Numerical approximation of distributed order reaction–diffusion equations

Morgado, Maria Luísa; Rebelo, Magda

doi:10.1016/j.cam.2014.07.029

Cited by 105 publications

(58 citation statements)

References 33 publications

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“…Ye et al [35] have treated the time distributed-order and space Riesz fractional diffusion on bounded domains numerically, where the distributed integral was discretized by the mid-point quadrature rule and the time-fractional derivatives in the resultant multi-term fractional diffusion equation were approximated by the classical L1 formula. The similar technique was applied in the recent works [36,37] for the time-fractional diffusion equations of distributed order. The nearly first-order accuracy in time of algorithms in [35][36][37] was obtained due to the use of L1 formula for the approximation of involving time-fractional derivatives.…”

Section: Introductionmentioning

confidence: 83%

“…The similar technique was applied in the recent works [36,37] for the time-fractional diffusion equations of distributed order. The nearly first-order accuracy in time of algorithms in [35][36][37] was obtained due to the use of L1 formula for the approximation of involving time-fractional derivatives. A similar work can also be found in the recent literature [38], but the distributed integral was not discretized at all.…”

Section: Introductionmentioning

confidence: 83%

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Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

Gao

Sun

2015

J Sci Comput

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Two alternating direction implicit difference schemes are derived for twodimensional distributed-order fractional diffusion equations. It is proved that the schemes are unconditionally stable and convergent in a discrete L 1 (L ∞ ) norm with the convergence orders O(τ 2 | ln τ | + h 2 1 + h 2 2 + α 2 ) and O(τ 2 | ln τ | + h 4 1 + h 4 2 + α 4 ), respectively, where τ, h i (i = 1, 2) and α are the step sizes in time, space and distributed order. Several numerical examples are given to confirm the theoretical results.

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 83%

Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

Gao

Sun

2015

J Sci Comput

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“…For a proof of these results see [11] and [28]. These properties will be useful when deriving the stability and convergence of the proposed method.…”

Section: Numerical Solutionmentioning

confidence: 83%

Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

Ferrás

Ford

Morgado

et al. 2015

FCAA

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In this work we provide a new mathematical model for the Pennes' bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bioheat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.MSC 2010 : 35R11, 65N99, 92-08

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“…A fundamental solution of a fractional order distributed parameter system was presented in [31]. The numerical solutions of such systems were proposed in the literature by means of finite difference methods [32], spectral collocation methods and others [33,34]. However, it is worth pointing out that up to now, there is no concern about ILC for fractional order distributed parameter systems.…”

Section: Introductionmentioning

confidence: 99%

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

Lan

Cui

2018

Algorithms

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This paper presents a second order P-type iterative learning control (ILC) scheme with initial state learning for a class of fractional order linear distributed parameter systems. First, by analyzing the control and learning processes, a discrete system for P-type ILC is established, and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by introducing a new norm and using the generalized Gronwall inequality, which is less conservative than the existing one. Finally, the validity of the proposed method is verified by a numerical example.

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Numerical approximation of distributed order reaction–diffusion equations

Cited by 105 publications

References 33 publications

Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

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