Mathematical Methods in the Applied Sciences最新文献

{"title":"Mathematical Modeling of Thermoelectric Current of the Divertor Material of a Thermonuclear Reactor Under Pulsed Action","authors":"Galina G. Lazareva","doi":"10.1002/mma.70863","DOIUrl":"https://doi.org/10.1002/mma.70863","url":null,"abstract":"<div>\u0000 \u0000 <p>The article presents a mathematical model of the propagation of thermoelectric current occurring in a heated material. The model is based on solving an elliptic-type equation. This equation is a consequence of the system of equations of electrodynamics, which is written for the case of axial symmetry. Various formulas for determining the resistivity and thermal strength in vapors of a substance, which are coefficients of the equations of thermoelectric current, are considered. As a result of calculations, electric current distributions in a tungsten sample and an evaporated substance are obtained when the surface is heated by an electron beam. It is shown that the introduction of a magnetic field into the model leads to suppression of thermoelectric current outside the melt zone. The model parameters are taken from experiments at the Beam of Electrons for materials Test Applications (BETA) stand.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16676-16685"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149027591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Behaviors of the Non-Viscous Oldroyd-B Systems Without Damping Mechanism","authors":"Zhen Li","doi":"10.1002/mma.70894","DOIUrl":"https://doi.org/10.1002/mma.70894","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper is mainly concerned with the global well-posedness of the incompressible Oldroyd-B model without damping mechanism. More precisely, by fully exploiting the structure of the system and deriving some new refined estimates for the velocity, we demonstrate the existence of global solution for a class of partially large initial data in the critical Besov space.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16817-16833"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149028021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation of Functions by Modified Phillips \u0000 \u0000 \u0000 q\u0000 −\u0000 \u0000 $$ q- $$\u0000 Bernstein Operators of Order Two","authors":"Nilesh Zapadiya,&nbsp;Shailesh Kumar Srivastava","doi":"10.1002/mma.70907","DOIUrl":"https://doi.org/10.1002/mma.70907","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;In this paper, we introduce modified Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Bernstein operators of order two, which provide better uniform approximation than Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Bernstein operators and Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Durrmeyer Bernstein operators. In this study, we consider functions belonging to the class &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ Cleft(left[0,1right]right) $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and show that the modified Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Bernstein operators converge uniformly to all functions in this class using a strengthened version of the Korovkin theorem. Also, we prove a Voronovskaya-type theorem for modified Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Bernstein operators, which show better approximation than the Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Bernstein operators. Using the modulus of continuity, we show that the order of approximation of modified Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Bernstein operators is better than that of Phillips &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ q- $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;Bernst","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16834-16849"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149028025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamic Modeling of Malware Attack and Defense Using the Laplace-Adomian Decomposition Method","authors":"Dinesh Kumar Saini,&nbsp;Yerra Sankar Rao","doi":"10.1002/mma.70903","DOIUrl":"https://doi.org/10.1002/mma.70903","url":null,"abstract":"<div>\u0000 \u0000 <p>Malware propagation in computer networks poses a critical cybersecurity challenge that demands mathematically rigorous defense frameworks. In this paper, a fractional-order e-SIQR (susceptible, infected, quarantined, recovered) malware-spreading model using Caputo fractional derivatives of order <i>μ</i> (0 &lt; <i>μ</i> &lt; 1) is proposed to investigate the dynamics of malware infections that incorporate memory and hereditary properties of the natural infection process. The system has a total number of <i>N</i> nodes divided into four groups having an initial configuration of S(0) = 50, I(0) = 25, Q(0) = 25, and R(0) = 0 and parameter values comprising the infection rate <i>β</i> = 0.01, quarantine rate <i>γ</i> = 0.04, recovery rate <i>δ</i> = 0.05, and crash rates <i>α</i>₁ = 0.1 and <i>α</i>₂ = 0.2. The LADM is considered a semianalytic method that avoids linearization and discretization while quickly yielding a series solution. <i>R</i>₀ is obtained using the next-generation matrix technique. It has been shown that a malware-free equilibrium is locally asymptotically stable whenever <i>R</i>₀ &lt; 1. Two optimal control variables, <i>U</i>₁ (preventive shielding) and <i>U</i>₂ (quarantine screening), are formulated using Pontryagin's maximum principle to minimize infection while optimizing resource deployment. MATLAB simulations demonstrate that the LADM solution, truncated at three terms, closely matches the fourth-order Runge–Kutta method, with enhanced convergence at lower fractional orders reflecting memory-driven dynamics. Results confirm that strategic quarantine and anti-malware measures substantially reduce infection peaks and accelerate recovery, offering actionable guidance for cybersecurity professionals in designing layered network defenses.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16603-16621"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149029017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A New Approach to Least Squares Support Vector Regression Using the Legendre Operational Matrix for Solving Fractional Fredholm Integro-Differential Equations","authors":"Marzieh Pourbabaee,&nbsp;Abbas Saadatmandi","doi":"10.1002/mma.70871","DOIUrl":"https://doi.org/10.1002/mma.70871","url":null,"abstract":"<div>\u0000 \u0000 <p>In this study, a novel hybrid technique is introduced to address linear and nonlinear fractional Fredholm integro-differential equations (FFIDEs) by utilizing least squares support vector regression (LS-SVR) along with the Legendre operational matrix of fractional derivatives (OMFD). The unknown function is represented as a linear combination of shifted Legendre polynomials. Also, the Galerkin and collocation spectral methods are used. We then formulate an optimization problem and convert it into a set of algebraic equations. To address a linear FFIDE, we introduced a matrix representation of the dual problem and demonstrated the resulting linear systems. Also, a discussion on error estimates is included. Finally, to demonstrate the validity and applicability of the suggested methods, some test examples are presented.