Pub Date : 2024-10-17DOI: 10.1016/j.aml.2024.109339
Liang Li , YanMeng Wang , Jun Zhu
In the conventional hybrid WENO scheme, the computation of extremum points of high-order polynomials is required, which poses challenges when extending this methodology to the hybrid trigonometric WENO (TWENO) scheme due to the difficulties in determining the extremum points of high-order trigonometric polynomials. In this paper, we propose a novel hybrid strategy that circumvents the necessity of finding extremum points of high-degree polynomials, requiring only the extremum points of three lower-order polynomials. Based on this hybrid strategy, two hybrid TWENO schemes are designed, both of which significantly improve the numerical simulation performance of the TWENO scheme while saving approximately 80% of the computation time.
{"title":"A new hybrid trigonometric WENO scheme for hyperbolic conservation laws and highly oscillatory problems","authors":"Liang Li , YanMeng Wang , Jun Zhu","doi":"10.1016/j.aml.2024.109339","DOIUrl":"10.1016/j.aml.2024.109339","url":null,"abstract":"<div><div>In the conventional hybrid WENO scheme, the computation of extremum points of high-order polynomials is required, which poses challenges when extending this methodology to the hybrid trigonometric WENO (TWENO) scheme due to the difficulties in determining the extremum points of high-order trigonometric polynomials. In this paper, we propose a novel hybrid strategy that circumvents the necessity of finding extremum points of high-degree polynomials, requiring only the extremum points of three lower-order polynomials. Based on this hybrid strategy, two hybrid TWENO schemes are designed, both of which significantly improve the numerical simulation performance of the TWENO scheme while saving approximately 80% of the computation time.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109339"},"PeriodicalIF":2.9,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.aml.2024.109337
Jianxiong Cao , Wenhao Xu
The diffusive-viscous wave (DVW) equation is a widely used model to describe frequency dependent attenuation of seismic wave in fluid-saturated porous medium. In this paper taking power law frequency dependent attenuation into account, we first introduce a modified DVW equation (time fractional DVW equation) in which the first order temporal derivative of viscous term is replaced with a fractional order temporal derivative. In consideration of that most of the existing numerical simulations for seismic wave equations are based on time domain methods and truncation with some specific boundary conditions, we incorporate the absorbing boundary condition as complex-frequency-shifted (CFS) perfectly matched layer (PML) into the time fractional DVW equation, and then develop an adaptive-coefficient (AC) finite difference frequency domain (FDFD) method for numerical simulation. The corresponding analytical solution for homogeneous time fractional DVW equation is provided for model validation, and the effectiveness of the developed AC FDFD method is verified by some numerical examples including the homogeneous model and the layered model. Numerical results show that AC FDFD method is more accurate than the traditional 2nd-order FDFD method for numerical modeling of time fractional DVW equation with CFS PML absorbing boundary condition, while requiring similar computational costs.
{"title":"Adaptive-coefficient finite difference frequency domain method for time fractional diffusive-viscous wave equation arising in geophysics","authors":"Jianxiong Cao , Wenhao Xu","doi":"10.1016/j.aml.2024.109337","DOIUrl":"10.1016/j.aml.2024.109337","url":null,"abstract":"<div><div>The diffusive-viscous wave (DVW) equation is a widely used model to describe frequency dependent attenuation of seismic wave in fluid-saturated porous medium. In this paper taking power law frequency dependent attenuation into account, we first introduce a modified DVW equation (time fractional DVW equation) in which the first order temporal derivative of viscous term is replaced with a fractional order temporal derivative. In consideration of that most of the existing numerical simulations for seismic wave equations are based on time domain methods and truncation with some specific boundary conditions, we incorporate the absorbing boundary condition as complex-frequency-shifted (CFS) perfectly matched layer (PML) into the time fractional DVW equation, and then develop an adaptive-coefficient (AC) finite difference frequency domain (FDFD) method for numerical simulation. The corresponding analytical solution for homogeneous time fractional DVW equation is provided for model validation, and the effectiveness of the developed AC FDFD method is verified by some numerical examples including the homogeneous model and the layered model. Numerical results show that AC FDFD method is more accurate than the traditional 2nd-order FDFD method for numerical modeling of time fractional DVW equation with CFS PML absorbing boundary condition, while requiring similar computational costs.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109337"},"PeriodicalIF":2.9,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.aml.2024.109338
Chen Yang, Chun-Lei Tang
We are devoted to studying the ground states of trapped attractive Bose–Einstein condensates in a bounded domain , which can be described by minimizers of -critical constraint Gross–Pitaevskii energy functional. It has been shown that there is a threshold such that minimizers exist if and only if the interaction strength satisfies . In present paper, we prove that when the trapping potential attains its flattest global minimum only at the boundary of , the mass of minimizers must concentrate near the boundary of as . This result extends the work of Luo and Zhu (2019).
