Partial Differential Equations in Applied Mathematics最新文献_第3页

Traveling synchronized asymmetric two-waves in the propagation of the KdV and mKdV equations incorporating time-space dispersion terms 含时空色散项的KdV和mKdV方程的行同步非对称双波传播

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-25 DOI: 10.1016/j.padiff.2025.101319

Marwan Alquran, Imad Jaradat

Joseph and Egri revised the standard Korteweg-de Vries equation by replacing its third-order space dispersion term by space-time dispersions aiming to adjust the wave speed and preserve frequency stability. The aim of the current study is twofold. First, it demonstrates that the Joseph-Egri equation exhibits dynamical behavior similar to the Boussinesq model, particularly in the propagation of synchronized asymmetric two-wave structures. Second, it presents and investigates a modified extension of the Joseph-Egri equation, inspired by similar modifications of the Korteweg-de Vries and Benjamin-Bona-Mahony equations. Effective schemes, including the trigonometric and hyperbolic rational functions method and the extended tanh-coth expansion method, are implemented to extract diverse explicit solutions for both models. These analytical results are further supported by 2D and 3D plots. The obtained findings reveal distinct physical structures for the Joseph-Egri and its modified counterparts. Specifically, the Joseph-Egri equation supports cusp soliton, bell-shaped soliton, and periodic concave-pattern solutions, while the modified Joseph-Egri equation admits kink soliton, periodic kink-pattern, and periodic concave-convex wave solutions. We believe the reported results will contribute to a deeper understanding of the synchronized propagation of bidirectional waves in various nonlinear media and related physical applications.

Joseph和Egri修改了标准的Korteweg-de Vries方程，将其三阶空间色散项替换为时空色散项，旨在调整波速并保持频率稳定性。目前这项研究的目的是双重的。首先，它证明了Joseph-Egri方程表现出与Boussinesq模型相似的动力学行为，特别是在同步非对称双波结构的传播中。其次，在Korteweg-de Vries方程和Benjamin-Bona-Mahony方程的类似修正的启发下，提出并研究了Joseph-Egri方程的修正扩展。采用三角和双曲有理函数法、扩展tanh-coth展开法等有效的方法来提取两种模型的多种显式解。这些分析结果进一步得到了二维和三维图的支持。所获得的发现揭示了不同的物理结构，约瑟夫- egri和其改良的同类。其中，Joseph-Egri方程支持尖孤子、钟形孤子和周期凹型解，而改进的Joseph-Egri方程支持扭结孤子、周期扭结孤子和周期凹凸波解。我们相信报告的结果将有助于深入理解各种非线性介质中双向波的同步传播及其相关的物理应用。

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引用次数: 0

Caputo approach to transmission dynamics of paragonimiasis in humans, snails, and crustaceans 人类、蜗牛和甲壳类动物中吸虫病传播动力学的卡普托方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-24 DOI: 10.1016/j.padiff.2025.101313

Stephen Edward

Paragonimiasis, a parasitic infection caused by trematodes of the genus Paragonimus, is characterized by chronic cough, chest pain, and hemoptysis. This study introduces a novel fractional-order modelling technique to capture the impact of memory effects and control strategies on the transmission dynamics of Paragonimiasis. We develop a system of Caputo fractional differential equations supplemented with constant control measures–specifically, human treatment to reduce infection prevalence and snail molluscicide to target intermediate hosts. We analyze the fundamental properties of the model, including positivity, boundedness, existence, and uniqueness of solutions. Local stability of equilibrium states is investigated via Routh’s criterion, while Ulam-Hyer’s stability concepts confirm global stability. Numerical solutions obtained through the Adams-Bashforth-Moulton predictor-corrector method highlight the considerable influence of memory on disease progression. The results underscore the importance of integrating memory effects into control strategies: enhanced human treatment combined with improved snail molluscicide can significantly mitigate infection levels. These findings suggest that fractional derivatives offer a flexible, detailed lens for examining nonlocal disease dynamics processes and refining practical intervention approaches.

