Pub Date : 2025-01-16DOI: 10.1016/j.aml.2025.109466
Chengdong Liu , Yimin Wei , Pengpeng Xie
Dual number matrices play a significant role in engineering applications such as kinematics and dynamics. The matrix exponential is ubiquitous in screw-based kinematics. In this paper, we develop an explicit formula for the dual matrix exponential. The result is closely related to the Fréchet derivative, which can be formed by the standard part and dual part of the original matrix. We only need to compute the exponential of a real matrix. Then, we give a formula of computing the dual quaternion matrix exponential. Our results are illustrated through a practical example from robotic kinematics.
{"title":"On the relation between the exponential of real matrices and that of dual matrices","authors":"Chengdong Liu , Yimin Wei , Pengpeng Xie","doi":"10.1016/j.aml.2025.109466","DOIUrl":"10.1016/j.aml.2025.109466","url":null,"abstract":"<div><div>Dual number matrices play a significant role in engineering applications such as kinematics and dynamics. The matrix exponential is ubiquitous in screw-based kinematics. In this paper, we develop an explicit formula for the dual matrix exponential. The result is closely related to the Fréchet derivative, which can be formed by the standard part and dual part of the original matrix. We only need to compute the exponential of a real matrix. Then, we give a formula of computing the dual quaternion matrix exponential. Our results are illustrated through a practical example from robotic kinematics.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109466"},"PeriodicalIF":2.9,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990379","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 : 2025-01-15DOI: 10.1016/j.aml.2025.109462
Haibin Chen , Guanglu Zhou , Hong Yan
We consider multilinear systems which arise in various applications, such as data mining and numerical differential equations. In this paper, we show that the multilinear system with a nonsingular -tensor can be formulated equivalently into a geometric programming (GP) problem which can be solved by the barrier-based interior point method with a worst-case polynomial-time complexity. To the best of our knowledge, there is not a complexity analysis for the existing algorithms of the multilinear systems. Numerical results are reported to show the efficiency of the proposed GP method.
我们考虑了在数据挖掘和数值微分方程等各种应用中出现的多线性系统。在本文中，我们展示了具有非正弦 M 张量的多线性系统可以等价地表述为一个几何程序设计（GP）问题，该问题可以通过基于障碍的内点法求解，并具有最坏情况下的多项式时间复杂度。据我们所知，现有的多线性系统算法还没有复杂度分析。报告的数值结果表明了所提出的 GP 方法的效率。
{"title":"Geometric programming for multilinear systems with nonsingular M-tensors","authors":"Haibin Chen , Guanglu Zhou , Hong Yan","doi":"10.1016/j.aml.2025.109462","DOIUrl":"10.1016/j.aml.2025.109462","url":null,"abstract":"<div><div>We consider multilinear systems which arise in various applications, such as data mining and numerical differential equations. In this paper, we show that the multilinear system with a nonsingular <span><math><mi>M</mi></math></span>-tensor can be formulated equivalently into a geometric programming (GP) problem which can be solved by the barrier-based interior point method with a worst-case polynomial-time complexity. To the best of our knowledge, there is not a complexity analysis for the existing algorithms of the multilinear systems. Numerical results are reported to show the efficiency of the proposed GP method.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109462"},"PeriodicalIF":2.9,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990387","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 : 2025-01-14DOI: 10.1016/j.aml.2025.109461
Lingjun Liu , Guiqin Qiu , Shu Wang , Lingda Xu
This paper investigates the large-time asymptotic behavior of contact waves in 1-D compressible Navier–Stokes equations. We derive the optimal decay rate for generic initial perturbations, meaning the perturbation’s integral does not need to be zero. It is well-known that generic perturbations in Navier–Stokes equations generate diffusion waves, implying that the optimal decay rate for contact waves in the -norm is . However, the presence of diffusion waves introduces error terms, leading to energy growth in the anti-derivatives of the perturbations. Furthermore, studying contact waves depends on certain structural conditions, which hold for the original system but not for its derivative systems. This makes it challenging to obtain accurate estimates for the energy of the derivatives.
In this paper, we refine the estimates for both anti-derivatives and the original perturbations. We then introduce an innovative transformation to ensure that the structural conditions continue to hold for the system of derivatives. With this approach, we achieve better estimates for the derivatives, leading to the optimal decay rates. This result improves upon the well-known findings of Huang et al. (2008), and the method has the potential for application in more general systems.
