Partial Differential Equations in Applied Mathematics最新文献_第5页

Heat transfer analysis of Ethylene Glycol based hybrid nanofluid (Au–Ag) flow over a porous medium with gyrotactic microorganisms: Levenberg–Marquardt backpropagation approach 乙二醇基混合纳米流体（Au-Ag）在多孔介质上与回旋微生物流动的传热分析：Levenberg-Marquardt反向传播方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-15 DOI: 10.1016/j.padiff.2025.101292

R. Shobika , B. Vennila , K. Loganathan

The proposed study employs the Levenberg–Marquardt backpropagation approach with artificial neural networks to examine the heat transfer in hybrid nanofluid flow over a porous embedded vertical stretching sheet in a Darcy–Forchheimer medium. This study seeks to investigate the interplay between gyrotactic microorganisms, magnetic fields, mixed convection, and temperature in hybrid nanofluids including Silver (Ag), Gold (Au), and the base fluid Ethylene Glycol

C_{2} H_{6} O_{2}

utilizing the Cattaneo–Christov heat flux model. It improves our comprehension of their behavior and potential uses. This intricate system of highly non-linear governing equations is simplified to a set of ordinary differential equations by similarity transformations and solved numerically using the Bvp4c method. Alongside the numerical method, Artificial Neural Networks (ANNs) are utilized to precisely illustrate intricate patterns, with an Mean Square Error (MSE) of 0.00043 and strengthening the impact of the numerical findings. This study demonstrates that the utilization of Au–Ag/

C_{2} H_{6} O_{2}

hybrid nanoparticles enhances thermal conductivity, augments volume fraction, and indicates that the application of a magnetic field and thermal radiation markedly improves the dispersion of microorganisms and the formation of hybrid nanofluids, resulting in elevated heat transfer rates. Especially, ANN-based regressor for sensitivity analysis is employed to forecast essential physical parameters, including the skin friction coefficient, Nusselt number, Sherwood number, and Density of Microorganisms, while also assessing the significance of factors affecting nanofluid properties, thereby demonstrating excellent concordance with prior studies and validating the robustness of the proposed model.

该研究采用Levenberg-Marquardt反向传播方法和人工神经网络来研究混合纳米流体在达西-福希海默介质中多孔嵌入垂直拉伸片上的传热。本研究旨在利用Cattaneo-Christov热流密度模型，研究包括银（Ag）、金（Au）和基液乙二醇C2H6O2在内的混合纳米流体中回旋微生物、磁场、混合对流和温度之间的相互作用。它提高了我们对它们的行为和潜在用途的理解。通过相似变换将这一复杂的高度非线性控制方程组简化为一组常微分方程，并采用Bvp4c方法进行数值求解。除了数值方法，人工神经网络（ann）被用来精确地说明复杂的模式，均方误差（MSE）为0.00043，并加强了数值结果的影响。本研究表明，使用Au-Ag /C2H6O2杂化纳米颗粒可以提高导热性，增加体积分数，并表明磁场和热辐射的应用显著改善了微生物的分散和杂化纳米流体的形成，从而提高了传热速率。特别是，基于神经网络的敏感性分析回归因子用于预测基本物理参数，包括皮肤摩擦系数，Nusselt数，Sherwood数和微生物密度，同时还评估了影响纳米流体性质的因素的重要性，从而证明了与先前研究的良好一致性，并验证了所提出模型的稳健性。

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引用次数: 0

Higher dimensional nonlinear model arising to the diversity of fields: Dynamics of wave structures with M-fractional derivative 由场的多样性引起的高维非线性模型：具有m阶导数的波结构动力学

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-15 DOI: 10.1016/j.padiff.2025.101284