吸虫病是由吸虫病属吸虫引起的一种寄生虫感染，其特征是慢性咳嗽、胸痛和咯血。本研究引入了一种新的分数阶建模技术，以捕捉记忆效应和控制策略对肺吸虫病传播动力学的影响。我们开发了一个卡普托分数阶微分方程系统，辅以恒定的控制措施-具体来说，人类治疗以减少感染流行，并针对中间宿主使用杀螺剂。我们分析了该模型的基本性质，包括解的正性、有界性、存在性和唯一性。通过Routh准则研究了平衡态的局部稳定性，而Ulam-Hyer稳定性概念证实了平衡态的全局稳定性。通过Adams-Bashforth-Moulton预测校正方法获得的数值解突出了记忆对疾病进展的重要影响。结果强调了将记忆效应整合到控制策略中的重要性：强化人类治疗结合改良的蜗牛杀螺剂可以显著降低感染水平。这些发现表明，分数衍生物为检查非局部疾病动力学过程和改进实际干预方法提供了灵活、详细的视角。

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引用次数: 0

Intrinsic dynamics of lumps and multi-soliton solutions to the higher dimensional Boussinesq model 块的内在动力学和高维Boussinesq模型的多孤子解

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-21 DOI: 10.1016/j.padiff.2025.101320

M. Belal Hossen , Md. Towhiduzzaman , Mst. Shekha Khatun , Harun-Or- Roshid , Md. Amanat Ullah

This research investigates diverse wave behaviors of innovative higher dimensional Boussinesq model (BM) based on Hirota bi-linear technique. From this, firstly we derive lump and multiple soliton solutions. The study explores various dynamic behaviors, including interactions involving one up to four solitons. Additionally, the study analyzes breather waves, twofold periodic wave, periodic line lump wave, and the interactions among bell solitons. Other interactions analyze include lump wave with periodic wave, 1-stripe soliton and 2-stripe solitons. Many of these dynamic properties not yet explored in previous research. The trajectories of these solutions are visualized using Maple software, providing deeper insights into the model's dynamical behavior.

本文研究了基于Hirota双线性技术的创新型高维Boussinesq模型（BM）的多种波动行为。在此基础上，首先导出了整体解和多孤子解。该研究探索了各种动态行为，包括涉及一个到四个孤子的相互作用。此外，研究还分析了呼吸波、双周期波、周期线块状波以及钟孤子之间的相互作用。其他相互作用的分析包括块波与周期波、1条孤子和2条孤子。许多这些动态特性在以前的研究中尚未被探索。这些解决方案的轨迹使用Maple软件可视化，提供对模型动态行为的更深入的见解。

引用次数: 0

Construction of an infinite-dimensional family of exact solutions of the Klein–Gordon equation by the hypercomplex method 用超复方法构造Klein-Gordon方程无穷维精确解族

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-20 DOI: 10.1016/j.padiff.2025.101312

Vitalii Shpakivskyi

The Klein–Gordon equation is one of the fundamental equations of mathematical physics. Therefore, it is important to have exact solutions to this equation. There are many methods for constructing exact solutions to the Klein–Gordon equation. Naturally, that different methods give different exact solutions. In this paper, by the hypercomplex method, we construct an infinite-dimensional new family of exact solutions of the polynomial–exponential type to the Klein–Gordon equation.

克莱恩-戈登方程是数学物理的基本方程之一。因此，得到这个方程的精确解是很重要的。有许多方法可以构造Klein-Gordon方程的精确解。当然，不同的方法给出不同的精确解。本文利用超复方法，构造了Klein-Gordon方程的无穷维多项式-指数型精确解族。

引用次数: 0

Generalised Kolmogorov-Petrovskii-Piskunov equation of fractional order: Power series and shifted Legendre collocation methods 分数阶广义Kolmogorov-Petrovskii-Piskunov方程：幂级数和移位Legendre配置方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-20 DOI: 10.1016/j.padiff.2025.101318