{"title":"Optimal decay rate to the contact discontinuity for Navier–Stokes equations under generic perturbations","authors":"Lingjun Liu , Guiqin Qiu , Shu Wang , Lingda Xu","doi":"10.1016/j.aml.2025.109461","DOIUrl":"10.1016/j.aml.2025.109461","url":null,"abstract":"<div><div>This paper investigates the large-time asymptotic behavior of contact waves in 1-D compressible Navier–Stokes equations. We derive the optimal decay rate for generic initial perturbations, meaning the perturbation’s integral does not need to be zero. It is well-known that generic perturbations in Navier–Stokes equations generate diffusion waves, implying that the optimal decay rate for contact waves in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm is <span><math><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>. However, the presence of diffusion waves introduces error terms, leading to energy growth in the anti-derivatives of the perturbations. Furthermore, studying contact waves depends on certain structural conditions, which hold for the original system but not for its derivative systems. This makes it challenging to obtain accurate estimates for the energy of the derivatives.</div><div>In this paper, we refine the estimates for both anti-derivatives and the original perturbations. We then introduce an innovative transformation to ensure that the structural conditions continue to hold for the system of derivatives. With this approach, we achieve better estimates for the derivatives, leading to the optimal decay rates. This result improves upon the well-known findings of Huang et al. (2008), and the method has the potential for application in more general systems.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109461"},"PeriodicalIF":2.9,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990414","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 : 2025-01-14DOI: 10.1016/j.aml.2025.109457
Jie Ding , Fei Xu , Zhi Ling
This paper demonstrates the fundamental properties of a conservative reaction–diffusion system. The solution of the system exists globally and is unique, as well as uniformly converges to its constant equilibrium as time tends to infinity. In addition, the steady-state system only has a constant solution under a mass conservation condition.
{"title":"Stability analysis of a conservative reaction–diffusion system with rate controls","authors":"Jie Ding , Fei Xu , Zhi Ling","doi":"10.1016/j.aml.2025.109457","DOIUrl":"10.1016/j.aml.2025.109457","url":null,"abstract":"<div><div>This paper demonstrates the fundamental properties of a conservative reaction–diffusion system. The solution of the system exists globally and is unique, as well as uniformly converges to its constant equilibrium as time tends to infinity. In addition, the steady-state system only has a constant solution under a mass conservation condition.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109457"},"PeriodicalIF":2.9,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990279","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 : 2025-01-13DOI: 10.1016/j.aml.2025.109460
Qinjiao Gao , Zhengjie Sun , Zongmin Wu
This paper presents an innovative energy-equidistributed moving mesh strategy for simulating Hamiltonian partial differential equations (PDEs) characterized by solitons and rapid temporal variations. A novel framework, named the Energy Equidistribution Principles (EEPs), is introduced, highlighting the critical role of energy conservation in achieving accurate simulations. Building on EEPs, three kinds of energy-equidistributed moving mesh PDEs (EMMPDEs) are proposed, each grounded in different methodologies. These strategies are rigorously examined in terms of their convergence conditions and rates. Both theoretical analysis and numerical experiments demonstrate that the proposed EMMPDEs offer superior robustness and effectiveness in long-term simulations, compared to traditional arc-length-equidistributed MMPDEs.
{"title":"Energy-equidistributed moving mesh strategies for simulating Hamiltonian partial differential equations","authors":"Qinjiao Gao , Zhengjie Sun , Zongmin Wu","doi":"10.1016/j.aml.2025.109460","DOIUrl":"10.1016/j.aml.2025.109460","url":null,"abstract":"<div><div>This paper presents an innovative energy-equidistributed moving mesh strategy for simulating Hamiltonian partial differential equations (PDEs) characterized by solitons and rapid temporal variations. A novel framework, named the Energy Equidistribution Principles (EEPs), is introduced, highlighting the critical role of energy conservation in achieving accurate simulations. Building on EEPs, three kinds of energy-equidistributed moving mesh PDEs (EMMPDEs) are proposed, each grounded in different methodologies. These strategies are rigorously examined in terms of their convergence conditions and rates. Both theoretical analysis and numerical experiments demonstrate that the proposed EMMPDEs offer superior robustness and effectiveness in long-term simulations, compared to traditional arc-length-equidistributed MMPDEs.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109460"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990370","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 : 2025-01-13DOI: 10.1016/j.aml.2025.109459
Xuemei Li
In this paper, we study boundary estimates for solution sets of fully nonlinear parabolic inequalities on domains, which generalize results for elliptic equations in Li and Li (2023).
