Partial Differential Equations in Applied Mathematics最新文献

英文中文

Investigation of lump, breather and multi solitonic wave solutions to fractional nonlinear dynamical model with stability analysis 分式非线性动力学模型的肿块波、呼吸波和多孤子波解的研究与稳定性分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-15 DOI: 10.1016/j.padiff.2024.100955

In the current research, the new extended direct algebraic method (NEDAM) and the symbolic computational method, along with different test functions, the Hirota bilinear method, are capitalized to secure soliton and lump solutions to the

(2 + 1)

-dimensional fractional telecommunication system. Consequently, we derive soliton solutions with sophisticated structures, such as mixed trigonometric, rational, hyperbolic, unique, periodic, dark-bright, bright-dark, and hyperbolic. We also developed a lump-type solution that includes rogue waves and breathers for curiosity’s intellect. These features are important for controlling extreme occurrences in optical communications. Additionally, we investigate modulation instability (MI) in the context of nonlinear optical fibres. Understanding MI is essential for developing systems that may either capitalize on its positive features or mitigate its adverse effects. Also, a comprehensive sensitivity analysis of the observed model is carried out to evaluate the influence of different factors. 3D surfaces and 2D visuals, contours, and density plots of the outcomes are represented with the help of a computer application. Our findings demonstrate the potential of using soliton theory and advanced nonlinear analysis methods to enhance the performance of telecommunication systems.

在当前的研究中，我们利用新扩展直接代数法（NEDAM）和符号计算法，以及不同的测试函数和 Hirota 双线性方法，来确保 (2+1)-dimensional 分数电信系统的孤子和块解。因此，我们得出了具有复杂结构的孤子解，如混合三角函数解、有理函数解、双曲线解、唯一解、周期解、暗-亮解、亮-暗解和双曲线解。我们还开发了一种包含流氓波和呼吸器的块状解，以满足好奇心的智力需求。这些特征对于控制光通信中的极端现象非常重要。此外，我们还研究了非线性光纤的调制不稳定性（MI）。了解调制不稳定性对于开发可利用其积极特性或减轻其不利影响的系统至关重要。此外，我们还对观测到的模型进行了全面的敏感性分析，以评估不同因素的影响。在计算机应用程序的帮助下，结果的三维表面和二维视觉效果、等值线和密度图都得到了体现。我们的研究结果证明了利用孤子理论和先进的非线性分析方法提高电信系统性能的潜力。

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引用次数: 0

An iterative approach for addressing monotone inclusion and fixed point problems with generalized demimetric mappings 解决广义非计量映射的单调包含和定点问题的迭代方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-14 DOI: 10.1016/j.padiff.2024.100953

Throughout this study, we present a new algorithm for finding the common solution of a finite family of monotone inclusion and the fixed point problem of a finite family of generalized demimetric operators in the context of a real Hilbert space and show its strong convergence. Moreover, we utilize our result to solve the minimization problem.

在本研究中，我们提出了一种新算法，用于在实希尔伯特空间的背景下寻找有限单调包含族的公共解和有限广义非计量算子族的定点问题，并证明了其强大的收敛性。此外，我们还利用我们的结果解决了最小化问题。

引用次数: 0

New modifications of natural transform iterative method and q-homotopy analysis method applied to fractional order KDV-Burger and Sawada–Kotera equations 应用于分数阶 KDV-Burger 和 Sawada-Kotera 方程的自然变换迭代法和 q-homotopy 分析法的新修正

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-11 DOI: 10.1016/j.padiff.2024.100950

This manuscript presents enhanced versions of two methods: the natural transform iterative method (NTIM) and the q-homotopy analysis method (q-HAM). These methods harness concepts from Fractional Calculus, particularly leveraging the Caputo fractional derivative operator, to successfully manage the complexities of fractional-order systems. To validate their accuracy and efficiency, we applied the proposed techniques to FPDEs like the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations. Our outcomes, which closely resemble the exact solutions, demonstrate how useful NTIM and q-HAM are for solving difficult FPDEs and improving the study of fractional calculus.

