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学者姓名：李娴娟
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Abstract ：
采用分数阶物理信息网络(fPINN)求解时间分数阶Nernst-Plank方程，并对其解决时间分数阶N-P的正问题与反问题的准确性和有效性进行说明.在这基础上，分析离散化时间分数阶算子所导致的离散误差、采样误差、神经网络优化误差对最终求解的影响.同时，分析离散误差与取样误差的关系，并发现当固定离散误差后存在最好的训练点集大小使得求解误差最低.最后，展示神经网络求解反问题的准确性与效率.
Keyword ：
分数阶物理信息神经网络 分数阶物理信息神经网络 时间分数阶Nernst-Plank方程 时间分数阶Nernst-Plank方程 深度学习 深度学习 误差分析 误差分析
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| GB/T 7714 | 徐国泰 , 李娴娟 , 宋方应 . 基于深度学习的分数阶Nernst-Plank方程求解 [J]. | 福州大学学报(自然科学版) , 2024 , 52 (04) : 379-386 . |
| MLA | 徐国泰 等. "基于深度学习的分数阶Nernst-Plank方程求解" . | 福州大学学报(自然科学版) 52 . 04 (2024) : 379-386 . |
| APA | 徐国泰 , 李娴娟 , 宋方应 . 基于深度学习的分数阶Nernst-Plank方程求解 . | 福州大学学报(自然科学版) , 2024 , 52 (04) , 379-386 . |
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In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method. © The Author(s) 2021.
Keyword ：
Convergence of numerical methods Convergence of numerical methods Domain decomposition methods Domain decomposition methods Efficiency Efficiency Finite difference method Finite difference method Iterative methods Iterative methods Spectroscopy Spectroscopy
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| GB/T 7714 | Li, Xianjuan , Su, Yanhui . A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations [J]. | Journal of Algorithms and Computational Technology , 2021 , 15 . |
| MLA | Li, Xianjuan 等. "A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations" . | Journal of Algorithms and Computational Technology 15 (2021) . |
| APA | Li, Xianjuan , Su, Yanhui . A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations . | Journal of Algorithms and Computational Technology , 2021 , 15 . |
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| GB/T 7714 | Li, Xianjuan , Mao, Zhiping , Wang, Nan et al. A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation ? [J]. | COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING , 2020 , 366 . |
| MLA | Li, Xianjuan et al. "A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation ?" . | COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 366 (2020) . |
| APA | Li, Xianjuan , Mao, Zhiping , Wang, Nan , Song, Fangying , Wang, Hong , Karniadakis, George Em . A fast solver for spectral elements applied to fractional differential equations using hierarchical matrix approximation ? . | COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING , 2020 , 366 . |
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In this paper, we consider the numerical solution of the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. A schema combining a finite difference approach in the time direction and a spectral method in the space direction is proposed and analyzed. The main contribution of this work is threefold: 1) We construct a finite difference/Legendre spectral schema for discretization of the fractional Cable equation. 2) We give a detailed analysis of the proposed schema by providing some stability and error estimates. Based on this analysis, the convergence of the method is rigourously established. We prove that the overall schema is unconditionally stable, and the numerical solution converges to the exact one with order O(4t2-max{a,ß}), where 4t is the time step size, a and ß are two different exponents between 0 and 1 involved in the fractional derivatives. 3) Finally, some numerical experiments are carried out to support the theoretical claims. © MODSIM 2009.All rights reserved.
Keyword ：
Cables Cables Circuit simulation Circuit simulation Software engineering Software engineering
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| GB/T 7714 | Lin, Yumin , Li, Xianjuan , Xu, Chuanju . Finite difference/spectral approximations for the fractional cable equation [C] . 2020 : 455-461 . |
| MLA | Lin, Yumin et al. "Finite difference/spectral approximations for the fractional cable equation" . (2020) : 455-461 . |
| APA | Lin, Yumin , Li, Xianjuan , Xu, Chuanju . Finite difference/spectral approximations for the fractional cable equation . (2020) : 455-461 . |
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针对含有对数自由能的空间分数阶Allen-Cahn方程,提出在空间上使用二阶中心差分,时间上采用二阶Crank-Nicolson差分格式的数值方法.在此基础上,阐明其数值解在合理的时间步长的限制下是唯一可解的,且满足极大值原理与离散能量稳定性.基于极大值原理,进一步探讨相应的误差分析.
