Approximate inversion method for time‐fractional subdiffusion equations

Lu, Xiwen; Pang, Hong‐Kui; Sun, Hai-Wei; Vong, Seakweng

doi:10.1002/nla.2132

Cited by 21 publications

(11 citation statements)

References 24 publications

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“…The fractional derivative denoted by D is introduced in Caputo sense. 11,18,49,50 In addition, we suppose that the function g and functions 0 and 1 are such that the problems have a unique solution.…”

Section: Problem Statementmentioning

confidence: 99%

“…Problem

D_{t}^{ν} u false(x, t false) = F ((), u false(x, t false), \frac{\partial}{\partial ξ} u false(x, t false), …, \frac{\partial^{s}}{\partial ξ^{s}} u false(x, t false), u false(x - τ_{0}, t - τ_{0} false), …, u false(x - τ_{r}, t - τ_{r} false)),

l - 1 < ν \leq l, x \in [0, h_{1}], t \in [0, h_{2}],

with the supplementary conditions,

\begin{matrix} alignleft \\ align-1 u (x, t) & align-2 = g (x, t), x \in [- τ_{0}, 0], t \in [- τ_{0}, 0], \\ align-1 u (0, t) & align-2 = ϕ_{0} (t), u (h_{1}, t) = ϕ_{1} (t), 0 < t \leq h_{2}, \end{matrix}

where τ r >…> τ 0 >0 is the delay parameter, p i , q i ∈(0,1], i =0,1,…, l and F is an analytical function. The fractional derivative denoted by D ν is introduced in Caputo sense . In addition, we suppose that the function g and functions ϕ 0 and ϕ 1 are such that the problems have a unique solution....…”

Section: Introductionmentioning

confidence: 99%

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On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

Dehestani

Ordokhani

Razzaghi

2019

Numerical Linear Algebra App

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Summary A novel collocation method based on Genocchi wavelet is presented for the numerical solution of fractional differential equations and time‐fractional partial differential equations with delay. In this work, to achieve the approximate solution with height accuracy, we employed the operational matrix of integer derivative and the pseudo‐operational matrix of fractional derivative in Caputo sense. Also, based on Genocchi function properties, we presented delay and pantograph operational matrices of Genocchi wavelet functions (GWFs). Due to operational and pseudo‐operational matrices, the equations under this study can be turned into nonlinear algebraic equations with the unknown GWF coefficients. For illustrating the upper bound of error for the proposed method, we estimate the error in the sense of Sobolev space. In addition, to demonstrate the efficacy of the pseudo‐operational matrix of fractional derivative, we investigate the upper bound of error for the mentioned matrix. Finally, the algorithm based on the proposed approach is implemented for some numerical experiments to confirm accuracy and applicability.

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Section: Problem Statementmentioning

confidence: 99%

“…Problem

D_{t}^{ν} u false(x, t false) = F ((), u false(x, t false), \frac{\partial}{\partial ξ} u false(x, t false), …, \frac{\partial^{s}}{\partial ξ^{s}} u false(x, t false), u false(x - τ_{0}, t - τ_{0} false), …, u false(x - τ_{r}, t - τ_{r} false)),

l - 1 < ν \leq l, x \in [0, h_{1}], t \in [0, h_{2}],

with the supplementary conditions,

\begin{matrix} alignleft \\ align-1 u (x, t) & align-2 = g (x, t), x \in [- τ_{0}, 0], t \in [- τ_{0}, 0], \\ align-1 u (0, t) & align-2 = ϕ_{0} (t), u (h_{1}, t) = ϕ_{1} (t), 0 < t \leq h_{2}, \end{matrix}

Section: Introductionmentioning

confidence: 99%

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

Dehestani

Ordokhani

Razzaghi

2019

Numerical Linear Algebra App

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“…In consequence of the nonlocal property and historical dependence of the fractional operators, the aforementioned numerical methods always require all previous function values, which leads to an average O(n) storage and computational cost O(n 2 ), where n is the total number of the time levels. To overcome this difficulty, many efforts have been made to speed up the evaluation of the CO fractional derivative [2,4,14,17,18,[23][24][25]42]. Nevertheless, the coefficient matrices of the numerical schemes for the VO fractional problems lose the Toeplitz-like structure and the VO fractional derivative is no longer a convolution operator.…”

Section: Introductionmentioning

confidence: 99%

Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

Zhang¹,

Fang²,

Sun³

2022

NMTMA

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In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on L2-1 σ formula and the exponential-sum-approximation technique. The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme (F L2-1 σ scheme) for the multi-dimensional VO time-fractional sub-diffusion equations. Theoretically, F L2-1 σ scheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1 σ scheme. Therefore, F L2-1 σ scheme is strictly proved to be unconditionally stable and convergent. A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

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“…Because of this, before solving the linear system ( 1 To remedy the situation, fast kernel compression methods are proposed in [2,12,17,26] Besides the time-stepping method, PinT method is another type of popular methods for (1.5), which solves {ũ n |n = 1, 2, ..., N } in a parallel way. In [9,20,24,25], PinT algorithms are developed by employing a fast diagonalizable approximation to T in the all-at-once system (1.6). The so approximated all-at-once system is block diagonalizable with each eigen-block corresponding to a complex scalar shifted spatial system.…”

Section: Introductionmentioning

confidence: 99%

A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel

Lin,

Ng,

Zhi

2021

Preprint

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In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. Such a problem has been intensively studied in recent years thanks to the numerous real world applications. However, due to the non-local property of the time evolution, solving the equation in PinT manner is difficult. We propose to use a two-sided preconditioning technique for the all-at-once discretization of the equation. Our preconditioner is constructed by replacing the variable diffusion coefficients with a constant coefficient to obtain a constant-coefficient all-at-once matrix. We split a square root of the constant Laplacian operator out of the constantcoefficient all-at-once matrix as a right preconditioner and take the remaining part as a left preconditioner, which constitutes our two-sided preconditioning. Exploiting the diagonalizability of the constant-Laplacian matrix and the triangular Toeplitz structure of the temporal discretization matrix, we obtain efficient representations of inverses of the right and the left preconditioners, because of which the iterative solution can be fast updated in a PinT manner. Theoretically, the condition number of the two-sided preconditioned matrix

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Approximate inversion method for time‐fractional subdiffusion equations

Cited by 21 publications

References 24 publications

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay

Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel

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