本文研究了全非线性抛物不等式 ut-M+(D2u,λ,Λ)≤f(x,t)≤ut-M-(D2u,λ,Λ) 在 C1,α 域上的解集的边界 W2,δ 估计，它概括了 Li 和 Li (2023) 中关于椭圆方程的结果。
{"title":"W2,δ estimates for fully nonlinear parabolic inequalities on C1,α domains","authors":"Xuemei Li","doi":"10.1016/j.aml.2025.109459","DOIUrl":"10.1016/j.aml.2025.109459","url":null,"abstract":"<div><div>In this paper, we study boundary <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>δ</mi></mrow></msup></math></span> estimates for solution sets of fully nonlinear parabolic inequalities <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></mrow><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></mrow></mrow></math></span> on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> domains, which generalize results for elliptic equations in Li and Li (2023).</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109459"},"PeriodicalIF":2.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990371","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 : 2025-01-11DOI: 10.1016/j.aml.2025.109463
Xiaotong Gao , Yan Gu , Bo Yu
Accurate and efficient treatment of domain integrals is critical for obtaining reliable and precise boundary element method (BEM) solutions in dynamic or time-dependent problems. Despite the success of existing techniques for handling domain integrals, significant challenges still remain, especially in time-dependent BEM analyses where time-dependent fundamental solutions often result in integrands with oscillations or near-singularities, particularly when small time steps are used. To address these issues, this study introduces an improved scaled coordinate transformation BEM (SCT-BEM), combined with a non-linear coordinate transformation, to enhance the robustness of domain integral evaluations in transient time-domain BEM. The proposed method is straightforward to implement, requiring minimal modifications to existing BEM frameworks, and significantly improves both the robustness and accuracy of domain integral evaluations in transient time-domain BEM.
要在动态或时间相关问题中获得可靠而精确的边界元法（BEM）解，准确而高效地处理域积分至关重要。尽管现有的域积分处理技术取得了成功，但仍然存在重大挑战，特别是在时变 BEM 分析中，时变基本解往往会导致积分出现振荡或接近奇异值，尤其是在使用小时间步长时。为解决这些问题，本研究引入了改进的比例坐标变换 BEM（SCT-BEM），并结合非线性坐标变换，以增强瞬态时域 BEM 中域积分评估的鲁棒性。所提出的方法简单易行，只需对现有的 BEM 框架进行最小限度的修改，就能显著提高瞬态时域 BEM 中域积分评估的稳健性和准确性。
{"title":"A novel time-domain SCT-BEM for transient heat conduction analysis","authors":"Xiaotong Gao , Yan Gu , Bo Yu","doi":"10.1016/j.aml.2025.109463","DOIUrl":"10.1016/j.aml.2025.109463","url":null,"abstract":"<div><div>Accurate and efficient treatment of domain integrals is critical for obtaining reliable and precise boundary element method (BEM) solutions in dynamic or time-dependent problems. Despite the success of existing techniques for handling domain integrals, significant challenges still remain, especially in time-dependent BEM analyses where time-dependent fundamental solutions often result in integrands with oscillations or near-singularities, particularly when small time steps are used. To address these issues, this study introduces an improved scaled coordinate transformation BEM (SCT-BEM), combined with a non-linear coordinate transformation, to enhance the robustness of domain integral evaluations in transient time-domain BEM. The proposed method is straightforward to implement, requiring minimal modifications to existing BEM frameworks, and significantly improves both the robustness and accuracy of domain integral evaluations in transient time-domain BEM.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109463"},"PeriodicalIF":2.9,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990372","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 : 2025-01-11DOI: 10.1016/j.aml.2025.109458
Yu-Xia Hao, Guo-Bao Zhang
This work is devoted to studying the propagation direction of the following nonlocal dispersal epidemic model (0.1)where . By discussing the case and using the monotone dependence of the wave speed of traveling wave solutions on parameters, we state the sufficient conditions for the speed and under some calculations and analysis. Compared to the known works for classical diffusive epidemic models, we have to overcome difficulties due to the appearance of nonlocal dispersal operators in the current paper.
{"title":"Propagation direction of traveling waves for a class of nonlocal dispersal bistable epidemic models","authors":"Yu-Xia Hao, Guo-Bao Zhang","doi":"10.1016/j.aml.2025.109458","DOIUrl":"10.1016/j.aml.2025.109458","url":null,"abstract":"<div><div>This work is devoted to studying the propagation direction of the following nonlocal dispersal epidemic model <span><span><span>(0.1)</span><span><math><mfenced><mrow><mtable><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>u</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mfenced><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><mi>J</mi><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mi>d</mi><mi>y</mi><mo>−</mo><mi>u</mi></mrow></mfenced><mo>−</mo><mi>u</mi><mo>+</mo><mi>α</mi><mi>v</mi><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>v</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mfenced><mrow><msub><mrow><mo>∫</mo></mrow><mrow><mi>R</mi></mrow></msub><mi>J</mi><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mi>v</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mi>d</mi><mi>y</mi><mo>−</mo><mi>v</mi></mrow></mfenced><mo>−</mo><mi>β</mi><mi>v</mi><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><mi>α</mi><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn>0</mn></mrow></math></span>. By discussing the case <span><math><mrow><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span> and using the monotone dependence of the wave speed of traveling wave solutions on parameters, we state the sufficient conditions for the speed <span><math><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>c</mi><mo><</mo><mn>0</mn></mrow></math></span> under some calculations and analysis. Compared to the known works for classical diffusive epidemic models, we have to overcome difficulties due to the appearance of nonlocal dispersal operators in the current paper.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109458"},"PeriodicalIF":2.9,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990373","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 : 2025-01-10DOI: 10.1016/j.aml.2024.109453
Roberto Cavoretto, Alessandra De Rossi, Adeeba Haider
In this paper, we study a direct discretization technique based on a radial basis function partition of unity (RBF-PU) method, which is built to numerically solve partial differential equations (PDEs). Unlike commonly used shape parameter free polyharmonic spline kernels, in this work we focus on local radial kernels depending on the shape parameter associated with the basis functions. The resulting scheme generally leads to more flexibility and accuracy, in particular when a polynomial term is added to the local RBF expansion. To emphasize the benefits deriving from use of the direct approach, we also compare it with the RBF finite difference (RBF-FD) method both in terms of computational efficiency and accuracy. Numerical results show the method performance by solving some elliptic PDE problems.
