Alzheimer’s disease (AD) is characterized by the progressive deposition of -amyloid (A) plaques in the brain, where the A oligomers have been confirmed to produce the critical cytotoxicity during the disease process. In this study, a model is established to describe the effect of A oligomers on the interplay between A and Ca2＋. Mathematical analysis demonstrates the existence and stability of the equilibria and the conditions under which backward bifurcation and saddle–node bifurcation occur are proposed. In addition, the aggregate reproduction number is introduced to characterize the progression of AD. These results may offer valuable insights for studying AD-related medical strategies.
{"title":"A study in Alzheimer’s disease model for pathological effect of oligomers on the interplay between β-amyloid and Ca2＋","authors":"Mingyan Dong , Yongxin Zhang , Gui-Quan Sun , Zun-Guang Guo , Jiao Zhang","doi":"10.1016/j.aml.2024.109407","DOIUrl":"10.1016/j.aml.2024.109407","url":null,"abstract":"<div><div>Alzheimer’s disease (AD) is characterized by the progressive deposition of <span><math><mi>β</mi></math></span>-amyloid (A<span><math><mi>β</mi></math></span>) plaques in the brain, where the A<span><math><mi>β</mi></math></span> oligomers have been confirmed to produce the critical cytotoxicity during the disease process. In this study, a model is established to describe the effect of A<span><math><mi>β</mi></math></span> oligomers on the interplay between A<span><math><mi>β</mi></math></span> and Ca<sup>2＋</sup>. Mathematical analysis demonstrates the existence and stability of the equilibria and the conditions under which backward bifurcation and saddle–node bifurcation occur are proposed. In addition, the aggregate reproduction number <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is introduced to characterize the progression of AD. These results may offer valuable insights for studying AD-related medical strategies.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109407"},"PeriodicalIF":2.9,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790149","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 : 2024-11-30DOI: 10.1016/j.aml.2024.109410
Qian Wang
The classical Haddock conjecture is extended to a kind of non-autonomous neutral functional differential equations (NFDEs) incorporating time-varying delays in this paper. By using the Dini derivative theory and inequality analyses, without requiring the strictly monotonically increasing property of the delay feedback function, it is demonstrated that every solution of the considered NFDEs is bounded and converges to a constant, which fully refines and generalizes the existing findings.
{"title":"New asymptotic study on the non-autonomous NFDEs involving Haddock conjecture","authors":"Qian Wang","doi":"10.1016/j.aml.2024.109410","DOIUrl":"10.1016/j.aml.2024.109410","url":null,"abstract":"<div><div>The classical Haddock conjecture is extended to a kind of non-autonomous neutral functional differential equations (NFDEs) incorporating time-varying delays in this paper. By using the Dini derivative theory and inequality analyses, without requiring the strictly monotonically increasing property of the delay feedback function, it is demonstrated that every solution of the considered NFDEs is bounded and converges to a constant, which fully refines and generalizes the existing findings.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109410"},"PeriodicalIF":2.9,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790148","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 : 2024-11-29DOI: 10.1016/j.aml.2024.109406
Jinrui Zhang , Haijun Hu , Chuangxia Huang
In this paper, we consider a class of Fisher–KPP equations with delays appearing in both diffusion and reaction terms. By employing some differential inequality analyses, we prove that the delayed Fisher–KPP equation possesses a pair of quasi-upper and quasi-lower solutions which have absolutely continuous derivatives. Based on this, we apply the monotone iteration method and the Perron’s theorem to establish a sufficient criterion ensuring the existence of wave fronts. Our proof corrects the previous related research.
{"title":"Wave fronts for a class of delayed Fisher–KPP equations","authors":"Jinrui Zhang , Haijun Hu , Chuangxia Huang","doi":"10.1016/j.aml.2024.109406","DOIUrl":"10.1016/j.aml.2024.109406","url":null,"abstract":"<div><div>In this paper, we consider a class of Fisher–KPP equations with delays appearing in both diffusion and reaction terms. By employing some differential inequality analyses, we prove that the delayed Fisher–KPP equation possesses a pair of quasi-upper and quasi-lower solutions which have absolutely continuous derivatives. Based on this, we apply the monotone iteration method and the Perron’s theorem to establish a sufficient criterion ensuring the existence of wave fronts. Our proof corrects the previous related research.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109406"},"PeriodicalIF":2.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823131","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 : 2024-11-28DOI: 10.1016/j.aml.2024.109404
Huanhuan Lu , Yufeng Zhang
By complexifying the independent variables of the Kaup–Kuperschmidt (KK) equation, we derive the 4+2 integrable extension of the KK equation and its Lax pair. The construction of the associated nonlinear Fourier transform pair comprising both direct and inverse transforms is accomplished by conducting a spectral analysis of the -independent part of the Lax pair. In the final section, the nonlinear Fourier transform pair will be used, after also taking into account the appropriate time evolution, for solving the Cauchy initial value problem of the three-spatial-dimensions KK equation with two temporal variables.