本手稿介绍了两种方法的增强版本：自然变换迭代法（NTIM）和 q-同调分析法（q-HAM）。这些方法利用分数微积分的概念，特别是利用卡普托分数导数算子，成功地处理了分数阶系统的复杂性。为了验证其准确性和效率，我们将所提出的技术应用于分数阶 KDV-Burger 和五阶 Sawada-Kotera 方程等 FPDE。我们的结果与精确解非常相似，这表明 NTIM 和 q-HAM 对于解决困难的 FPDE 和改进分数微积分研究非常有用。

引用次数: 0

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation 对流卡恩-希利亚德方程的存在性、稳定性和二维不变流形数

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-09 DOI: 10.1016/j.padiff.2024.100946

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.

In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

我们研究了众所周知的非线性卡恩-希利亚德演化方程的广义版本，并补充了周期性边界条件。我们研究了空间均匀平衡态附近的局部分岔。我们证明了所研究的边界值问题存在有限或可数平衡态集的可能性，在满足适当条件的情况下，平衡态集附近存在二维不变流形，其中充满了在演化变量中具有周期性的解。此外，我们还推导出了这些周期解的渐近公式。最后，我们研究了不变流形和属于不变流形的解的稳定性。为了分析分岔问题，我们使用了无穷维相动力系统理论中的方法，即不变流形方法和正常形式方法。

引用次数: 0

A two-strain COVID-19 co-infection model with strain 1 vaccination 接种 1 号菌株疫苗的双菌株 COVID-19 协同感染模型

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-09 DOI: 10.1016/j.padiff.2024.100945

COVID-19 has caused substantial morbidity and mortality worldwide. Previous models of strain 1 vaccination with re-infection when vaccinated, as well as infection with strain 2 did not consider co-infected classes. To fill this gap, a two co-circulating COVID-19 strains model with strain 1 vaccination, and co-infected is formulated and theoretically analyzed. Sufficient conditions for the stability of the disease-free equilibrium and single-strain 1 and -strain 2 endemic equilibria are obtained. Results show as expected that (1) co-infected classes play a role in the transmission dynamics of the disease (2) a high efficacy vaccine could effectively help mitigate the spread of co-infection with both strain 1 and 2 compared to the low-efficacy vaccine. Sensitivity analysis reveals that the main drivers of the effective reproduction number

R_{e}

are primarily the effective contact rate for strain 2 (

β_{2}

), the strain 2 recovery rate (

τ_{2}

), and the vaccine efficacy or infection reduction due to the vaccine (

η

). Thus, implementing intervention measures to mitigate the spread of COVID-19 should not ignore the co-infected individuals who can potentially spread both strains of the disease.

COVID-19 在全球造成了大量的发病和死亡。以往的 1 号菌株疫苗接种后再感染以及 2 号菌株感染的模型没有考虑共同感染的类别。为了填补这一空白，我们建立了一个包含 1 号菌株接种和共同感染的两种共同循环 COVID-19 菌株模型，并对其进行了理论分析。得到了无病平衡和单一菌株 1 及菌株 2 流行平衡稳定的充分条件。结果如预期所示：(1) 共感染人群在疾病的传播动态中发挥了作用；(2) 与低效疫苗相比，高效疫苗可有效缓解 1 号和 2 号菌株共感染的传播。敏感性分析表明，有效繁殖数 Re 的主要驱动因素是毒株 2 的有效接触率 (β2)、毒株 2 的恢复率 (τ2) 和疫苗效力或疫苗造成的感染减少 (η)。因此，在采取干预措施以减少 COVID-19 的传播时，不应忽视合并感染者，因为他们有可能同时传播两种毒株。

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引用次数: 0

Analysis of boundary layer flow of a Jeffrey fluid over a stretching or shrinking sheet immersed in a porous medium 杰弗里流体在浸入多孔介质的伸缩片上的边界层流动分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-06 DOI: 10.1016/j.padiff.2024.100951

Heat transfer optimization is critical in many applications, such as heat exchangers, electric coolers, solar collectors, and nuclear reactors. The current work looks at the thermohydraulic behavior of Jeffery fluid flow along a plane containing a magnetic field, a non-uniform heat source/sink, and a porous media. Numerical solutions are derived using the Runge-Kutta 4th-order approach and the shooting method. Graphs show how Prandtl number (Pr), thermal stratification (e₁), Jeffery parameter (λ₁), porous parameter (λ₂), magnetic field (M), and heat generation/absorption (γ, a, b) affect velocity and temperature profiles. The results show that thermal stratification increases fluid velocity and temperature, whereas heat source/sink parameters have the reverse effect on heat transfer, and raising the Jeffrey parameter reduces velocity and increases boundary layer thickness. There is extremely high agreement with experimental data from the literature. This work illustrates the utility of hydromagnetic properties in modelling fluid flow over stretching/shrinking sheets in porous media.