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16872-16892"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149028028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Well Posedness and Polynomial Decay of Solution to the Initial-Boundary Value Problem in Thermodiffusion for a Solid Body","authors":"Salim Messaoudi,&nbsp;Jerzy Gawinecki,&nbsp;Stanisław Kachel,&nbsp;Łukasz Kiszkowiak,&nbsp;Jarosław Łazuka","doi":"10.1002/mma.70897","DOIUrl":"https://doi.org/10.1002/mma.70897","url":null,"abstract":"<div>\u0000 <p>In this paper, we consider a hyperbolic-parabolic system of equations describing the thermodiffusion in a solid body. The theory of thermodiffusion is a combination of the theory of heat conduction and diffusion with the theory of elasticity. In this work, we establish the well-posedness of an initial-boundary value problem for a system describing this phenomenon, using the semigroup theory. Moreover, we prove a polynomial decay of solutions satisfying certain properties.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16622-16635"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149029124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Uniqueness of Positive Solutions for an Iterative System of Higher-Order Hadamard Fractional Equations With Three-Point Boundary Conditions","authors":"Chengbo Zhai,&nbsp;Zhuan Gao","doi":"10.1002/mma.70900","DOIUrl":"https://doi.org/10.1002/mma.70900","url":null,"abstract":"<div>\u0000 <p>The aim of this paper is to prove the uniqueness of positive solutions for an iterative system of Hadamard fractional problems under three-point boundary conditions. Different from some methods in literature, we use some fixed point theorems for monotone operators in ordered Banach spaces and then get the unique positive solutions for the iterative system. Finally, by constructing two examples, the feasibility and effectiveness of the obtained research result is verified.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16636-16648"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149029109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence, Multiplicity of Radial \u0000 \u0000 \u0000 p\u0000 \u0000 $$ p $$\u0000 -\u0000 \u0000 \u0000 k\u0000 \u0000 $$ k $$\u0000 -Convex Solutions for \u0000 \u0000 \u0000 p\u0000 \u0000 $$ p $$\u0000 -\u0000 \u0000 \u0000 k\u0000 \u0000 $$ k $$\u0000 -Hessian Systems With Gradient Terms","authors":"Guotao Wang,&nbsp;Huimin Li","doi":"10.1002/mma.70902","DOIUrl":"https://doi.org/10.1002/mma.70902","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;In this article, we mainly consider a class of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ p $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ k $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-Hessian system with gradient terms \u0000\u0000 &lt;/p&gt;&lt;div&gt;&lt;span&gt;&lt;span&gt;&lt;!--FIGURE--&gt;&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mtable&gt;\u0000 &lt;mtr&gt;\u0000 &lt;mtd&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mfenced&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 ","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16709-16718"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149027592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotical Profiles of A Delayed SIRV Foot-and-Mouth Disease Model with Age-Dependent Recovery","authors":"Dongxue Yan,&nbsp;Xinyu Bai,&nbsp;Hui Cao,&nbsp;Yu Cao","doi":"10.1002/mma.70909","DOIUrl":"https://doi.org/10.1002/mma.70909","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;Foot-and-mouth disease (FMD) is a highly contagious viral disease affecting cloven-hoofed animals, causing significant economic losses and posing a major global concern for livestock industries. In order to reveal the transmission dynamics of FMD, this paper formulates a novel FMD model with age structure for immunologically recovered animals and a time delay. The time delay occurs in the transition from immune to recovered animals. By reformulating the model as an abstract Cauchy problem, we demonstrate the existence of equilibria. The basic reproduction number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {mathcal{R}}_0 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; determines the FMD model's threshold dynamics: the virus-free equilibrium &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {E}_0 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is globally asymptotically stable when &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {mathcal{R}}_0&lt;1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. When &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {mathcal{R}}_0&gt;1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the endemic equilibrium &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {E}_{ast } $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; becomes unstable, indicating potential Hopf bifurcation","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16767-16789"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149027796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-Order Conservative Crank–Nicolson Difference Scheme for the Hirota–Satsuma Coupled KdV Equation","authors":"Xiaofeng Wang","doi":"10.1002/mma.70904","DOIUrl":"https://doi.org/10.1002/mma.70904","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper introduces a high-order, conservative Crank–Nicolson finite difference scheme for the Hirota–Satsuma coupled KdV (HS-KdV) equation, achieving fourth-order accuracy in space and second-order accuracy in time. Unlike existing low-order, non-conservative, or computationally expensive methods, the nonlinear system is solved via fixed-point iteration at each time step, exactly preserving discrete mass and energy. We rigorously prove solvability, conservation, stability, and convergence with order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>τ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ Oleft({tau}^2+{h}^4right) $$</annotation>\u0000 </semantics></math> in the discrete <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ {L}^2 $$</annotation>\u0000 </semantics></math> norm. By employing a novel cut-off function technique combined with mathematical induction, we derive optimal <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ {L}^{infty } $$</annotation>\u0000 </semantics></math>-norm error estimates without requiring a priori boundedness of the numerical solution. Extensive numerical experiments involving single, double, and triple solitary wave interactions confirm the theoretical convergence rates and exact preservation of invariants, demonstrating the scheme's robustness and superior conservation properties in simulating long-term, complex HS-KdV wave dynamics.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 15","pages":"16649-16675"},"PeriodicalIF":2.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149027833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}