{"title":"Nonlocal ∂̄ formalism for the three-spatial-dimensions Kaup–Kuperschmidt equation with two temporal variables","authors":"Huanhuan Lu , Yufeng Zhang","doi":"10.1016/j.aml.2024.109404","DOIUrl":"10.1016/j.aml.2024.109404","url":null,"abstract":"<div><div>By complexifying the independent variables of the Kaup–Kuperschmidt (KK) equation, we derive the 4+2 integrable extension of the KK equation and its Lax pair. The construction of the associated nonlinear Fourier transform pair comprising both direct and inverse transforms is accomplished by conducting a spectral analysis of the <span><math><mi>t</mi></math></span>-independent part of the Lax pair. In the final section, the nonlinear Fourier transform pair will be used, after also taking into account the appropriate time evolution, for solving the Cauchy initial value problem of the three-spatial-dimensions KK equation with two temporal variables.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109404"},"PeriodicalIF":2.9,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142756228","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 : 2024-11-28DOI: 10.1016/j.aml.2024.109405
Reinhard Racke , Shuji Yoshikawa
We prove a decay estimate for solutions to a transmission problem for wave equations with different propagation speeds in an infinite waveguide. The problem represents the wave propagation in unbounded and layered composite materials in which different properties are given. It is a new composition of a waveguide problem and a transmission problem, motivated by a unit cell model for CFRP. The proof is based on splitting variables, partial eigenfunction expansions in the bounded cross section, and on an explicit Weyl type estimate for the eigenvalues.
{"title":"A transmission problem for wave equations in infinite waveguides","authors":"Reinhard Racke , Shuji Yoshikawa","doi":"10.1016/j.aml.2024.109405","DOIUrl":"10.1016/j.aml.2024.109405","url":null,"abstract":"<div><div>We prove a decay estimate for solutions to a transmission problem for wave equations with different propagation speeds in an infinite waveguide. The problem represents the wave propagation in unbounded and layered composite materials in which different properties are given. It is a new composition of a waveguide problem and a transmission problem, motivated by a unit cell model for CFRP. The proof is based on splitting variables, partial eigenfunction expansions in the bounded cross section, and on an explicit Weyl type estimate for the eigenvalues.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109405"},"PeriodicalIF":2.9,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790147","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 : 2024-11-26DOI: 10.1016/j.aml.2024.109382
Chaoyue Guan, Yuli Sun, Jing Niu
In this paper, we introduce novel class of Legendre spectral volume (LSV) methods for solving Allen–Cahn equations. Each spectral volume (SV) is refined with Gauss–Legendre points to define an arbitrary order control volume (CV). Moreover, the second derivative is handled using the direct discontinuous Galerkin (DDG) approach. Furthermore, four numerical experiments are detailed including 1D and 2D Allen–Cahn equations with Neumann and periodic boundary conditions. These experiments demonstrate the stability and accuracy in capturing phase transitions of the approach. Meanwhile, we also show the LSV methods can maintain physical properties such as energy dissipation and uniform boundedness. It is worth mentioning that we observe that the LSV methods achieve both optimal convergence and superconvergence as the numerical flux parameter is carefully selected.
{"title":"Legendre spectral volume methods for Allen–Cahn equations by the direct discontinuous Galerkin formula","authors":"Chaoyue Guan, Yuli Sun, Jing Niu","doi":"10.1016/j.aml.2024.109382","DOIUrl":"10.1016/j.aml.2024.109382","url":null,"abstract":"<div><div>In this paper, we introduce novel class of Legendre spectral volume (LSV) methods for solving Allen–Cahn equations. Each spectral volume (SV) is refined with <span><math><mi>k</mi></math></span> Gauss–Legendre points to define an arbitrary order control volume (CV). Moreover, the second derivative is handled using the direct discontinuous Galerkin (DDG) approach. Furthermore, four numerical experiments are detailed including 1D and 2D Allen–Cahn equations with Neumann and periodic boundary conditions. These experiments demonstrate the stability and accuracy in capturing phase transitions of the approach. Meanwhile, we also show the LSV methods can maintain physical properties such as energy dissipation and uniform boundedness. It is worth mentioning that we observe that the LSV methods achieve both optimal convergence and superconvergence as the numerical flux parameter is carefully selected.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109382"},"PeriodicalIF":2.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748130","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 : 2024-11-25DOI: 10.1016/j.aml.2024.109402
Wei Zhang , Jialing Zhang
<div><div>We consider this equation <span><span><span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mfenced><mrow><mi>p</mi><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mi>k</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>∈</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>. Here <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> denotes the <span><math><mi>k</mi></math></span>th elementary symmetric function of the eigenvalues of <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span>, and <span><span><span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><mo>∇</mo><mi>u</mi><mo>⊗</mo><mo>∇</mo><mi>u</mi><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mi>I</mi><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mo>∇</mo><mi>u</mi></mrow></math></span> denotes the gradient of <span><math><mi>u</mi></math></span> and <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow></math></span> denotes the Hessian of <span><math><mi>u</mi></math></span>. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y
{"title":"Harnack type inequality and Liouville theorem for subcritical fully nonlinear equations","authors":"Wei Zhang , Jialing Zhang","doi":"10.1016/j.aml.2024.109402","DOIUrl":"10.1016/j.aml.2024.109402","url":null,"abstract":"<div><div>We consider this equation <span><span><span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mfenced><mrow><mi>p</mi><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mi>k</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>∈</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>. Here <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> denotes the <span><math><mi>k</mi></math></span>th elementary symmetric function of the eigenvalues of <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span>, and <span><span><span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><mo>∇</mo><mi>u</mi><mo>⊗</mo><mo>∇</mo><mi>u</mi><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mi>I</mi><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mo>∇</mo><mi>u</mi></mrow></math></span> denotes the gradient of <span><math><mi>u</mi></math></span> and <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow></math></span> denotes the Hessian of <span><math><mi>u</mi></math></span>. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109402"},"PeriodicalIF":2.9,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748132","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 : 2024-11-25DOI: 10.1016/j.aml.2024.109401
Huaijun Yang
In this paper, a linearized Euler Galerkin scheme is studied and the unconditionally optimal error estimate in -norm is obtained for Schrödinger equation with cubic nonlinearity without any time-step restriction. The key to the analysis is to bound the -norm between the numerical solution and the Ritz projection of the exact solution by mathematical induction for two cases rather than the error splitting technique used in the previous work. Finally, some numerical results are presented to confirm the theoretical analysis.