在热交换器、电冷却器、太阳能集热器和核反应堆等许多应用中，传热优化至关重要。目前的研究着眼于杰弗里流体沿含有磁场、非均匀热源/散热器和多孔介质的平面流动的热流体力学行为。使用 Runge-Kutta 四阶方法和射击法得出了数值解。图表显示了普朗特数 (Pr)、热分层 (e1)、杰弗里参数 (λ1)、多孔参数 (λ2)、磁场 (M) 和发热/吸热 (γ、a、b) 如何影响速度和温度曲线。结果表明，热分层会增加流体速度和温度，而热源/沉降参数对热传递的影响相反，提高杰弗里参数会降低速度并增加边界层厚度。与文献中的实验数据具有极高的一致性。这项工作说明了水磁特性在模拟多孔介质中拉伸/收缩片上的流体流动时的实用性。

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引用次数: 0

A new solution approach to proportion delayed and heat like fractional partial differential equations 比例延迟和类热分式偏微分方程的新求解方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-05 DOI: 10.1016/j.padiff.2024.100948

The importance of fractional partial differential equations (FPDEs) may be observed in many fields of science and engineering. On the same hand their solutions and the approaches for the same are also very important to notice due to the effectiveness of the methods and accuracy of the results. This work discusses the diverse estimated analytic description of fractional partial differential equations (with proportion delay and heat like equation), applying the Iterative Laplace Transform Method. The specified method represents a significant advancement in the tool case of applied mathematicians and scientists. Its ability to efficiently and accurately solve complex differential equations, especially FPDEs. Here in this work, the solution of four test problems of FPDEs related to proportion delay and heat like equations is obtained for testing the validity and asset of the Iterative Laplace Transform Method. Further their numerical and graphical interpretations are also mentioned.

分数偏微分方程（FPDEs）在科学和工程学的许多领域都具有重要意义。另一方面，由于方法的有效性和结果的准确性，它们的解法和解决方法也非常重要。本著作讨论了分式偏微分方程（带比例延迟和类热方程）的各种估计分析描述，并应用了迭代拉普拉斯变换方法。该方法代表了应用数学家和科学家在工具方面的重大进步。它能够高效、准确地求解复杂微分方程，尤其是 FPDE。在这项工作中，为了测试迭代拉普拉斯变换方法的有效性和资产，我们获得了与比例延迟和热方程相关的 FPDEs 的四个测试问题的解法。此外，还提到了它们的数值和图形解释。

引用次数: 0

Optimal control strategies for infectious disease management: Integrating differential game theory with the SEIR model 传染病管理的最佳控制策略：将微分博弈论与 SEIR 模型相结合

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-04 DOI: 10.1016/j.padiff.2024.100943

The rapid spread of infectious diseases poses a critical threat to global public health. Traditional frameworks, such as the Susceptible–Exposed–Infectious–Recovered (SEIR) model, have been crucial in elucidating disease dynamics. Nonetheless, these models frequently overlook the strategic interactions between public health authorities and individuals. This research extends the classic SEIR model by incorporating differential game theory to analyze optimal control strategies. By modeling the conflicting objectives of public health authorities aiming to minimize infection rates and intervention costs, and individuals seeking to reduce their infection risk and inconvenience, we derive a Nash equilibrium that provides a balanced approach to disease management. Using Picard’s iterative method, we solve the extended model to determine dynamic, optimal control strategies, revealing oscillatory behavior in public health interventions and individual preventive measures. This comprehensive approach offers valuable insights into the dynamic interactions essential for effective infectious disease control.

传染病的快速传播对全球公共卫生构成了严重威胁。传统的框架，如 "易感-暴露-传染-康复"（SEIR）模型，对于阐明疾病动态至关重要。然而，这些模型往往忽略了公共卫生机构与个人之间的战略互动。本研究结合微分博弈论分析最优控制策略，对经典的 SEIR 模型进行了扩展。公共卫生机构的目标是最大限度地降低感染率和干预成本，而个人的目标是减少感染风险和不便，通过模拟这两种目标之间的冲突，我们得出了一种纳什均衡，为疾病管理提供了一种平衡的方法。利用皮卡尔迭代法，我们求解了扩展模型，以确定动态的最优控制策略，揭示了公共卫生干预和个人预防措施中的振荡行为。这种综合方法为有效控制传染病所必需的动态互动提供了宝贵的见解。