Richard Olu Awonusika, Abayomi Samuel Oke

The classical Kolmogorov-Petrovskii-Piskunov (KPP) equation describes physical phenomena such as combustion, chemical reaction, evolution of dominant genes, and propagation of nerve pulses. In this paper, we present solutions of a time-fractional order generalised KPP equation using a power series method and Legendre collocation method. The fractional order derivative is described in the Caputo sense. The proposed power series method assumes that the solution of the governing problem can be represented by a fractional power series in time variable, with space-variable expansion coefficients. The nonlinear term is assumed to be analytic in the unknown, and thus, admits a power series representation. The generalised Cauchy product is applied to transform the power series in the unknown to one in the time-dependent variable. An explicit recursion formula for the variable expansion coefficient is then constructed. On the other hand, the shifted Legendre collocation method assumes that the solution of the proposed problem can be expressed as a shifted Legendre polynomial series, with constant expansion coefficients to be determined. Collocating at the shifted Legendre-Gauss nodes, we obtain a set of nonlinear algebraic equations. These algebraic equations are then solved for the unknown Legendre expansion coefficients using Newton’s iteration method. Convergence analyses of both methods are presented. Examples of the proposed problem involving quadratic, cubic, and exponential nonlinearities are considered to demonstrate the efficiency, accuracy, and reliability of the proposed techniques. The results obtained from both methods are in excellent agreement with the exact solutions.

经典的Kolmogorov-Petrovskii-Piskunov （KPP）方程描述了燃烧、化学反应、优势基因的进化和神经脉冲的传播等物理现象。本文利用幂级数法和勒让德配置法给出了一类时间分数阶广义KPP方程的解。分数阶导数是在卡普托意义上描述的。所提出的幂级数方法假定控制问题的解可以用一个时变分数阶幂级数来表示，其展开系数是空变的。假定非线性项在未知情况下是解析的，因此可以用幂级数表示。应用广义柯西积将未知的幂级数变换为时变的幂级数。然后构造了变膨胀系数的显式递推公式。另一方面，移位勒让德配置法假定所提问题的解可以表示为移位的勒让德多项式级数，其展开系数为常数。在移位的legende - gauss节点处，我们得到了一组非线性代数方程。然后用牛顿迭代法求解未知的勒让德展开系数。给出了两种方法的收敛性分析。所提出的问题涉及二次、三次和指数非线性的例子被认为是为了证明所提出的技术的效率、准确性和可靠性。两种方法得到的结果与精确解非常吻合。

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引用次数: 0

Invariant formulation of nonclassical symmetries and explicit solutions of Rosenau-Hyman equation along with bifurcation analysis Rosenau-Hyman方程非经典对称的不变量表述和显式解及其分岔分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-19 DOI: 10.1016/j.padiff.2025.101314

M.A. El-Shorbagy , Sonia Akram , Mati Ur Rahman , Hossam A. Nabwey

This study focuses on the Rosenau–Hyman equation, which is a fundamental model in nonlinear wave dynamics, and investigates it through the lens of nonclassical symmetry analysis. The approach employs symbolic computation to derive determining equations and uncover new invariant formulations, from which several explicit exact solutions are constructed. To further understand the system’s behavior, dynamical tools such as bifurcation analysis, sensitivity tests, Lyapunov exponents, and phase portraits are applied, highlighting the presence of stability transitions, multistability, and chaotic regimes. In addition, travelling wave solutions are obtained using the enhanced modified extended tanh function method (eMETFM), providing complementary wave structures. The findings deepen our understanding of nonlinear dispersive wave propagation and soliton interactions, with particular relevance to shallow water dynamics. More broadly, the developed solutions and their graphical interpretations contribute valuable insights for theoretical studies and applied research in fluid dynamics and wave modeling.