{"title":"Mass concentration near the boundary for attractive Bose–Einstein condensates in bounded domains","authors":"Chen Yang, Chun-Lei Tang","doi":"10.1016/j.aml.2024.109338","DOIUrl":"10.1016/j.aml.2024.109338","url":null,"abstract":"<div><div>We are devoted to studying the ground states of trapped attractive Bose–Einstein condensates in a bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, which can be described by minimizers of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-critical constraint Gross–Pitaevskii energy functional. It has been shown that there is a threshold <span><math><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>></mo><mn>0</mn></mrow></math></span> such that minimizers exist if and only if the interaction strength <span><math><mi>a</mi></math></span> satisfies <span><math><mrow><mi>a</mi><mo><</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>. In present paper, we prove that when the trapping potential <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> attains its flattest global minimum only at the boundary of <span><math><mi>Ω</mi></math></span>, the mass of minimizers must concentrate near the boundary of <span><math><mi>Ω</mi></math></span> as <span><math><mrow><mi>a</mi><mo>↗</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>. This result extends the work of Luo and Zhu (2019).</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109338"},"PeriodicalIF":2.9,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.aml.2024.109335
Mingxin Wang
It is of interest to understand effects of variable coefficients and degenerate diffusion on the longtime behaviors of solutions of reaction diffusion equations. Recently, Yang and Yao (2024) studied a classical prey-predator model and proved that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties. In this note we shall simplify the proof of Yang and Yao (2024) and delete the condition (4).
了解可变系数和退化扩散对反应扩散方程解的长期行为的影响是很有意义的。最近，Yang 和 Yao（2024）研究了一个经典的猎物-捕食者模型，证明了满足适当条件的猎物扩散系数和可变系数的退化不会影响动力学性质。在本注释中，我们将简化 Yang 和 Yao (2024) 的证明，删除条件 (4)。
{"title":"Note on the diffusive prey-predator model with variable coefficients and degenerate diffusion","authors":"Mingxin Wang","doi":"10.1016/j.aml.2024.109335","DOIUrl":"10.1016/j.aml.2024.109335","url":null,"abstract":"<div><div>It is of interest to understand effects of variable coefficients and degenerate diffusion on the longtime behaviors of solutions of reaction diffusion equations. Recently, Yang and Yao (2024) studied a classical prey-predator model and proved that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties. In this note we shall simplify the proof of Yang and Yao (2024) and delete the condition (4).</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109335"},"PeriodicalIF":2.9,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.aml.2024.109333
Jianguo Huang, Wenxuan Wang
This paper focuses on proposing and analyzing a new locking-free lowest order virtual element method for the linear elasticity problem in three dimensions. A virtual element function on a polyhedron is harmonic, while it is continuous piecewise linear corresponding to an auxiliary triangulation on the boundary . Such construction requires no further three-dimensional partition of . Under some reasonable mesh assumptions, we derive the inverse inequality, the norm equivalence and the error estimate of the interpolation operator for the underlying virtual element. Using these results combined with a rigorous analysis, we establish a robust error estimate in norm for the proposed method. Finally, we perform numerical results to demonstrate theoretical findings.
本文主要针对三维线性弹性问题提出并分析了一种新的无锁定最低阶虚元方法。多面体 K 上的虚元函数是谐函数，而与边界 ∂K 上的辅助三角剖分相对应的虚元函数是连续的片断线性函数。在一些合理的网格假设下，我们推导出了底层虚元的逆不等式、规范等价性和插值算子的误差估计。利用这些结果并结合严格的分析，我们为所提出的方法建立了 H1 准则的稳健误差估计。最后，我们用数值结果来证明理论结论。
{"title":"A locking-free virtual element method for 3D linear elasticity problems","authors":"Jianguo Huang, Wenxuan Wang","doi":"10.1016/j.aml.2024.109333","DOIUrl":"10.1016/j.aml.2024.109333","url":null,"abstract":"<div><div>This paper focuses on proposing and analyzing a new locking-free lowest order virtual element method for the linear elasticity problem in three dimensions. A virtual element function on a polyhedron <span><math><mi>K</mi></math></span> is harmonic, while it is continuous piecewise linear corresponding to an auxiliary triangulation on the boundary <span><math><mrow><mi>∂</mi><mi>K</mi></mrow></math></span>. Such construction requires no further three-dimensional partition of <span><math><mi>K</mi></math></span>. Under some reasonable mesh assumptions, we derive the inverse inequality, the norm equivalence and the error estimate of the interpolation operator for the underlying virtual element. Using these results combined with a rigorous analysis, we establish a robust error estimate in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm for the proposed method. Finally, we perform numerical results to demonstrate theoretical findings.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109333"},"PeriodicalIF":2.9,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.aml.2024.109336
Seakweng Vong , Han Xue , Yuanyuan Zhang , Zhongsheng Yao
This paper studies the stability of time delay systems. The Lyapunov-Krasovskii functional (LKF) method is used for our study, in which a novel negative-determination criterion for cubic polynomials is proposed. An improved stability criterion of time delay system is obtained by the new method. The effectiveness of the proposed method is verified by some numerical examples and reduced conservativeness can be obtained.