Usman Younas , Jan Muhammad , Ejaz Hussain

In this work, we investigates the dynamics of waves of a higher dimensional nonlinear partial differential equation known as P-type (3+1)-dimensional model. This model is employed for modeling plasma waves and instabilities in plasma physics. In the context of quantum field theory and other domains, the (3+1)-dimensional p-type model is a theoretical construct that is employed to investigate a diverse array of physical processes. In addition to magnetism and the conventional theory of particle physics, this model elucidates the specific proper ties of materials and spontaneous processes in solid-state structures. In this study, we utilize the M-fractional derivative and an appropriate wave transformation for converting the governing equation into an ordinary differential equation, thereby attaining the desired exact solutions. The generalized Arnous method, F-expansion approach, and Kumar–Malik method are employed to acquire solutions. By employing these techniques, a variety of solutions are attained, such as combined, bright, dark, bright and dark, mixed, and singular solitons. The model under investigation contains a significant number of soliton solution structures. Moreover, we represent the behaviors of the solutions in 2D and 3D graphs using the appropriate parameter values. The findings presented in this study can improve the nonlinear dynamical characteristics of a specific system and validate the efficacy of the used methodologies. Our findings provide useful insights into the intricacy of nonlinear equations, enhancing prior research on the subject through the introduction of innovative techniques and the discovery of a significant number of solutions that have wide-ranging relevance.

在这项工作中，我们研究了被称为p型（3+1）维模型的高维非线性偏微分方程的波的动力学。该模型用于模拟等离子体波和等离子体物理中的不稳定性。在量子场论和其他领域的背景下，（3+1）维p型模型是一种用于研究各种物理过程的理论结构。除了磁性和粒子物理的传统理论，该模型阐明了固体结构中材料和自发过程的特定适当联系。在本研究中，我们利用m阶导数和适当的波动变换将控制方程转化为常微分方程，从而获得所需的精确解。采用广义Arnous法、f展开法和Kumar-Malik法求解。通过使用这些技术，可以获得多种解决方案，例如组合孤子、亮孤子、暗孤子、明暗孤子、混合孤子和奇异孤子。所研究的模型包含大量的孤子解结构。此外，我们用适当的参数值在二维和三维图形中表示解的行为。本研究的发现可以改善特定系统的非线性动力学特性，并验证所使用方法的有效性。我们的发现为非线性方程的复杂性提供了有用的见解，通过引入创新技术和发现大量具有广泛相关性的解决方案，加强了对该主题的先前研究。

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引用次数: 0

Mathematical model of immune response to Hepatitis C virus (HCV) disease 丙型肝炎病毒（HCV）疾病免疫反应的数学模型

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-13 DOI: 10.1016/j.padiff.2025.101275

Amna H.A. Ibrahim, Hermane Mambili Mamboundou

This paper presents a mathematical model that comprehensively analyzes the dynamics of Hepatitis C Virus (HCV) infection. The model based on a system of nonlinear differential equations captures the interactions between liver cells (hepatocytes), the Hepatitis C virus, immune cells, and cytokines dynamics. We establish the well-posedness of the model within a biologically feasible region. Using the next-generation method, we calculate the basic reproduction number,

ℜ_{0}

, a threshold parameter that determines whether the infection will spread or die. A sensitivity analysis is also performed to identify the parameters that most significantly influence this number. We derive the conditions for the stability of disease-free and endemic equilibrium. The model is then used to investigate the system’s behavior under various scenarios: a weak immune response, the absence of T helper cell support, and a strong immune response. Our simulations show that the lack of interleukin-2 (IL-2) significantly affects the activation of cytotoxic T lymphocyte (CTLs). These results underscore the importance of including T helper cells, Interferon

- γ

(IFN-

γ

) and IL-2 for an accurate representation of the dynamics of hepatitis C virus infection. Ultimately, this study deepens our understanding of the dynamics of HCV infection and simplifies how specific immune components shape the course of the disease.