Keyword ：
Allen-Cahn方程 Allen-Cahn方程 分数阶导数 分数阶导数 极大值原理 极大值原理 离散能量稳定性 离散能量稳定性 误差分析 误差分析
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| GB/T 7714 | 王齐鹏 , 李娴娟 . 具有对数自由能的空间分数阶Allen-Cahn方程的数值分析 [J]. | 福州大学学报（自然科学版） , 2019 , 47 (2) : 167-172 . |
| MLA | 王齐鹏 et al. "具有对数自由能的空间分数阶Allen-Cahn方程的数值分析" . | 福州大学学报（自然科学版） 47 . 2 (2019) : 167-172 . |
| APA | 王齐鹏 , 李娴娟 . 具有对数自由能的空间分数阶Allen-Cahn方程的数值分析 . | 福州大学学报（自然科学版） , 2019 , 47 (2) , 167-172 . |
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We investigate the stability of Rayleigh-Taylor (RT) problem of the stratified incompressible viscoelastic fluids under the rotation and the gravity in a horizontal periodic domain, in which the rotation axis is parallel to the direction of gravity, the two fluids are immiscible, and the heavier fluid lies on the lighter one. We establish a stability condition for the RT problem. Moreover, we prove that, under the stability condition, the RT problem enjoys a unique strong solution, which exponentially decays with respect to time. In addition, we note that the stability condition is independent of rotation angular velocity, and the rotation has no destabilizing effect.
Keyword ：
Horizontally periodic domain Horizontally periodic domain Rayleigh-Taylor instability Rayleigh-Taylor instability Rotation Rotation Viscoelastic fluid Viscoelastic fluid
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| GB/T 7714 | Jiang, Yi , Li, Xianjuan , Zhao, Youyi . On the stability of Rayleigh-Taylor problem for stratified rotating viscoelastic fluids [J]. | BOUNDARY VALUE PROBLEMS , 2018 . |
| MLA | Jiang, Yi et al. "On the stability of Rayleigh-Taylor problem for stratified rotating viscoelastic fluids" . | BOUNDARY VALUE PROBLEMS (2018) . |
| APA | Jiang, Yi , Li, Xianjuan , Zhao, Youyi . On the stability of Rayleigh-Taylor problem for stratified rotating viscoelastic fluids . | BOUNDARY VALUE PROBLEMS , 2018 . |
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利用一阶有限差分-谱方法求解Allen-Cahn方程ut-Δu+1ε2 f(u)=0并进行严格的误差分析,其中交面宽度ε是一个很小的参数.误差分析结果表明:若初始解u0的正则性受ε-σ控制,当时间步长δt充分小以及多项式阶数N充分大时,全离散格式的误差界也受ε-σ控制.该误差分析有效改进了误差界受eε1/2控制的结果.
Keyword ：
Allen-Cahn方程 Allen-Cahn方程 有限差分 有限差分 误差分析 误差分析 谱方法 谱方法
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| GB/T 7714 | 李娴娟 . 有限差分-谱方法求解Allen-Cahn方程的误差分析 [J]. | 福州大学学报（自然科学版） , 2017 , 45 (3) : 307-311 . |
| MLA | 李娴娟 . "有限差分-谱方法求解Allen-Cahn方程的误差分析" . | 福州大学学报（自然科学版） 45 . 3 (2017) : 307-311 . |
| APA | 李娴娟 . 有限差分-谱方法求解Allen-Cahn方程的误差分析 . | 福州大学学报（自然科学版） , 2017 , 45 (3) , 307-311 . |
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The Ladyzenskaja-Babuska-Brezzi (LBB) condition is a necessary condition for the well-posedness of discrete saddle point problems stemming from discretizing the Stokes equations. In this paper, we prove the LBB condition and provide the (optimal) lower bound for this condition for the triangular spectral method proposed by L. Chen, J. Shen, and C. Xu in [3]. Then this lower bound is used to derive an error estimate for the pressure. Some numerical examples are provided to confirm the theoretical estimates.