{"title":"New negative-determination conditions for cubic polynomials with applications to time-varying delay systems","authors":"Seakweng Vong , Han Xue , Yuanyuan Zhang , Zhongsheng Yao","doi":"10.1016/j.aml.2024.109336","DOIUrl":"10.1016/j.aml.2024.109336","url":null,"abstract":"<div><div>This paper studies the stability of time delay systems. The Lyapunov-Krasovskii functional (LKF) method is used for our study, in which a novel negative-determination criterion for cubic polynomials is proposed. An improved stability criterion of time delay system is obtained by the new method. The effectiveness of the proposed method is verified by some numerical examples and reduced conservativeness can be obtained.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109336"},"PeriodicalIF":2.9,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-08DOI: 10.1016/j.aml.2024.109334
Xinyan Zhou, Xiaoli Wang, Guohong Zhang
In the paper, we investigate a diffusive predator–prey model with nonlocal intraspecific prey competition and spatial memory under Neumann boundary conditions. Through stability and bifurcation analysis, we find that the memory-based diffusion coefficient and the spatiotemporal diffusive delay have important effects on the dynamics of the model. By using the spatiotemporal diffusive delay as a bifurcation parameter, the critical values are determined for the stability of the positive constant steady state and the associated Hopf bifurcation. We find that the system may admit no stability switch, one stability switch and multiple stability switches.
{"title":"Dynamics of a non-local intraspecific competition predator–prey model with memory effect","authors":"Xinyan Zhou, Xiaoli Wang, Guohong Zhang","doi":"10.1016/j.aml.2024.109334","DOIUrl":"10.1016/j.aml.2024.109334","url":null,"abstract":"<div><div>In the paper, we investigate a diffusive predator–prey model with nonlocal intraspecific prey competition and spatial memory under Neumann boundary conditions. Through stability and bifurcation analysis, we find that the memory-based diffusion coefficient and the spatiotemporal diffusive delay have important effects on the dynamics of the model. By using the spatiotemporal diffusive delay as a bifurcation parameter, the critical values are determined for the stability of the positive constant steady state and the associated Hopf bifurcation. We find that the system may admit no stability switch, one stability switch and multiple stability switches.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109334"},"PeriodicalIF":2.9,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-05DOI: 10.1016/j.aml.2024.109332
Zekun Wang , Hongjia Chen , Zhongming Teng , Xiang Wang
<div><div>The calculation of the dominant eigenvalues of a symmetric matrix <span><math><mi>A</mi></math></span> together with its eigenvectors, followed by the calculation of the deflation of <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>ρ</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow></msubsup></mrow></math></span> corresponds to one step of the Wielandt deflation technique, where <span><math><mi>ρ</mi></math></span> is a shift and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> are eigenvectors of <span><math><mi>A</mi></math></span>. In this paper, we investigate how the eigenspace of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> changes when <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is perturbed to <span><math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>ρ</mi><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow></msubsup></mrow></math></span>, where <span><math><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></math></span> are approximate eigenvectors of <span><math><mi>A</mi></math></span>. We establish the bounds for the angle of eigenspaces of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub></math></span> based on the Davis-Kahan theorem. Moreover, in the practical implementation for singular value decomposition, once one or several singular triplets converge to a preset accuracy, they should be deflated by <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>B</mi><mo>−</mo><mi>γ</mi><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>H</mi></mrow></msubsup></mrow></math></span> with <span><math><mi>γ</mi></math></span> being a shift, <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> are singular vectors of <span><math><mi>B</mi></math></span>, so that they will not be re-computed. We investigate how the singular subspaces of <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>B</mi><mo>−</mo><mi>γ</mi><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi>V</m
计算对称矩阵 A 的主特征值及其特征向量，然后计算 A1=A-ρUkUkT 的放缩，相当于维兰德放缩技术的一个步骤，其中 ρ 是位移，Uk 是 A 的特征向量。本文研究了当 A1 被扰动为 A˜1=A-ρU˜kU˜kT 时，A1 的特征空间如何变化，其中 U˜k 是 A 的近似特征向量。此外，在奇异值分解的实际应用中，一旦一个或几个奇异三元组收敛到预设精度，就应按 B1=B-γWkVkH 放空，其中γ 是移位，Wk 和 Vk 是 B 的奇异向量，这样它们就不会被重新计算。我们研究当 B1 被扰动为 B˜1=B-γW˜kV˜kH 时，B1=B-γWkVkH 的奇异子空间如何变化，W˜k 和 V˜k 是 B 的近似奇异向量。