本文提出了一个全面分析丙型肝炎病毒（HCV）感染动态的数学模型。该模型基于非线性微分方程系统，捕获肝细胞（肝细胞）、丙型肝炎病毒、免疫细胞和细胞因子动力学之间的相互作用。我们在一个生物可行的区域内建立了模型的适定性。使用新一代方法，我们计算基本繁殖数，即确定感染是否会传播或死亡的阈值参数。还进行了敏感性分析，以确定对该数字影响最大的参数。导出了无病平衡和地方病平衡稳定的条件。然后使用该模型来研究系统在各种情况下的行为：弱免疫反应，缺乏T辅助细胞支持和强免疫反应。我们的模拟表明，白细胞介素-2 （IL-2）的缺乏显著影响细胞毒性T淋巴细胞（ctl）的激活。这些结果强调了T辅助细胞、干扰素-γ (IFN-γ)和IL-2对于准确表征丙型肝炎病毒感染动力学的重要性。最终，这项研究加深了我们对HCV感染动力学的理解，简化了特定免疫成分如何塑造疾病的过程。

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引用次数: 0

Numerical solution of the variable-order time fractional advection reaction–diffusion equation via combination of a Newton’s polynomial and Cubic B-spline method 结合牛顿多项式和三次b样条法求解变阶时间分数阶平流反应扩散方程

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-10 DOI: 10.1016/j.padiff.2025.101289

A.S.V. Ravi Kanth, Varela Pavankalyan

This work presents a numerical technique based on the cubic B-spline function for solving the variable-order time fractional advection reaction–diffusion equation in the sense of the Caputo derivative. Newton’s interpolation formulation has been employed to approximate the variable-order time-fractional derivative, while the cubic B-spline functions are utilized for spatial discretization. The proposed methodology demonstrated unconditionally stable and convergence of order

(Δ r^{4 - ϑ (ς, r)} + h^{2})

through the Von Neumann analysis. Numerical investigations that confirm theoretical conclusions using data visualizations and tables to illustrate efficiency and accuracy. Furthermore, the comparative findings demonstrate that the novel discretization methodology outperforms the other techniques present in the literature in terms of accuracy.

本文提出了一种基于三次b样条函数的Caputo导数意义下变阶时间分数阶平流反应扩散方程的数值求解方法。采用牛顿插值公式逼近变阶时间分数阶导数，采用三次b样条函数进行空间离散化。该方法通过Von Neumann分析证明了阶（Δr4−￣(ς,r)+h2）的无条件稳定性和收敛性。使用数据可视化和表格来证实理论结论的数值调查，以说明效率和准确性。此外，比较结果表明，新的离散化方法优于其他技术目前在准确性方面的文献。

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引用次数: 0

Pseudo-planar deformations of a linearized elastic solid 线性化弹性固体的拟平面变形

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-10 DOI: 10.1016/j.padiff.2025.101301

E. Momoniat , C. Harley

The equations of motion for the pseudo-planar motions of a classical linearized elastic solid and an incompressible linearized elastic solid undergoing non-uniform rotation about a vertical axis are derived. The pseudo-planar motions for both a classical linearized and an incompressible linearized elastic solid are determined numerically. For a classical linearized elastic solid, the non-uniform rotation is time-dependent and is specified. We derive a wave equation that models the non-uniform rotation for an incompressible linearized elastic solid. A pressure Poisson equation is derived and depends on the time derivative of the non-uniform rotation. The locus of the equations of motion coupled with the pseudo-planar motions of a cylindrical solid are plotted and the results are discussed. We show that the pseudo-planar motions of a classical linearized elastic solid with zero rotation are translations of the pseudo-planes about the locus. The pseudo-plane motions for classical and incompressible linearized elastic solids undergo translations and rotations about the locus. The motions are bound and stable when the pressure is symmetric. Unsymmetric pressure, which is just the mechanical pressure, results in a distortion of the pseudo-planar curves.

