.

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“…As we know, this approach replaces the appropriate estimate for each derivative in the di erential equations based on nodal values and estimates derivatives of a known function only by values of the function itself at these discrete points. We recall the time when fractional di usion equations were scrutinized in both analytical and numerical frames by several authors (see for example [25][26][27][28]). But, according to fundamental role of this class of equations in science and engineering, we will talk about it again.…”

Section: Introductionmentioning

confidence: 99%

A reliable implicit difference scheme for treatments of fourth-order fractional sub-diffusion equation

Sayevand

Arjang

2017

Scientia Iranica

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KEYWORDSAbstract. In this paper, a reliable implicit di erence scheme is proposed to analyze the fractional fourth-order subdi usion equation on a bounded domain. The time-fractional derivative operator is characterized in the Ji Huan He's sense, and the space derivative is approximated by the ve-point centered formula. The numerical parameters, i.e. consistency, stability, and convergence analyses of the considered scheme, are proven.

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“…Tadjeran et al [26] developed the solution of fractional diffusion equation. Liu et al [19] provided the stability and convergence of the difference schemes for the fractional advection-diffusion equation. Sousa [23] proposed an approximation of the Caputo fractional derivative of order 1 < α 2.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of time-fractional hyperbolic partial differential equations

Arshed¹

2017

J. Math. Computer Sci.

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In this study, a numerical scheme is developed for the solution of time-fractional hyperbolic partial differential equation. In the proposed scheme, cubic B-spline collocation is used for space discretization and time discretization is obtained by using central difference formula. Caputo fractional derivative is used for time-fractional derivative. The stability and convergence of the developed scheme, have also been proved. The numerical examples support the theoretical results.

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Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

Cited by 390 publications

References 5 publications

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

A reliable implicit difference scheme for treatments of fourth-order fractional sub-diffusion equation

Numerical study of time-fractional hyperbolic partial differential equations

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