{"title":"On perturbations for spectrum and singular value decompositions followed by deflation techniques","authors":"Zekun Wang , Hongjia Chen , Zhongming Teng , Xiang Wang","doi":"10.1016/j.aml.2024.109332","DOIUrl":"10.1016/j.aml.2024.109332","url":null,"abstract":"<div><div>The calculation of the dominant eigenvalues of a symmetric matrix <span><math><mi>A</mi></math></span> together with its eigenvectors, followed by the calculation of the deflation of <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>ρ</mi><msub><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow></msubsup></mrow></math></span> corresponds to one step of the Wielandt deflation technique, where <span><math><mi>ρ</mi></math></span> is a shift and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> are eigenvectors of <span><math><mi>A</mi></math></span>. In this paper, we investigate how the eigenspace of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> changes when <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is perturbed to <span><math><mrow><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>ρ</mi><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow></msubsup></mrow></math></span>, where <span><math><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></math></span> are approximate eigenvectors of <span><math><mi>A</mi></math></span>. We establish the bounds for the angle of eigenspaces of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub></math></span> based on the Davis-Kahan theorem. Moreover, in the practical implementation for singular value decomposition, once one or several singular triplets converge to a preset accuracy, they should be deflated by <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>B</mi><mo>−</mo><mi>γ</mi><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>H</mi></mrow></msubsup></mrow></math></span> with <span><math><mi>γ</mi></math></span> being a shift, <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> are singular vectors of <span><math><mi>B</mi></math></span>, so that they will not be re-computed. We investigate how the singular subspaces of <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>B</mi><mo>−</mo><mi>γ</mi><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow></msub><msubsup><mrow><mi>V</m","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109332"},"PeriodicalIF":2.9,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.aml.2024.109331
Xiaokun Li , Shengdong Zhao , Wenzhen Qu
The localized Fourier collocation method (LFCM) is a newly developed meshless approach for solving certain types of partial differential equations (PDEs). The main idea of this method is to break down the problem domain into a series of overlapping small regions, where the solution within each sub-domain is approximated using Fourier series expansions. The rapid convergence and high computational accuracy make the method particularly effective for handing complex geometries and boundary conditions. This paper presents the first application of LFCM to transient heat conduction problems. The Houbolt method is employed for the time discretization. Several benchmark examples with complex geometries and diverse initial/boundary conditions are well-studied to illustrate the flexibility and accuracy of the new method.
{"title":"Localized Fourier collocation method for 2D transient heat conduction problems","authors":"Xiaokun Li , Shengdong Zhao , Wenzhen Qu","doi":"10.1016/j.aml.2024.109331","DOIUrl":"10.1016/j.aml.2024.109331","url":null,"abstract":"<div><div>The localized Fourier collocation method (LFCM) is a newly developed meshless approach for solving certain types of partial differential equations (PDEs). The main idea of this method is to break down the problem domain into a series of overlapping small regions, where the solution within each sub-domain is approximated using Fourier series expansions. The rapid convergence and high computational accuracy make the method particularly effective for handing complex geometries and boundary conditions. This paper presents the first application of LFCM to transient heat conduction problems. The Houbolt method is employed for the time discretization. Several benchmark examples with complex geometries and diverse initial/boundary conditions are well-studied to illustrate the flexibility and accuracy of the new method.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109331"},"PeriodicalIF":2.9,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.aml.2024.109330
Jian Liu, Qiguang An
This article investigates the existence of solutions to quasilinear degenerate elliptic equation with Hardy singular coefficient, in which the weighted function is unbounded (singular), then we cannot use the classical space , so we have to find another space to deal with the difficulties caused by singularities or degeneracies. New criteria for the existence of at least one and at least two generalized solutions are established via variational methods and critical point theorems provided that the nonlinearity satisfies appropriate hypotheses.
{"title":"Analysis of degenerate p-Laplacian elliptic equations involving Hardy terms: Existence and numbers of solutions","authors":"Jian Liu, Qiguang An","doi":"10.1016/j.aml.2024.109330","DOIUrl":"10.1016/j.aml.2024.109330","url":null,"abstract":"<div><div>This article investigates the existence of solutions to quasilinear degenerate elliptic equation with Hardy singular coefficient, in which the weighted function <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is unbounded (singular), then we cannot use the classical space <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, so we have to find another space <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><mi>ω</mi><mo>,</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> to deal with the difficulties caused by singularities or degeneracies. New criteria for the existence of at least one and at least two generalized solutions are established via variational methods and critical point theorems provided that the nonlinearity satisfies appropriate hypotheses.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109330"},"PeriodicalIF":2.9,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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