本文以非线性波动动力学的基本模型Rosenau-Hyman方程为研究对象，从非经典对称分析的角度对其进行了研究。该方法采用符号计算来推导确定方程并揭示新的不变公式，并从中构造出几个显式精确解。为了进一步了解系统的行为，应用了诸如分岔分析、灵敏度测试、李雅普诺夫指数和相位画像等动力学工具，突出了稳定性转变、多稳定性和混沌状态的存在。此外，使用增强修正扩展tanh函数法（eMETFM）获得行波解，提供互补波结构。这些发现加深了我们对非线性色散波传播和孤子相互作用的理解，特别是与浅水动力学有关。更广泛地说，开发的解决方案及其图形解释为流体动力学和波浪建模的理论研究和应用研究提供了宝贵的见解。

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引用次数: 0

Improved explicit finite difference method for extended shallow water partial differential equation 扩展浅水偏微分方程的改进显式有限差分法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-17 DOI: 10.1016/j.padiff.2025.101316

Saeed Ahmed Rajput , Shakeel Ahmed Kamboh , Khuda Bux Amur , Afaque Ahmed Bhutto

The shallow water and Boussinesq equations, being highly nonlinear coupled partial differential equations, form a significant basis for the simulation of flow phenomena within the surface and subsurface domains. The application of these equations extends into several fields in the area of engineering and science, including the simulation of Kelvin wake waves, scenarios involving dam break, propagation of flood waves and flow over a bump. In this paper, the shallow water equation for surface flow regions and the Boussinesq partial differential equation for subsurface flow regions are unified into a single set of extended shallow water nonlinear partial differential equations which is applicable for both flow domains. These equations are solved numerically using an improved two-step Lax-Wendroff method with sixth-order accuracy. The method is validated by benchmarking against existing experimental data, which shows a percentage error of less than 1%, confirming its high accuracy and reliability. Moreover, the method is applied to two test cases. The first case is flow over a bump, where the relative error for surface and subsurface flow regions is as low as 10⁻⁸ at t = 100. In the second case, Kelvin wake wave is investigated with the improved sixth-order Lax-Wendroff scheme predicting an arm angle of 20.6955°, while a theoretical particle angle of 19.47° was observed. The results show that this method is good enough and useful for the simulation of surface and subsurface flows phenomena.

浅水方程和Boussinesq方程是高度非线性耦合的偏微分方程，是模拟地表和地下区域内流动现象的重要基础。这些方程的应用扩展到工程和科学领域的几个领域，包括开尔文尾流的模拟，涉及大坝溃坝的场景，洪水波的传播和流过凸起的水流。本文将表面流区的浅水方程和地下流区的Boussinesq偏微分方程统一为一组适用于两个流域的扩展浅水非线性偏微分方程。采用改进的六阶精度的两步Lax-Wendroff方法对这些方程进行了数值求解。通过对已有实验数据的基准测试，验证了该方法的准确性和可靠性，误差小于1%。此外，还将该方法应用于两个测试用例。第一种情况是通过凸起的流动，在t = 100时，表面和地下流动区域的相对误差低至10−8。在第二种情况下，采用改进的六阶Lax-Wendroff格式研究开尔文尾流，预测臂角为20.6955°，而理论粒子角为19.47°。结果表明，该方法可以很好地模拟地表和地下流动现象。

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引用次数: 0

Dynamics and modeling of Malaria disease with vector mortality rate and host transmission by using Piecewise Fractional Operator 基于分段分数算子的疟疾病媒死亡率和宿主传播动力学与建模

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-16 DOI: 10.1016/j.padiff.2025.101309

Muhammad Farman , Saba Jamil , Evren Hincal , Ali Akgul , Muhammad Umer Saleem , Dumitru Baleanu