导出了经典线性化弹性固体和不可压缩线性化弹性固体绕垂直轴非均匀旋转时的拟平面运动方程。用数值方法确定了经典线性化弹性固体和不可压缩线性化弹性固体的拟平面运动。对于经典的线性化弹性固体，非均匀旋转是时间相关的，并且是指定的。我们推导了一个波动方程来模拟不可压缩线性化弹性固体的非均匀旋转。导出了压力泊松方程，该方程依赖于非均匀旋转的时间导数。绘制了与柱体拟平面运动耦合的运动方程轨迹，并对结果进行了讨论。我们证明了经典线性化弹性固体的伪平面运动是伪平面围绕轨迹的平移。经典和不可压缩线性化弹性固体的伪平面运动经历了围绕轨迹的平移和旋转。当压力对称时，运动是有约束的和稳定的。不对称压力即机械压力，会导致拟平面曲线的畸变。

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引用次数: 0

Dynamic complexity in fractional multispecies ecological systems: A Caputo derivative approach 分数多物种生态系统的动态复杂性：卡普托导数方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-10 DOI: 10.1016/j.padiff.2025.101293

Sonal Jain , Kolade M. Owolabi , Edson Pindza , Eben Mare

In this study, a novel implicit numerical approach is introduced by combining finite-difference techniques with innovative L1 schemes. This method is designed to solve time-fractional reaction–diffusion systems occurring in one and two dimensions. Specifically, the focus is on ecological systems with mixed boundary conditions, which are commonly found in biological and chemical processes. This research focuses on the spatiotemporal behavior of a predator–prey model with a Holling III functional response, taking into account the presence of prey refuges. This study revealed that this model does not exhibit a Turing pattern, which is typically associated with diffusion-driven instability. Consequently, this investigation explored alternative non-Turing patterns using extensive numerical simulations. In scenarios involving two-dimensional subdiffusion, the study observed a variety of spatiotemporal dynamics within the diffusive prey–predator model. When prey refuge availability was low, the system displayed a circular pattern that gradually expanded over time to encompass the entire spatial domain. As the availability of refugees decreased, the system transitioned from a spiral to a chaotic pattern. Furthermore, the research revealed that, as the ratio of predator-to-prey diffusion rates increased, the system exhibited a subdiffusive spiral pattern, which then transformed into a spot-like pattern. Eventually, these spots merged to form stripe-like patterns as the ratio increased. This investigation highlights the rich and intricate dynamics that can emerge in fractional predator–prey interactions when considering both spatial and temporal factors. To further confirm the complexity of the dynamical behaviors, Lyapunov exponents were estimated numerically.

本文将有限差分技术与新颖的L1格式相结合，提出了一种新的隐式数值方法。该方法用于求解一维和二维的时间分数反应扩散系统。具体来说，重点是具有混合边界条件的生态系统，这在生物和化学过程中很常见。本研究主要关注具有Holling III功能反应的捕食者-猎物模型的时空行为，并考虑猎物避难所的存在。这项研究表明，该模型不表现出图灵模式，这通常与扩散驱动的不稳定性有关。因此，本研究利用广泛的数值模拟探索了替代的非图灵模式。在涉及二维亚扩散的情况下，研究在扩散捕食模型中观察到各种时空动态。当猎物庇护所的可用性较低时，该系统显示出一个圆形模式，随着时间的推移逐渐扩展到整个空间域。随着难民数量的减少，该系统从螺旋形转变为混乱的模式。此外，研究还发现，随着捕食者对猎物扩散率的增加，该系统呈现出一个亚扩散的螺旋模式，然后转变为一个点状模式。最终，随着比例的增加，这些斑点合并形成条纹状图案。这项研究强调了在考虑空间和时间因素时，在分数捕食者-猎物相互作用中可能出现的丰富而复杂的动态。为了进一步确认动力学行为的复杂性，对Lyapunov指数进行了数值估计。

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引用次数: 0

Building novel solitary wave solutions for the generalized non-linear (3+1)-dimensional wave equation with gas bubbles in fluids using an analytic method 用解析法建立流体中含气泡的广义非线性（3+1）维波动方程的孤波解