{"title":"A shape-parameterized RBF-partition of unity technique for PDEs","authors":"Roberto Cavoretto, Alessandra De Rossi, Adeeba Haider","doi":"10.1016/j.aml.2024.109453","DOIUrl":"10.1016/j.aml.2024.109453","url":null,"abstract":"<div><div>In this paper, we study a direct discretization technique based on a radial basis function partition of unity (RBF-PU) method, which is built to numerically solve partial differential equations (PDEs). Unlike commonly used shape parameter free polyharmonic spline kernels, in this work we focus on local radial kernels depending on the shape parameter associated with the basis functions. The resulting scheme generally leads to more flexibility and accuracy, in particular when a polynomial term is added to the local RBF expansion. To emphasize the benefits deriving from use of the direct approach, we also compare it with the RBF finite difference (RBF-FD) method both in terms of computational efficiency and accuracy. Numerical results show the method performance by solving some elliptic PDE problems.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109453"},"PeriodicalIF":2.9,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-09DOI: 10.1016/j.aml.2025.109456
Chen Wang, Yue Shi, Weiao Yang, Xiangpeng Xin
This paper investigates an integrable nonlocal Kaup–Newell (NKN) equation with variable coefficients. Utilizing Lax pair theory, the construction of the variable coefficient NKN equation is presented for the first time, alongside a systematic analysis employing the Darboux transform technique. This approach explicitly derives the form of the nth-order Darboux transform, which is presented for the first time. The article offers a thorough explanation of the derivation process for the second-order Darboux transform using Cramer’s rule, further extending this to propose a general formula for the th Darboux transform applicable to multi-parameter scenarios. By applying a zero-seed solution, the exact solution of the variable coefficient NKN equation is obtained. To explore the influence of different coefficient functions on the solutions, specific coefficient functions are selected, and their corresponding graphical representations are analyzed, uncovering a range of solution types, including single soliton solutions, multi-solitons, rogue wave solutions, mixed twisted soliton solutions and breather wave solutions. Through the comprehensive analysis of these solutions, the study underscores the significant enhancement in modeling accuracy when time- and space-dependent coefficients are incorporated into the NKN equations, particularly in the context of simulating the dynamic behavior of nonlinear waves in real-world applications.
{"title":"Darboux transformations and exact solutions of nonlocal Kaup–Newell equations with variable coefficients","authors":"Chen Wang, Yue Shi, Weiao Yang, Xiangpeng Xin","doi":"10.1016/j.aml.2025.109456","DOIUrl":"10.1016/j.aml.2025.109456","url":null,"abstract":"<div><div>This paper investigates an integrable nonlocal Kaup–Newell (NKN) equation with variable coefficients. Utilizing Lax pair theory, the construction of the variable coefficient NKN equation is presented for the first time, alongside a systematic analysis employing the Darboux transform technique. This approach explicitly derives the form of the nth-order Darboux transform, which is presented for the first time. The article offers a thorough explanation of the derivation process for the second-order Darboux transform using Cramer’s rule, further extending this to propose a general formula for the <span><math><mi>n</mi></math></span>th Darboux transform applicable to multi-parameter scenarios. By applying a zero-seed solution, the exact solution of the variable coefficient NKN equation is obtained. To explore the influence of different coefficient functions on the solutions, specific coefficient functions are selected, and their corresponding graphical representations are analyzed, uncovering a range of solution types, including single soliton solutions, multi-solitons, rogue wave solutions, mixed twisted soliton solutions and breather wave solutions. Through the comprehensive analysis of these solutions, the study underscores the significant enhancement in modeling accuracy when time- and space-dependent coefficients are incorporated into the NKN equations, particularly in the context of simulating the dynamic behavior of nonlinear waves in real-world applications.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109456"},"PeriodicalIF":2.9,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990375","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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