Malaria remains one of the most persistent vector-borne diseases, requiring advanced modeling tools to capture its complex transmission dynamics. This study introduces a novel piecewise Caputo fractional operator with a singular kernel to model malaria transmission between humans and mosquito vectors. The proposed operator allows the system to switch between different fractional dynamics across subintervals, thereby capturing sudden behavioral or environmental changes an ability not present in traditional fractional- or integer-order models. The existence and uniqueness of the systems solutions are rigorously established using the Arzelà–Ascoli and Schauder fixed-point theorems, ensuring mathematical validity. The basic reproduction number is derived via the next-generation matrix approach and analyzed through sensitivity indices to identify key epidemiological parameters influencing transmission. Furthermore, the generalized Ulam–Hyers stability confirms robustness under small perturbations. Numerical simulations based on the Newton polynomial scheme reveal crossover behavior between subintervals and demonstrate that lower fractional orders intensify memory effects, leading to delayed but more stable epidemic responses. Overall, the piecewise Caputo framework enhances the modeling of malaria dynamics by integrating memory-dependent and regime-switching properties, offering a more biologically realistic approach to designing and timing intervention strategies.

疟疾仍然是最顽固的病媒传播疾病之一，需要先进的建模工具来捕捉其复杂的传播动态。本文引入了一种新颖的带奇异核的分段Caputo分数算子来模拟人与蚊子媒介之间的疟疾传播。所提出的算子允许系统在子区间的不同分数动态之间切换，从而捕获突然的行为或环境变化，这是传统分数阶或整数阶模型所不具备的能力。利用Arzelà-Ascoli和Schauder不动点定理严格地建立了系统解的存在唯一性，保证了系统解的数学有效性。通过新一代矩阵法导出基本繁殖数，并通过敏感性指数进行分析，确定影响传播的关键流行病学参数。此外，广义Ulam-Hyers稳定性证实了小扰动下的鲁棒性。基于牛顿多项式格式的数值模拟揭示了子区间之间的交叉行为，并表明较低分数阶强化了记忆效应，导致延迟但更稳定的流行病响应。总的来说，分段Caputo框架通过整合记忆依赖和状态切换特性增强了疟疾动力学的建模，为设计和定时干预策略提供了一种更现实的生物学方法。

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引用次数: 0

On nonlinear ordinary differential system for infectious disease: A neuro-swarming intelligence scheme 传染病的非线性常微分系统：一种神经群智能方案

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-16 DOI: 10.1016/j.padiff.2025.101317

Farhad Muhammad Riaz , Junaid Ali Khan , Khalil Ur Rehman , Wasfi Shatanawi

It is believed that nonlinear ordinary differential systems are essential for epidemic modeling because they can narrate the complex and unpredictable aspects of the spread of infectious diseases. Therefore, one can use nonlinear differential systems to propose control measures for the spread of diseases, and it remains a challenging task for researchers to obtain the best solution for such nonlinear systems. The present article offers the best solution remedy for nonlinear differential systems. To be more specific, to address the nonlinear dynamics of the spread of COVID-19, we propose an intelligent computational framework based on single-layer feed-forward artificial neural networks (FF-ANNs) and the optimization techniques of global and local search approaches. The SEIR-NDC model is solved by using a global-local search strategy called PSONM, which combines Particle Swarm Optimization (PSO) and the Nelder-Mead Simplex (NM). The differential nonlinear mathematical model based on SEIR-NDC and initial conditions is used in the hybrid PSONM combination to optimize an error-based fitness function. Ten neurons are used to demonstrate the numerical performance of the SEIR-NDC nonlinear model using ANN methods in conjunction with PSO-SQP. The correctness of the developed scheme is testified through the comparative analysis of the reference solution and the obtained outcomes. The absolute error performances are reported within appropriate ranges for every class of the SEIR-NDC model. It is found that the AE lies in the range 10^–14 and 10^–17. The statistical analysis is presented to verify the developed scheme's stability, convergence, and robustness. The statistical measures, i.e., mean square error falls between 10^–8 and 10^–13, while the mean absolute deviation falls between 10^–8 and 10^–11. We believe that the outcomes of the present analysis will be a helping hand in encountering nonlinear differential systems that are subject to practical applications.

