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}
Pub Date : 2025-01-07DOI: 10.1016/j.aml.2025.109454
Minjie Dong , Xiubin Wang
In this work, we theoretically calculate higher-order freak wave solutions of the AB system through a Darboux transformation by a separation of variable method. Furthermore, the dynamics of first-order and second-order freak wave solutions are discussed with some illustrative graphics. In particular, we observe the emergence of a four peaky-shaped freak wave in the second component, which contrasts with the previously reported four eye-shaped waves.
{"title":"Higher-order freak waves of the AB system revisited via a variable separation method","authors":"Minjie Dong , Xiubin Wang","doi":"10.1016/j.aml.2025.109454","DOIUrl":"10.1016/j.aml.2025.109454","url":null,"abstract":"<div><div>In this work, we theoretically calculate higher-order freak wave solutions of the AB system through a Darboux transformation by a separation of variable method. Furthermore, the dynamics of first-order and second-order freak wave solutions are discussed with some illustrative graphics. In particular, we observe the emergence of a four peaky-shaped freak wave in the second component, which contrasts with the previously reported four eye-shaped waves.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109454"},"PeriodicalIF":2.9,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968144","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 : 2025-01-07DOI: 10.1016/j.aml.2025.109455
Giuseppina D’Aguì , Valeria Morabito , Patrick Winkert
This paper investigates nonlinear differential problems involving the -Laplace operator and subject to Neumann boundary value conditions whereby the right-hand side consists of a nonlinearity which is highly discontinuous. Using variational methods suitable for nonsmooth functionals, we prove the existence of at least two nontrivial weak solutions of such problems.
{"title":"Elliptic Neumann problems with highly discontinuous nonlinearities","authors":"Giuseppina D’Aguì , Valeria Morabito , Patrick Winkert","doi":"10.1016/j.aml.2025.109455","DOIUrl":"10.1016/j.aml.2025.109455","url":null,"abstract":"<div><div>This paper investigates nonlinear differential problems involving the <span><math><mi>p</mi></math></span>-Laplace operator and subject to Neumann boundary value conditions whereby the right-hand side consists of a nonlinearity which is highly discontinuous. Using variational methods suitable for nonsmooth functionals, we prove the existence of at least two nontrivial weak solutions of such problems.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109455"},"PeriodicalIF":2.9,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968145","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}
We characterize the limit solutions of loop solitons for an integrable compound equation which is a mix of the Wadati-Konno-Ichikawa (WKI) equation and the short-pulse (SP) equation. We do so by taking an ingenious limit on the -function derived from Hirota’s bilinear equations of the mKdV-SG (modified Korteweg–de Vries and sine-Gordon) equation. By virtue of a hodograph transformation, we compute the limit solution of 2-loop (noted as ) soliton and discuss the interactions between two solitons in detail. One singlevalued and two nonsinglevalued limit solutions of 2-breather solution are presented at last.
{"title":"Limit solutions of loop solitons for a compound WKI-SP equation","authors":"Gaizhu Qu , Junyang Zhang , Xiaorui Hu , Shoufeng Shen","doi":"10.1016/j.aml.2024.109452","DOIUrl":"10.1016/j.aml.2024.109452","url":null,"abstract":"<div><div>We characterize the limit solutions of loop solitons for an integrable compound equation which is a mix of the Wadati-Konno-Ichikawa (WKI) equation and the short-pulse (SP) equation. We do so by taking an ingenious limit on the <span><math><mi>τ</mi></math></span>-function derived from Hirota’s bilinear equations of the mKdV-SG (modified Korteweg–de Vries and sine-Gordon) equation. By virtue of a hodograph transformation, we compute the limit solution of 2-loop (noted as <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn><mo>−</mo><mi>l</mi><mi>o</mi><mi>o</mi><mi>p</mi></mrow></msub></math></span>) soliton and discuss the interactions between two <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn><mo>−</mo><mi>l</mi><mi>o</mi><mi>o</mi><mi>p</mi></mrow></msub></math></span> solitons in detail. One singlevalued and two nonsinglevalued limit solutions of 2-breather solution are presented at last.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109452"},"PeriodicalIF":2.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968148","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 : 2025-01-04DOI: 10.1016/j.aml.2024.109450
Xun Wang , Sanling Yuan , Jin Zhao
In this paper, we present the Euler equation of steady periodic equatorial water waves in two-layer flows with different densities and generalise the two Stokes’ definitions for the velocity of the wave propagation. We further demonstrate that the excess potential energy density of nonlinear equatorial two-layer waves is always positive, while the excess kinetic energy density is negative.
{"title":"Energy of steady periodic equatorial water waves in two-layer flows","authors":"Xun Wang , Sanling Yuan , Jin Zhao","doi":"10.1016/j.aml.2024.109450","DOIUrl":"10.1016/j.aml.2024.109450","url":null,"abstract":"<div><div>In this paper, we present the Euler equation of steady periodic equatorial water waves in two-layer flows with different densities and generalise the two Stokes’ definitions for the velocity of the wave propagation. We further demonstrate that the excess potential energy density of nonlinear equatorial two-layer waves is always positive, while the excess kinetic energy density is negative.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109450"},"PeriodicalIF":2.9,"publicationDate":"2025-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968149","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 : 2025-01-04DOI: 10.1016/j.aml.2024.109443
Rafael Granero-Belinchón
In this note we derive a new nonlocal and nonlinear dispersive equations capturing the main dynamics of a circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. This equation that we termed the model shares some common structure with the Camassa–Holm equation but has additional nonlocal effects. For this new PDE we study the well-posedness together with the existence of periodic traveling waves. Furthermore, we also show some numerical simulations suggesting the finite time singularity formation.
