Time discretization of fractional subdiffusion equations via fractional resolvent operators

Ponce, Rodrigo

doi:10.1016/j.camwa.2020.04.024

Cited by 13 publications

(17 citation statements)

References 39 publications

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“…where ρ is a small positive constant. The operators d n and D n can be seen as the discrete approximation of the fractional resolvent family R α,1 (t) and R α,α (t) at t = t n (it is also called discrete fractional resolvent family [28,36]). We can see that these two operators serve as the discrete Mittag-Leffler functions.…”

Section: 22)mentioning

confidence: 99%

“…The discrete fractional resolvent sequence for time fractional difference equations with step size h = 1 has been an important tool to study the qualitative properties of the solutions to fractional difference equations, such as the ℓ p -maximum regularity and the existence and uniqueness [28]. This concept has recently been extended to arbitrary step size h > 0 in [36] and was used to construct numerical schemes for linear sub-diffusion equations. One main advantage of the α-resolvent sequence is that it allows us to write numerical solutions in terms of discrete constant variation formulas, exactly like the continuous case given in (4.1).…”

Section: Discrete Fractional Resolvent Family and Poisson Transformationmentioning

confidence: 99%

“…Definition 10. [36] For any α ∈ (0, 1) and sequence v = (v 0 , v 1 , ...), the α-fractional sum of v with a stepsize h > 0 is given by J α h (v n ) := h α n j=0 k α n−j v j for n ∈ N 0 , where the coefficients k α 0 = 1 and k α n = Γ(α+n) Γ(α)Γ(1+n) for n ≥ 1. Furthermore, the Caputo α-fractional difference is given by…”

Section: Discrete Fractional Resolvent Family and Poisson Transformationmentioning

confidence: 99%

“…The α-fractional sum and difference operators have several nice properties, of which the so-called Poisson transformation is one that was studied in [28] and extended in [36]. For fixed h > 0 and n ∈ N 0 , the discrete Poisson distribution is given by…”

Section: Discrete Fractional Resolvent Family and Poisson Transformationmentioning

confidence: 99%

“…Numerical solutions to these schemes can also be expressed by the discrete constant variation formula involving a discrete fractional resolvent operator. A significant advantage of this approach is that the discrete fractional resolvent operator can be directly connected with the continuous fractional resolvent operator through the Poisson transformation [28,36]. Therefore, we can use various properties of Poisson transformation and continuous operator to give an optimal estimate of the decay rate for the discrete fractional resolvent operator in a simple and straightforward manner.…”

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confidence: 99%

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Mittag--Leffler stability of numerical solutions to time fractional ODEs

Wang¹,

Zou²

2021

Preprint

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The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3 (1990), 216-240), we are able to prove that both L1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on α-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation and the decay rate of the continuous fractional resolvent operator. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations, fractional linear system and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

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Section: 22)mentioning

confidence: 99%

Section: Discrete Fractional Resolvent Family and Poisson Transformationmentioning

confidence: 99%

Section: Discrete Fractional Resolvent Family and Poisson Transformationmentioning

confidence: 99%

Section: Discrete Fractional Resolvent Family and Poisson Transformationmentioning

confidence: 99%

mentioning

confidence: 99%

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Mittag--Leffler stability of numerical solutions to time fractional ODEs

Wang¹,

Zou²

2021

Preprint

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Identification of the order in fractional discrete systems

Ponce

2024

Math Methods in App Sciences

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Discretization of C0$$ {C}_0 $$‐semigroups and discrete semigroups of operators in Banach spaces

Ponce

2023

Math Methods in App Sciences

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In this paper, we introduce the notion of a 𝜏-discrete semigroup {T n 𝜏 } n∈N 0 generated by a closed linear operator A in a Banach space X. We show that {T n 𝜏 } n∈N 0 allows us to write the solution to an abstract discrete difference equation of first order as a discrete variation of parameters formula. Moreover, we study the main properties of {T n 𝜏 } n∈N 0 and its relation with the well-known notion of discrete semigroups. Finally, we characterize uniform exponential stability of a C 0 -semigroup {T(t)} t≥0 in terms of the 𝜏-discrete semigroup {T n 𝜏 } n∈N 0 .

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Time discretization of fractional subdiffusion equations via fractional resolvent operators

Cited by 13 publications

References 39 publications

Mittag--Leffler stability of numerical solutions to time fractional ODEs

Mittag--Leffler stability of numerical solutions to time fractional ODEs

Identification of the order in fractional discrete systems

Discretization of C0$$ {C}_0 $$‐semigroups and discrete semigroups of operators in Banach spaces

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