人们认为，非线性常微分系统对于流行病建模是必不可少的，因为它们可以描述传染病传播的复杂和不可预测的方面。因此，人们可以利用非线性微分系统来提出控制疾病传播的措施，但如何获得这种非线性系统的最佳解仍然是研究人员面临的一个挑战。本文给出了非线性微分系统的最佳解补救方法。具体而言，为了解决COVID-19传播的非线性动力学问题，我们提出了一种基于单层前馈人工神经网络（ff - ann）和全局和局部搜索方法优化技术的智能计算框架。SEIR-NDC模型采用粒子群算法（PSO）和Nelder-Mead单纯形算法（NM）相结合的全局局部搜索策略PSONM进行求解。将基于SEIR-NDC和初始条件的微分非线性数学模型应用于混合PSONM组合中，对基于误差的适应度函数进行优化。采用10个神经元对SEIR-NDC非线性模型进行了数值模拟，并结合PSO-SQP方法对模型进行了仿真。通过对参考解和所得结果的对比分析，验证了所提方案的正确性。在适当的范围内报告了每一类SEIR-NDC模型的绝对误差性能。发现声发射分布在10-14和10-17范围内。通过统计分析验证了所提方案的稳定性、收敛性和鲁棒性。统计度量，即均方误差在10-8 ~ 10-13之间，平均绝对偏差在10-8 ~ 10-11之间。我们相信，本分析的结果将有助于遇到实际应用的非线性微分系统。

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引用次数: 0

Buoyancy-induced nanofluid circulation in a novel configuration of a porous square cavity 一种新型多孔方腔结构中浮力诱导的纳米流体循环

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-14 DOI: 10.1016/j.padiff.2025.101315

Muhammad Faisal , Talha Anwar , Farah Javed

Efficient thermal management is vital in modern mechanical and energy systems, where conventional engine oils often exhibit limited heat transfer capabilities. This study investigates the enhancement of thermal convection in engine oil by dispersing molybdenum tetrasulfide nanoparticles (MoS₄) to form a high-performance nanofluid. The natural convection behavior of this nanofluid is analyzed within a square porous cavity featuring uniformly heated horizontal walls and isothermally cooled vertical walls. The governing equations are developed using scaling variables and the Boussinesq approximation and solved numerically through the finite element method. The effects of nanoparticle volume fraction (0–0.07), Rayleigh number (10³–10⁶), and Darcy number (10⁻⁵–10⁻²) are systematically examined. Results show that increasing the MoS₄ nanoparticle concentration substantially enhances convective heat transfer, with the average Nusselt number rising by up to 28 % and the peak stream function reaching 17.0 at a volume fraction of 0.07 under low Darcy and Rayleigh conditions. These findings demonstrate that even minimal nanoparticle addition can significantly improve the heat transport capability of engine oils in porous enclosures. The study introduces a novel combination of molybdenum tetrasulfide-based nanofluids and porous media analysis, extending beyond prior work by quantifying the coupled effects of nanoparticle concentration and porous resistance on buoyancy-driven flow performance.

在现代机械和能源系统中，高效的热管理是至关重要的，传统的发动机油通常表现出有限的传热能力。本研究通过分散四硫化钼纳米颗粒（MoS₄）形成高性能纳米流体来增强机油中的热对流。在具有均匀加热的水平壁面和等温冷却的垂直壁面的方形多孔腔内，分析了这种纳米流体的自然对流行为。利用尺度变量和Boussinesq近似建立了控制方程，并通过有限元方法进行了数值求解。系统地考察了纳米颗粒体积分数（0-0.07）、瑞利数（103-10⁶）和达西数（10 -10⁻2）的影响。结果表明：在低达西和瑞利条件下，当体积分数为0.07时，mos4纳米颗粒浓度的增加显著增强了对流换热，平均Nusselt数提高了28%，峰值流函数达到17.0；这些发现表明，即使是最小的纳米颗粒添加也可以显著提高多孔外壳中发动机油的传热能力。该研究引入了基于四硫化钼的纳米流体和多孔介质分析的新组合，通过量化纳米颗粒浓度和多孔阻力对浮力驱动流动性能的耦合影响，扩展了之前的工作。

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引用次数: 0