{"title":"A new error analysis of a linearized Euler Galerkin scheme for Schrödinger equation with cubic nonlinearity","authors":"Huaijun Yang","doi":"10.1016/j.aml.2024.109401","DOIUrl":"10.1016/j.aml.2024.109401","url":null,"abstract":"<div><div>In this paper, a linearized Euler Galerkin scheme is studied and the unconditionally optimal error estimate in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm is obtained for Schrödinger equation with cubic nonlinearity without any time-step restriction. The key to the analysis is to bound the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm between the numerical solution and the Ritz projection of the exact solution by mathematical induction for two cases rather than the error splitting technique used in the previous work. Finally, some numerical results are presented to confirm the theoretical analysis.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109401"},"PeriodicalIF":2.9,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723174","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 : 2024-11-25DOI: 10.1016/j.aml.2024.109403
R. Katani , S. McKee
This paper presents a novel product integration method that provides an appropriate numerical solution for nonlinear weakly singular Volterra integral equations (WSVIEs). Extensive research in the literature has focused on studying the existence and uniqueness of solutions to these equations. However, when solving the WSVIEs, the solution may exhibit a singular behavior near the initial point of the integration interval, which can pose challenges for numerical computation. In these cases, we propose a smoothing change of variables that transforms the equation into one with a smooth solution, while still being weakly singular. We provide a convergence analysis and determine the order of convergence. The effectiveness of the proposed method is then demonstrated through the solution of various test problems.
{"title":"A product integration method for nonlinear second kind Volterra integral equations with a weakly singular kernel (with application to fractional differential equations)","authors":"R. Katani , S. McKee","doi":"10.1016/j.aml.2024.109403","DOIUrl":"10.1016/j.aml.2024.109403","url":null,"abstract":"<div><div>This paper presents a novel product integration method that provides an appropriate numerical solution for nonlinear weakly singular Volterra integral equations (WSVIEs). Extensive research in the literature has focused on studying the existence and uniqueness of solutions to these equations. However, when solving the WSVIEs, the solution may exhibit a singular behavior near the initial point of the integration interval, which can pose challenges for numerical computation. In these cases, we propose a smoothing change of variables that transforms the equation into one with a smooth solution, while still being weakly singular. We provide a convergence analysis and determine the order of convergence. The effectiveness of the proposed method is then demonstrated through the solution of various test problems.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109403"},"PeriodicalIF":2.9,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096967","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 : 2024-11-24DOI: 10.1016/j.aml.2024.109391
Zhao Yang , Tao Chen , Sanyang Liu
In this paper, we show that collocation matrices of the Cauchy–Bernstein basis can be decomposed as products of a Cauchy–Vandermonde matrix and a block diagonal matrix. A useful application of this result is that the explicit expression of the determinant for the collocation matrices is presented. Consequently, an algorithm is provided to accurately compute the determinants. Numerical experiments confirm the high accuracy of the algorithm.
{"title":"On decomposition of collocation matrices for the Cauchy–Bernstein basis and applications","authors":"Zhao Yang , Tao Chen , Sanyang Liu","doi":"10.1016/j.aml.2024.109391","DOIUrl":"10.1016/j.aml.2024.109391","url":null,"abstract":"<div><div>In this paper, we show that collocation matrices of the Cauchy–Bernstein basis can be decomposed as products of a Cauchy–Vandermonde matrix and a block diagonal matrix. A useful application of this result is that the explicit expression of the determinant for the collocation matrices is presented. Consequently, an algorithm is provided to accurately compute the determinants. Numerical experiments confirm the high accuracy of the algorithm.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109391"},"PeriodicalIF":2.9,"publicationDate":"2024-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142756235","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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