Discretized fractional substantial calculus

Chen, Minghua; Deng, Weihua

doi:10.1051/m2an/2014037

Cited by 16 publications

(12 citation statements)

References 14 publications

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“…Obviously, C D α,0 a u(t) = C D α a u(t). This type of derivative was first defined in the article [37] (see also [12,23,32]). Definition 4.…”

Section: Preliminariesmentioning

confidence: 99%

“…The definition of the tempered Ψ −Caputo fractional derivative covers the wellknown fractional derivatives for λ = 0, like the Caputo fractional derivative (Ψ (t) = t), the Caputo-Hadamard fractional derivative (Ψ (t) = ln t), the Caputo-Erdélyi-Kober fractional derivative (Ψ (t) = t σ ). The tempered Ψ −Caputo derivative with Ψ (t) = t, is defined and considered in the papers [12,23]. We need the following lemma, proved in [10, Theorem 1].…”

Section: It Is Clear Thatmentioning

confidence: 99%

“…Some definitions of such fractional derivatives can be found in the remarkable monographs [13,16,19,22,33,35,45,48] on fractional calculus and qualitative analysis of fractional-order differential equations. In the recently published papers [12,23,37] so-called tempered Riemann-Liouville and tempered Caputo fractional derivatives are defined. Fractional differential equations with these derivatives have very good properties from the point of view of stability of their solutions.…”

mentioning

confidence: 99%

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Differential Equations With Tempered Ψ-Caputo Fractional Derivative

Medveď

Brestovanská

2021

Mathematical Modelling and Analysis

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In this paper we define a new type of the fractional derivative, which we call tempered Ψ−Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the Ψ−Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered Ψ−fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of Ψ−Caputo fractional differential equations is proved and in the last section an example is presented.

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“…Obviously, C D α,0 a u(t) = C D α a u(t). This type of derivative was first defined in the article [37] (see also [12,23,32]). Definition 4.…”

Section: Preliminariesmentioning

confidence: 99%

Section: It Is Clear Thatmentioning

confidence: 99%

mentioning

confidence: 99%

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Differential Equations With Tempered Ψ-Caputo Fractional Derivative

Medveď

Brestovanská

2021

Mathematical Modelling and Analysis

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“…Definition 3 (fractional substantial derivative [16,40,6]) For n−1 < α < n, n ∈ N + , and λ(x) being any given function defined in space domain. The Riemann-Liouville fractional substantial derivative is defined by…”

Section: Remark 1 ([3])mentioning

confidence: 99%

“…Remark 2 The fractional substantial derivative (6) is equivalent to the Riemann-Liouville tempered fractional derivative (3) if λ(x) is a nonnegative constant function. In fact, using integration by parts leads to…”

Section: Remark 1 ([3])mentioning

confidence: 99%

Well-posedness and numerical algorithm for the tempered fractional differential equations

Can¹,

Deng²,

Zhao³

2019

Discrete &Amp; Continuous Dynamical Systems - B

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Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the wellposedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, any desired convergence order can be obtained and the computational cost linearly increases with time. And the effectiveness of the algorithm is numerically confirmed.

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Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutions

Wang

2022

Math Methods in App Sciences

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In this work, anisotropic bilinear finite element and second‐order temporal approximation are adopted to establish a fully discrete scheme for the fractional substantial diffusion equation with variable coefficient. We prove the proposed discrete scheme is unconditionally stable in the senses of L2$$ {L}&#x0005E;2 $$‐norm and H1$$ {H}&#x0005E;1 $$‐norm. By introducing a new projection, the optimal convergence error in L2$$ {L}&#x0005E;2 $$‐norm and the superclose property in H1$$ {H}&#x0005E;1 $$‐norm are obtained under the condition of u∈H3false(normalΩfalse)$$ u\in {H}&#x0005E;3\left(\Omega \right) $$, where u$$ u $$ is the exact solution of the problem. Through interpolation post processing technique, we arrive at the global superconvergence of the interpolation. Furthermore, an improved algorithm is proposed to solve the problem with nonsmooth solution. Finally, numerical tests are given to illustrate the validity and efficiency of our theoretical analysis.

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Discretized fractional substantial calculus

Cited by 16 publications

References 14 publications

Differential Equations With Tempered Ψ-Caputo Fractional Derivative

Differential Equations With Tempered Ψ-Caputo Fractional Derivative

Well-posedness and numerical algorithm for the tempered fractional differential equations

Superconvergence analysis of anisotropic finite element method for the time fractional substantial diffusion equation with smooth and nonsmooth solutions

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