Keyword ：
Inf-Sup Condition Inf-Sup Condition Stokes Equations Stokes Equations Triangular Spectral Method Triangular Spectral Method
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| GB/T 7714 | Su, Yanhui , Chen, Lizhen , Li, Xianjuan et al. On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations [J]. | COMPUTATIONAL METHODS IN APPLIED MATHEMATICS , 2016 , 16 (3) : 507-522 . |
| MLA | Su, Yanhui et al. "On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations" . | COMPUTATIONAL METHODS IN APPLIED MATHEMATICS 16 . 3 (2016) : 507-522 . |
| APA | Su, Yanhui , Chen, Lizhen , Li, Xianjuan , Xu, Chuanju . On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations . | COMPUTATIONAL METHODS IN APPLIED MATHEMATICS , 2016 , 16 (3) , 507-522 . |
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In this paper we present and analyze Chebyshev and Legendre pseudo-spectral methods for the second kind Volterra integral equations with weakly singular kernel (x - s)(-mu), 0 < mu < 1. The proposed methods are based on the Gauss-type quadrature formula for approximating the integral operators involved in the equations. The present work is an extension of the earlier proposed spectral Jacobi-Galerkin method for the second kind Volterra integral equations with regular kernels (Xie et al. in J Sci Comput 53(2):414-434, [21]). Adetailed convergence analysis is carried out, and several error estimates in L-infinity and L-omega(2) norms are obtained. Numerical examples are considered to verify the theoretical predictions.
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| GB/T 7714 | Li, Xianjuan , Tang, Tao , Xu, Chuanju . Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods [J]. | JOURNAL OF SCIENTIFIC COMPUTING , 2016 , 67 (1) : 43-64 . |
| MLA | Li, Xianjuan et al. "Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods" . | JOURNAL OF SCIENTIFIC COMPUTING 67 . 1 (2016) : 43-64 . |
| APA | Li, Xianjuan , Tang, Tao , Xu, Chuanju . Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods . | JOURNAL OF SCIENTIFIC COMPUTING , 2016 , 67 (1) , 43-64 . |
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This work is concerned with spectral Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form (t - s)(-alpha). When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen & Tang, J. Comput. Appl. Math., 233 (2009), pp. 938-950]; due to technical reasons the results are restricted to 0 < mu < 1/2. In this work, we will improve the results to the general case 0 < mu < 1 and demonstrate that the numerical errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.
Keyword ：
Convergence analysis Convergence analysis Spectral-collocation methods Spectral-collocation methods Volterra integral equations Volterra integral equations
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| GB/T 7714 | Chen, Yanping , Li, Xianjuan , Tang, Tao . A NOTE ON JACOBI SPECTRAL-COLLOCATION METHODS FOR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS WITH SMOOTH SOLUTIONS [J]. | JOURNAL OF COMPUTATIONAL MATHEMATICS , 2013 , 31 (1) : 47-56 . |
| MLA | Chen, Yanping et al. "A NOTE ON JACOBI SPECTRAL-COLLOCATION METHODS FOR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS WITH SMOOTH SOLUTIONS" . | JOURNAL OF COMPUTATIONAL MATHEMATICS 31 . 1 (2013) : 47-56 . |
| APA | Chen, Yanping , Li, Xianjuan , Tang, Tao . A NOTE ON JACOBI SPECTRAL-COLLOCATION METHODS FOR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS WITH SMOOTH SOLUTIONS . | JOURNAL OF COMPUTATIONAL MATHEMATICS , 2013 , 31 (1) , 47-56 . |
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