A compact finite difference scheme for variable order subdiffusion equation

Cao, Jianxiong; Qiu, Yanan; Song, Guojie

doi:10.1016/j.cnsns.2016.12.022

Cited by 44 publications

(13 citation statements)

References 28 publications

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“…Some scientific journal considers manuscripts only if accuracy and convergence of numerical solutions are established by discussion of results on multiple grids. Recently, there have been many research studies on the second-order convergence schemes for parabolictype partial differential equations [1][2][3][4][5][6][7][8][9][10][11][12]. To demonstrate the second-order convergence, some authors [1][2][3] showed the convergence by refining the spatial and temporal steps at the same time; some authors in [4][5][6][7][8][9][10] showed the convergence by refining the spatial and temporal steps separately.…”

Section: Introductionmentioning

confidence: 99%

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Jeong

Lee

et al. 2019

Mathematical Problems in Engineering

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In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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Section: Introductionmentioning

confidence: 99%

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Jeong

Lee

et al. 2019

Mathematical Problems in Engineering

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“…It must be pointed out that there are various numerical methods in the numerical analysis for the proposed equations. For the fractional diffusion equation, Zhang et al 32 used the time‐discrete scheme based on the finite difference method, Hajipour et al 33 applied the Chebyshev wavelet approach for solving multiterm VO time‐fractional diffusion‐wave equation, Cao et al 34 provided the compact finite difference operator for solving VO‐fractional reaction–diffusion equation. For fractional subdiffusion equations, Yu et al 35 applied the compact finite difference scheme for VO‐fractional subdiffusion equations, Yaseen et al 24 proposed the generalized Laguerre spectral Petrov–Galerkin method, Ghaffari and Ghoreishi 36 proposed a cubic trigonometric B‐spline collocation approach, and so forth (see References 22,37‐39).…”

Section: Introductionmentioning

confidence: 99%

“…Problem 2: VO‐fractional subdiffusion equation: 34

\frac{\partial U (x, t)}{\partial t} = 𝒟_{0, t}^{1 - ν (x, t)} [κ_{1} \frac{\partial^{2} U (x, t)}{\partial x^{2}} - κ_{2} U (x, t)] + 𝒢 (x, t, U), (x, t) \in (0, 1) \times (0, 1],

with initial and boundary conditions

\begin{align} frakturU false(x, 0 false) & = φ false(x false), x \in false[0, 1 false], \\ frakturU false(0, t false) & = ω_{0} false(t false), frakturU false(1, t false) = ω_{1} false(t false), t \in false[0, 1 false], \end{align}

where

κ_{1}, κ_{2} > 0

are the reaction terms and

𝒟_{0, t}^{ν (x, t)}

is the VO Caputo fractional derivative of order

0 < ν_{\min} \leq ν (x, t) \leq ν_{\max} < 1...

…”

Section: Introductionmentioning

confidence: 99%

A novel direct method based on the Lucas multiwavelet functions for variable‐order fractional reaction‐diffusion and subdiffusion equations

Dehestani

Ordokhani

Razzaghi

2020

Numerical Linear Algebra App

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In this article, we study the numerical technique for variable‐order fractional reaction‐diffusion and subdiffusion equations that the fractional derivative is described in Caputo's sense. The discrete scheme is developed based on Lucas multiwavelet functions and also modified and pseudo‐operational matrices. Under suitable properties of these matrices, we present the computational algorithm with high accuracy for solving the proposed problems. Modified and pseudo‐operational matrices are employed to achieve the nonlinear algebraic equation corresponding to the proposed problems. In addition, the convergence of the approximate solution to the exact solution is proven by providing an upper bound of error estimate. Numerical experiments for both classes of problems are presented to confirm our theoretical analysis.

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“…In Ref. [11], Cao et al presented a Crank-Nicolson-type compact finite-difference scheme with the second-order temporal accuracy and the fourth-order spatial accuracy. Shen [34] considered the variable-order time-fractional diffusion equation, and discussed the stability, convergence, and solvability of the numerical scheme.…”

Section: Introductionmentioning

confidence: 99%

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Wei

Zhai

Zhang

2020

Commun. Appl. Math. Comput.

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The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L 2-convergence of the scheme are proved for all variable-order (t) ∈ (0, 1). The proposed method is of accuracyorder O(3 + h k+1) , where , h, and k are the temporal step size, the spatial step size, and the degree of piecewise P k polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

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A compact finite difference scheme for variable order subdiffusion equation

Cited by 44 publications

References 28 publications

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

A novel direct method based on the Lucas multiwavelet functions for variable‐order fractional reaction‐diffusion and subdiffusion equations

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

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