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-08 DOI: 10.1016/j.padiff.2025.101272

Abeer S. Khalifa , Niveen M. Badra , Hamdy M. Ahmed , Wafaa B. Rabie , Homan Emadifar , Karim K. Ahmed

This article investigates the generalized nonlinear (3+1)-dimensional wave equation using an analytical technique — the Extended F-expansion method — to derive a variety of exact wave solutions and analyze the dynamic behavior of distinct wave profiles. The study presents several types of soliton solutions, including dark, bright, singular, periodic, and singular periodic forms. To the best of our knowledge, these specific solutions have not been previously reported in the literature. By assigning appropriate values to the free parameters, the behavior of the obtained solutions is illustrated through two- and three-dimensional plots, as well as corresponding contour diagrams. The proposed analytical method not only contributes to the theoretical understanding of nonlinear wave phenomena but also demonstrates practical relevance in applied sciences, particularly in fluid mechanics and engineering contexts involving gas-liquid interactions.

本文研究了广义非线性（3+1）维波动方程，利用扩展f展开法导出了各种精确的波动解，并分析了不同波浪剖面的动力行为。该研究提出了几种类型的孤子解，包括暗、亮、奇异、周期和奇异周期形式。据我们所知，这些具体的解决方案在以前的文献中没有报道过。通过给自由参数赋适当的值，得到的解的行为通过二维和三维图以及相应的等高线图来说明。所提出的分析方法不仅有助于对非线性波动现象的理论理解，而且在应用科学，特别是在流体力学和涉及气液相互作用的工程背景下，展示了实际的相关性。

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引用次数: 0

Exploring solitary wave structures and bifurcation dynamics in the (2+1)-dimensional generalized Hietarinta equation 探索（2+1）维广义Hietarinta方程中的孤波结构和分岔动力学

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-08 DOI: 10.1016/j.padiff.2025.101283

Yeşim Sağlam Özkan , Esra Ünal Yılmaz

This study investigates the

(2 + 1)

-dimensional generalized Hietarinta equation, which models the propagation of waves on water surfaces in the presence of gravity and surface tension. Solitary wave solutions are obtained using the

e x p (- w (x))

method and the

F

-expansion method, and are expressed in terms of hyperbolic, trigonometric, exponential and rational functions. Two- and three-dimensional plots illustrate various wave structures, such as dark, kinked, and singular kinked waves, highlighting their dynamic behaviors under different parameter settings. Hamiltonian functions and bifurcation theory are employed to analyze phase portraits and nonlinear wave dynamics, including chaotic behavior. Numerical simulations has been conducted using Mathematica and Maple confirm the theoretical findings. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed techniques are effective, computationally efficient and reliable. In this context, considering previous studies, the findings of this research contribute to the existing literature.

本文研究了（2+1）维广义Hietarinta方程，该方程模拟了重力和表面张力作用下波浪在水面上的传播。用exp（- w(x)）法和f展开法得到了孤波解，并以双曲函数、三角函数、指数函数和有理函数表示。二维和三维图形分别描绘了暗波、扭结波和奇异扭结波等不同的波浪结构，突出了它们在不同参数设置下的动力学行为。利用哈密顿函数和分岔理论分析了相图和非线性波动动力学，包括混沌行为。使用Mathematica和Maple进行的数值模拟证实了理论发现。并将结果与文献中已有的结果进行了比较，以显示其独特性。所提出的技术是有效的，计算效率高，可靠的。在此背景下，考虑到以往的研究，本研究的发现有助于现有文献。

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引用次数: 0

A study on intuitionistic fuzzy neutral functional integro-differential PDEs with impulses 带有脉冲的直觉模糊中立泛函积分微分偏微分方程的研究

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-01 DOI: 10.1016/j.padiff.2025.101296

T. Gunasekar , K. Nithyanandhan , Hanumagowda B. N , Jagadish V. Tawade , Nashwan Adnan Othman , Barno Abdullaeva , Nadia Batool , Khayrilla Kurbonov