{"title":"On a Camassa–Holm type equation describing the dynamics of viscous fluid conduits","authors":"Rafael Granero-Belinchón","doi":"10.1016/j.aml.2024.109443","DOIUrl":"10.1016/j.aml.2024.109443","url":null,"abstract":"<div><div>In this note we derive a new nonlocal and nonlinear dispersive equations capturing the main dynamics of a circular interface separating a light, viscous fluid rising buoyantly through a heavy, more viscous, miscible fluid at small Reynolds numbers. This equation that we termed the <span><math><mrow><mi>g</mi><mo>−</mo></mrow></math></span>model shares some common structure with the Camassa–Holm equation but has additional nonlocal effects. For this new PDE we study the well-posedness together with the existence of periodic traveling waves. Furthermore, we also show some numerical simulations suggesting the finite time singularity formation.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109443"},"PeriodicalIF":2.9,"publicationDate":"2025-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968150","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 : 2025-01-04DOI: 10.1016/j.aml.2024.109451
Chuangxia Huang , Bingwen Liu
In this paper, we explore a modified diffusive Nicholson’s blowflies equation accompanying multiple pairs of time-varying delays which include distinct diapause and maturation effects. With the help of some differential inequality analyses, we obtain a criterion to assure the stability and exponential attraction of the addressed reaction–diffusion equation accompanying Neumann boundary conditions, which fully refines and generalizes some existing ones.
{"title":"Exponential stability of a diffusive Nicholson’s blowflies equation accompanying multiple time-varying delays","authors":"Chuangxia Huang , Bingwen Liu","doi":"10.1016/j.aml.2024.109451","DOIUrl":"10.1016/j.aml.2024.109451","url":null,"abstract":"<div><div>In this paper, we explore a modified diffusive Nicholson’s blowflies equation accompanying multiple pairs of time-varying delays which include distinct diapause and maturation effects. With the help of some differential inequality analyses, we obtain a criterion to assure the stability and exponential attraction of the addressed reaction–diffusion equation accompanying Neumann boundary conditions, which fully refines and generalizes some existing ones.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109451"},"PeriodicalIF":2.9,"publicationDate":"2025-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096578","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 : 2025-01-03DOI: 10.1016/j.aml.2024.109447
Jessika Camaño , Ricardo Oyarzúa , Miguel Serón , Manuel Solano
We revisit the continuous formulation introduced in Discacciati and Oyarzúa (2017) for the stationary Navier–Stokes/Darcy (NSD) coupled system and propose an equivalent scheme that does not require a Lagrange multiplier to enforce the continuity of normal velocities at the interface. Building on this formulation and following a similar approach to Kanschat and Rivière (2010), we derive a mass-conservative, –conforming finite element method for the NSD system.
{"title":"A strong mass conservative finite element method for the Navier–Stokes/Darcy coupled system","authors":"Jessika Camaño , Ricardo Oyarzúa , Miguel Serón , Manuel Solano","doi":"10.1016/j.aml.2024.109447","DOIUrl":"10.1016/j.aml.2024.109447","url":null,"abstract":"<div><div>We revisit the continuous formulation introduced in Discacciati and Oyarzúa (2017) for the stationary Navier–Stokes/Darcy (NSD) coupled system and propose an equivalent scheme that does not require a Lagrange multiplier to enforce the continuity of normal velocities at the interface. Building on this formulation and following a similar approach to Kanschat and Rivière (2010), we derive a mass-conservative, <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>div</mi><mo>)</mo></mrow></mrow></math></span>–conforming finite element method for the NSD system.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109447"},"PeriodicalIF":2.9,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968153","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 : 2025-01-03DOI: 10.1016/j.aml.2024.109449
Yelei Guo, Chinyin Qian
The global existence of solution for 2D inhomogeneous incompressible Navier–Stokes equations is established by Abidi et al. (2024), and the uniqueness of solution is also investigated under some additional conditions on initial density. The purpose of this paper is to obtain the uniqueness of the solution without any additional assumptions on the initial density in case of . The key strategy is to establish a new estimate of solution in Lagrangian coordinates.
Abidi et al.（2024）建立了二维非齐次不可压缩Navier-Stokes方程解的整体存在性，并在初始密度的附加条件下研究了解的唯一性。3 .本文的目的是在2≤p<的情况下，在不附加初始密度假设的情况下，得到解的唯一性；关键的策略是在拉格朗日坐标系中建立一个新的解的估计。
{"title":"Uniqueness of solution for incompressible inhomogeneous Navier–Stokes equations in dimension two","authors":"Yelei Guo, Chinyin Qian","doi":"10.1016/j.aml.2024.109449","DOIUrl":"10.1016/j.aml.2024.109449","url":null,"abstract":"<div><div>The global existence of solution for 2D inhomogeneous incompressible Navier–Stokes equations is established by Abidi et al. (2024), and the uniqueness of solution is also investigated under some additional conditions on initial density. The purpose of this paper is to obtain the uniqueness of the solution without any additional assumptions on the initial density in case of <span><math><mrow><mn>2</mn><mo>≤</mo><mi>p</mi><mo><</mo><mn>4</mn></mrow></math></span>. The key strategy is to establish a new estimate of solution in Lagrangian coordinates.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109449"},"PeriodicalIF":2.9,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968151","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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