引用次数: 0

Numerical simulation of an effective transform mechanism with convergence analysis of the fractional diffusion-wave equations 有效转换机制的数值模拟与分数扩散波方程的收敛分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-10-03 DOI: 10.1016/j.padiff.2024.100947

In the current study, we solve two very important mathematical models, such as the time fractional-order space-fractional telegraph and diffusion-wave equations using a reliable technique called the Adomian decomposition natural method (ADNM), which combines Adomian decomposition and natural transform. The diffusion wave equation describes the flood wave propagation, which is used in solving overland and open channel flow problems. For this reason, it is critical to fully understand and effectively solve the diffusion wave equations. Because telegraph equations are crucial for modeling and developing voltage or frequency transmission, they are widely used in physics and engineering. Furthermore, the designing process is greatly impacted by the uncertainty in the system parameters. For nonlinear ordinary differential equations based on the theorem of Banach fixed point, we provide existence and uniqueness theorem proofs. The present approach has been successfully used to explore exact solutions for time fractional-order and space fractional-order applications. The results show how effective and valuable the ADNM. This paper presents a methodology that will be used in future work to address similar nonlinear problems related to fractional calculus.

在本研究中，我们使用一种可靠的技术，即阿多米分解自然法（ADNM），结合阿多米分解和自然变换，求解了两个非常重要的数学模型，如时间分数阶空间分数电报方程和扩散波方程。扩散波方程描述了洪水波的传播，用于解决陆地和明渠水流问题。因此，充分理解并有效求解扩散波方程至关重要。由于电报方程对于电压或频率传输的建模和开发至关重要，因此在物理学和工程学中得到了广泛应用。此外，系统参数的不确定性对设计过程影响很大。对于基于巴拿赫定点定理的非线性常微分方程，我们提供了存在性和唯一性定理证明。本方法已成功用于探索时间分数阶和空间分数阶应用的精确解。结果显示了 ADNM 的有效性和价值。本文提出的方法将在未来的工作中用于解决与分数微积分相关的类似非线性问题。

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引用次数: 0

Thermal analysis of hybrid nano-fluids: Modeling and non-similar solutions 混合纳米流体的热分析：建模与非相似解

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2024-09-30 DOI: 10.1016/j.padiff.2024.100944

The thermal analysis of hybrid nano-fluids is a significant research area with diverse applications in industries such as paint, electronics, and mechanical engineering. Existing literature provides limited solutions to the governing equations for the flow of these fluids. Modeling and deriving non-similar solutions for these equations pose interesting and challenging mathematical problems. This study focuses on investigating heat transfer in the flow of two types of nano-fluids, specifically Al_{2O_{3/H_2O}} micropolar nano-fluid and Al₂O₃ + Ag/H₂O hybrid nano-fluid, near an isothermal sphere. Conservation laws are employed to formulate the mathematical problem, and by normalizing the variables, the governing equations are converted into a set of dimensionless partial differential equations. Non-similar solutions are then obtained using numerical methods. A comparative analysis is carried out to assess the influence of various parameters on different profiles and engineering quantities for both types of nano-fluids. Both linear and rotational velocities fall down near the surface of sphere with rising microstructure in hybrid nanofluid. The micro-rotation parameter rises the temperature profile while reduces the Nusselt number of both traditional Al₂O₃/water based nanofluid as well as hybrid nanofluid.

混合纳米流体的热分析是一个重要的研究领域，在涂料、电子和机械工程等行业有着广泛的应用。现有文献对这些流体流动的控制方程提供了有限的解决方案。这些方程的建模和非相似解的推导提出了有趣而具有挑战性的数学问题。本研究的重点是研究两种纳米流体（特别是 Al2O3/H2O 微极性纳米流体和 Al2O3 + Ag/H2O 混合纳米流体）在等温球附近流动时的传热问题。数学问题的表述采用了守恒定律，通过对变量进行归一化处理，将控制方程转换为一组无量纲偏微分方程。然后使用数值方法获得非相似解。对两种纳米流体进行了比较分析，以评估各种参数对不同剖面和工程量的影响。在混合纳米流体中，随着微结构的上升，线速度和旋转速度在球体表面附近都会下降。微旋转参数提高了温度曲线，同时降低了传统 Al2O3/水基纳米流体和混合纳米流体的努塞尔特数。

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