This paper investigates the existence and uniqueness of solutions for a nonlocal intuitionistic fuzzy impulsive integro-differential equation, employing intuitionistic fuzzy semigroups and the contraction mapping principle. Through a systematic theoretical framework, it establishes that, under certain conditions, a distinct solution is ensured. Additionally, the study expands its analysis to explore the existence results for intuitionistic fuzzy impulsive neutral integro-differential equations, broadening its research focus. This approach introduces a new perspective on understanding intuitionistic fuzzy integro-differential equations, introducing innovative methodologies and significant discoveries that advance theoretical exploration in this field. The findings underscore that, subject to specific assumptions, a singular fuzzy solution emerges for these problems marked by nonlocal conditions, effectively addressing crucial challenges in the analysis of fuzzy systems.

利用直觉模糊半群和压缩映射原理，研究了一类非局部直觉模糊脉冲积分微分方程解的存在唯一性。通过系统的理论框架，确立了在一定条件下，保证有一个独特的解。此外，本研究将其分析扩展到探索直觉模糊脉冲中立型积分微分方程的存在性结果，拓宽了研究的重点。这种方法引入了理解直觉模糊积分微分方程的新视角，引入了创新的方法和重大发现，推动了该领域的理论探索。研究结果强调，在特定的假设下，对于这些以非局部条件为特征的问题，一个单一的模糊解决方案出现了，有效地解决了模糊系统分析中的关键挑战。

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引用次数: 0

Machine learning analysis of tangent hyperbolic nanofluid with radiation and Arrhenius activation energy over falling cone under gravity 重力作用下落锥上具有辐射和Arrhenius活化能的正切双曲纳米流体的机器学习分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-09-01 DOI: 10.1016/j.padiff.2025.101280

Muhammad Zubair , Hamid Qureshi , Usman Khaliq , Taoufik Saidani , Waqar Azeem Khan

This study is a machine learning investigation of the advance level nanofluidic coolant through a cone in a two-dimensional transitory boundary layer. The model accounts for both radiation absorption and the Arrhenius activation energy. Synthetic datasets from governing mathematical model are used in Artificial Intelligence (AI) based Levenberg Marquardt Back Propagation algorithm (LM-BP). Multiple scenarios of Tangent Hyperbolic Nanofluidic (THNF) coolant are framed with variation of influencing characteristics like Magnetic field M, power law index n, permeability k, Radiation absorption Q, Prandtl ratio Pr, Brownian motion Nb, Lewis number Le and Chemical reaction parameter γ. Convergence parameters of AI-based feed routing Neural Network computing is presented through graphs and numerical tables. Results indicate that flow slows when the Lorentz force and surface permeability grow, but it gets stronger when thermal absorption and momentum to thermal diffusivity ratio Pr increase. Meanwhile, the temperature increases when thermal absorption rises and drops when thermal to mass diffusivity ratio Le increases so that temperature falls for greater chemical reaction influence.

本研究采用机器学习的方法研究了先进的纳米流控冷却剂在二维过渡边界层中的锥形流动。该模型同时考虑了辐射吸收和阿伦尼乌斯活化能。基于人工智能（AI）的Levenberg Marquardt反向传播算法（LM-BP）采用控制数学模型合成的数据集。研究了正切双曲型纳米流体（THNF）冷却剂的磁场M、幂律指数n、磁导率k、辐射吸收Q、普朗特比Pr、布朗运动Nb、路易斯数Le和化学反应参数γ等影响特性的变化。以图形和数值表的形式给出了基于人工智能的馈电路由神经网络计算的收敛参数。结果表明，随着洛伦兹力和表面渗透率的增大，流动速度减慢，但随着热吸收和动量与热扩散比Pr的增大，流动速度加快。同时，随着热吸收率的升高，温度升高；随着热质扩散比Le的增大，温度降低，化学反应影响更大。

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引用次数: 0