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

Lyu, Pin; Vong, Seakweng

doi:10.1002/num.22441

Cited by 13 publications

(4 citation statements)

References 41 publications

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“…For more related research, one can refer to [33][34][35]. In addition, in the numerical simulation of the fractional derivatives, a variety of interpolation approximation methods has been reported, such as Grünwald-Letnikov definition [36], the L1 scheme [37], the L2 − 1 σ scheme [38] and the fast algorithm [39].…”

Section: Introductionmentioning

confidence: 99%

Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition

Yang,

Liu,

Chen

et al. 2024

Fractal Fract

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The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed based on the artificial boundary method (ABM), addressing issues related to the semi-infinite boundary. We adopt the finite difference method (FDM) for deriving the numerical solution by employing the L1 scheme to approximate the fractional derivative. To confirm the precision of this method, a source term is added to establish an exact solution for verification purposes. A comparative evaluation of the ABC versus the direct truncated boundary condition (DTBC) is conducted, with their effectiveness and soundness being visually scrutinized and assessed. This study investigates the impact of the motion of plates at different fluid flow velocities, focusing on the effects of dynamic elements influencing flow mechanisms and velocity. This research’s primary conclusion is that a higher fractional parameter correlates with the fluid flow. As relaxation parameters decrease, the delay effect intensifies and the fluid velocity decreases.

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Section: Introductionmentioning

confidence: 99%

Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition

Yang,

Liu,

Chen

et al. 2024

Fractal Fract

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“…The stability and convergence of the scheme are also analysed under an assumption condition. Lyu and Vong [36] constructed a temporal nonuniform L2 formula (the same as the modified L1 formula above) for the Caputo derivative of order β (1 < β < 2). Based on this formula, a linearized difference scheme was presented for the time-fractional Benjamin-Bona-Mahony-type equation by the mathematical induction.…”

mentioning

confidence: 99%

“…In this paper, we construct the temporal nonuniform difference scheme [35] by combining the order of reduction with the modified L1 formula for the time fractional diffusion-wave. The aim of this paper is to present a method different from [35][36][37] for the convergence of the difference scheme. It relies on a useful discrete tool: the discrete complementary convolution (DCC) kernels [32] generated by the discrete convolution kernels of the modified L1 formula.…”

mentioning

confidence: 99%

Error estimate of the nonuniform $L1$ type formula for the time fractional diffusion-wave equation

Sun¹,

Chen²,

Zhang³

2023

Preprint

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In this paper, a temporal nonuniform L1 type difference scheme is built up for the time fractional diffusion-wave equation with the help of the order reduction technique. The unconditional convergence of the nonuniform difference scheme is proved rigorously in L 2 norm. Our main tool is the discrete complementary convolution kernels with respect to the coefficient kernels of the L1 type formula. The positive definiteness of the complementary convolution kernels is shown to be vital to the stability and convergence. To the best of our knowledge, this property is proved at the first time on the nonuniform time meshes. Two numerical experiments are presented to verify the accuracy and the efficiency of the proposed numerical methods.

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“…Moreover, presented scheme can be extended to the BBMB equation with homogeneous boundary conditions without any difficulty. In the future, extended our technique and idea to other nonlocal and nonlinear evolution equations [9,21,23,24,33,43] will be our on-going project.…”

mentioning

confidence: 99%

Uniform convergence and stability of linearized fourth-order conservative compact scheme for Benjamin-Bona-Mahony-Burgers' equation

Zhang,

Liu

2020

Preprint

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In the paper, a newly developed three-point fourth-order compact operator is utilized to construct an efficient compact finite difference scheme for the Benjamin-Bona-Mahony-Burgers' (BBMB) equation. Detailed derivation is carried out based on the reduction order method together with a three-level linearized technique. The conservative invariant, boundedness and unique solvability are studied at length. The uniform convergence is proved by the technical energy argument with the optimal convergence order O(τ 2 + h 4 ) in the sense of the maximum norm. The almost unconditional stability can be achieved based on the uniform boundedness of the numerical solution. The present scheme is very efficient in practical computation since only a system of linear equations with a symmetric circulant matrix needing to be solved at each time step. The extensive numerical examples verify our theoretical results and demonstrate the superiority of the scheme when compared with state-of-the-art those in the references.

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

Cited by 13 publications

References 41 publications

Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition

Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition

Error estimate of the nonuniform $L1$ type formula for the time fractional diffusion-wave equation

Uniform convergence and stability of linearized fourth-order conservative compact scheme for Benjamin-Bona-Mahony-Burgers' equation

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