Unconditional Convergence in Maximum-Norm of a Second-Order Linearized Scheme for a Time-Fractional Burgers-Type Equation

Vong, Seakweng; Lyu, Pin

doi:10.1007/s10915-018-0659-0

Cited by 22 publications

(12 citation statements)

References 32 publications

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“…Combining (3.23) and (3.25), we get Referring to the proof of Theorem 3.5 in [33], we discuss the following two cases:…”

Section: The Unconditional Convergencementioning

confidence: 96%

“…In a way similar to the proof of convergence, we can show that the numerical scheme (2.27)-(2.30) is unconditionally stable with respect to discrete H 1 -norm. Readers can refer to [33] for more details.…”

Section: The Unconditional Convergencementioning

confidence: 99%

“…Moreover, the problem under consideration is a nonlinear time-fractional equation, and classical nonlocal numerical schemes blending with iterative methods would take more expensive computation and have more complexity in the analysis. So it is really competitive and also necessary to construct linearized numerical methods for nonlinear time-fractional problems [12,16,17,18,33]. It is worth to mention that the solution of many time-fractional differential equations typically displays a weak singularity near the initial time [8,9,10,19,20,21,26,29,30], which leads to the loss of time accuracy for many related high-order numerical methods.…”

Section: Introductionmentioning

confidence: 99%

“…33) where Lemma 3.7 (b) and (c) have been used.Thus, (3.24) and (3.27)-(3.33) yielde q+1 + 2C α σe q+1 + (1 − σ)e q n + C(τ 2 + h 2 + ǫ) 2 . (3.34)Similar discussion with Case (II) in Theorem 3.5 of[33], we can get S q+1 ≤ S q or S q+1 = e q+1 + 2C α σe q+1 + (1 − σ)e q 1 inequality (3.12) is clarified according to (3.34)-(3.35). Consequently, we can apply Lemma 3.6 (Gronwall's inequality) and Lemma 3.7 (c) on (3.12) to conclude S n ≤ C(τ 2 + h 2 + ǫ) 2 , 0 ≤ n ≤ N.…”

mentioning

confidence: 99%

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A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation

Lyu

Liang

Wang

2020

Applied Numerical Mathematics

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In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L2-1 σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H 1 -norm. Numerical examples are provided to justify the efficiency.

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“…Combining (3.23) and (3.25), we get Referring to the proof of Theorem 3.5 in [33], we discuss the following two cases:…”

Section: The Unconditional Convergencementioning

confidence: 96%

Section: The Unconditional Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation

Lyu

Liang

Wang

2020

Applied Numerical Mathematics

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“…For nonlinear time‐fractional equations, the computation will be more expensive and the analysis may be more complicate if the nonlocal numerical scheme is blending with iterative methods. Therefore, it would be more competitive and also necessary to construct linearized numerical methods for nonlinear time‐fractional problems .…”

Section: Introductionmentioning

confidence: 99%

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony‐type equation with nonsmooth solutions

Lyu

Vong

2019

Numerical Methods Partial

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To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time‐fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time‐fractional Benjamin–Bona–Mahony‐type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H1‐norm. Numerical examples are provided to justify the accuracy.

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A computational procedure and analysis for multi‐term time‐fractional Burgers‐type equation

Kanth

Garg

2022

Math Methods in App Sciences

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This paper presents a new numerical algorithm dealing with multi-term time-fractional Burgers-type equation involving the Caputo derivative. The proposed method consists of temporal discretization of L2 formula and spatial discretization using the exponential B-splines. The semi implicit approach is applied to discretize the nonlinear term u𝜕 x u. We adopt the Von-Neumann method to study stability. We also establish the convergence analysis. The proposed method is employed to solve a few numerical examples in order to test its efficiency and accuracy. Comparisons with the recent works confirm the efficiency and robustness of the proposed method.

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Unconditional Convergence in Maximum-Norm of a Second-Order Linearized Scheme for a Time-Fractional Burgers-Type Equation

Cited by 22 publications

References 32 publications

A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation

A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony‐type equation with nonsmooth solutions

A computational procedure and analysis for multi‐term time‐fractional Burgers‐type equation

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