Dissipativity and contractivity for fractional-order systems

Wang, Dongling; Xiao, Aiguo

doi:10.1007/s11071-014-1868-1

Cited by 31 publications

(21 citation statements)

References 30 publications

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“…Figures 4.3 (a)-(c) show the numerical solutions for Case III, where the exact solution is not explicitly given. We can see that the solutions are bounded as theoretically expected [31]. Figure 4.3 (d) shows that the fast method is much more efficient than the direct method.…”

Section: Error Analysissupporting

confidence: 72%

A New Class of Semi-Implicit Methods with Linear Complexity for Nonlinear Fractional Differential Equations

Zeng¹,

Turner²,

Burrage³

et al. 2018

SIAM J. Sci. Comput.

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We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters. Subsequently, we develop an efficient strategy to calculate the discrete convolution for the approximation of the fractional operator in the semi-implicit method and we derive an error bound of the fast convolution. The memory requirement and computational cost of the present semi-implicit methods with a fast convolution are about O(N log n T ) and O(N n T log n T ), respectively, where N is a suitable positive integer and n T is the final number of time steps. Numerical simulations, including the solution of a system of two nonlinear fractional diffusion equations with different fractional orders in two-dimensions, are presented to verify the effectiveness of the semi-implicit methods.

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Section: Error Analysissupporting

confidence: 72%

A New Class of Semi-Implicit Methods with Linear Complexity for Nonlinear Fractional Differential Equations

Zeng¹,

Turner²,

Burrage³

et al. 2018

SIAM J. Sci. Comput.

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“…These results have made possible advances in the study of dissipativity and contractivity for fractional order systems [9], Mittag-Leffler stability of fractional Lorenz-family systems [10], and Lyapunov uniform stability for fractional order systems [11]. Other relevant results presented in [12,13] allow us to easily test the stability of a Caputo fractional system of order 0 ă α ă 1.…”

Section: Hernández-martínez)mentioning

confidence: 96%

“…Considering that Lemma 2 has been applied in [12,13,9,10,11], its generalization presented in Lemma 3 and Remark 2 may also have interesting consequences.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Lyapunov functions for a class of nonlinear systems using Caputo derivative

Fernández‐Anaya

Nava-Antonio

Jamous-Galante

et al. 2017

Communications in Nonlinear Science and Numerical Simulation

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“…It is very interesting and natural for us to understand if the fundamental results we have reviewed above about contractivity and dissipativity of the classical ODEs (2) can be established also for F-ODEs. We first studied the Caputo F-ODEs in [42] and established the contractivity and dissipativity under the same conditions as those for classical ODEs. More precisely, we obtained the following results [42].…”

mentioning

confidence: 99%

“…We first studied the Caputo F-ODEs in [42] and established the contractivity and dissipativity under the same conditions as those for classical ODEs. More precisely, we obtained the following results [42]. Lemma 1.…”

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confidence: 99%

Long-time behavior of numerical solutions to nonlinear fractional ODEs

2020

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In this work, we study the long time behaviors, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By mens of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grünwald-Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including some second order F-BDFs and 3-α order method, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods. .hk).1 As noted in [22], a numerical method preserves the contractivity in (4) is sometimes referred to be dissipative in the numerical literature, but this conflicts with the corresponding terminology in dynamical systems. At the same time, the authors in [22] give an accurate definition of dissipativity for ODEs, which mainly emphasizes the existence of a global attracting set. This definition was later widely accepted in the numerical literature, and we also follow this definition.

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Dissipativity and contractivity for fractional-order systems

Cited by 31 publications

References 30 publications

A New Class of Semi-Implicit Methods with Linear Complexity for Nonlinear Fractional Differential Equations

A New Class of Semi-Implicit Methods with Linear Complexity for Nonlinear Fractional Differential Equations

Lyapunov functions for a class of nonlinear systems using Caputo derivative

Long-time behavior of numerical solutions to nonlinear fractional ODEs

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