Partial Differential Equations in Applied Mathematics最新文献_第9页

Thermal radiation effect on fractional MHD Couette Flow of Jeffrey fluid in a vertical channel with activation energy and Joule Heating 具有活化能和焦耳加热的垂直通道中Jeffrey流体分数阶MHD Couette流动的热辐射效应

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-04 DOI: 10.1016/j.padiff.2025.101251

Paul M. Matao , Jumanne Mng’ang’a , B. Prabhakar Reddy

This study investigates the consequence of thermal radiation on the fractional magnetohydrodynamic (MHD) Couette flow of a Jeffrey fluid in a vertical channel, incorporating the influences of activation energy and Joule heating. The mathematical model is derived using appropriate governing equations that account for the non-Newtonian behavior of the Jeffrey fluid, combined with the impacts of thermal radiation, magnetic field, and activation energy mechanisms. The classical mathematical framework has been transformed into a system of fractal fractional-order derivatives using the Caputo–Fabrizio derivative operator. To solve these systems, the finite difference technique was employed. The behavior of fluid flow fields in response to several significant parameters was analyzed and represented graphically. It is ascertained that velocity distribution upsurges as Hall current parameter rises, while a more substantial effect from the Jeffrey fluid parameter results in a decrease in the velocity field. Additionally, thermal field profiles exhibited higher values in response to increased thermal radiation and Joule heating parameters, whereas the temperature distribution showed a decline with improving in Hall current parameter values. The concentration field improved with higher activation energy parameter values, in contrast to the opposite trend observed with temperature difference and chemical reaction parameters. Furthermore, it is remarked that fractal fractional-order derivatives operator produced a more pronounced boundary layer compared to both fractional and classical models. It is ascertained that the Nusselt number showing a 15.7% improvement in thermal efficiency as thermal radiation varied from 2 to 4. These findings are important for applications in geothermal energy extraction, and biomedical engineering.

考虑活化能和焦耳加热的影响，研究了热辐射对Jeffrey流体在垂直通道中分数磁流体动力学（MHD） Couette流动的影响。数学模型是使用适当的控制方程推导出来的，该方程考虑了杰弗里流体的非牛顿行为，并结合了热辐射、磁场和活化能机制的影响。利用Caputo-Fabrizio导数算子将经典数学框架转化为分形分数阶导数系统。为了求解这些系统，采用了有限差分技术。分析了流体流场对几个重要参数的响应特性，并用图形表示。确定了随着霍尔电流参数的增大，速度分布呈上升趋势，而杰弗里流体参数对速度场的影响更大，导致速度场减小。热场分布随着热辐射和焦耳加热参数的增大而增大，而温度分布随着霍尔电流参数的增大而减小。随着活化能参数的增大，浓度场得到改善，而随着温度差异和化学反应参数的增大，浓度场呈现相反的趋势。此外，还指出分形分数阶导数算子与分数阶和经典模型相比产生了更明显的边界层。可以确定，当热辐射在2到4之间变化时，努塞尔数显示热效率提高15.7%。这些发现对于地热能提取和生物医学工程的应用具有重要意义。

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引用次数: 0

Analytical new soliton solutions and stability analysis of the (2 + 1)-dimensional time-fractional nonlinear GZKBBM equation （2 + 1）维时间分数阶非线性GZKBBM方程的解析新孤子解及稳定性分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-03 DOI: 10.1016/j.padiff.2025.101256

Nazia Parvin , Hasibun Naher , M. Ali Akbar

In this study, we investigate the soliton solutions of the (2 + 1)-dimensional time-fractional nonlinear generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation using the extended sinh-Gordon expansion approach. This equation is useful in modeling the hydro-magnetic waves in cold plasma, acoustic waves in harmonic crystals, shallow water waves, and acoustic gravity waves. By utilizing the suggested approach, we derive some rich structured soliton solutions, including bell-shaped soliton, anti-bell-shaped soliton, anti-peakon, periodic soliton and singular solitons of the model. These solutions are expressed in hyperbolic and trigonometric forms, and their dynamical behaviors are illustrated through 3D and 2D plots for various values of the fractional parameter

β

and other physical parameters. The impact of the time-fractional derivative on the introduced model is examined using the beta derivative framework, which provides a more general and flexible way to enhance the accuracy of the solutions. The stability of the model is also examined through the linear stability theory, confirming that all analytical findings are stable. The results unambiguously demonstrate that the extended sinh-Gordon expansion approach is compatible, reliable, and efficient for investigating various nonlinear evolution equations in fields of applied science and engineering.

本文利用扩展的sinh-Gordon展开方法研究了（2 + 1）维时分数阶非线性广义Zakharov-Kuznetsov-Benjamin-Bona-Mahony方程的孤子解。该方程可用于模拟冷等离子体中的磁流体波、谐波晶体中的声波、浅水波和声引力波。利用该方法，我们得到了模型的钟形孤子、反钟形孤子、反峰子、周期孤子和奇异孤子等丰富的结构孤子解。这些解以双曲和三角形式表示，并通过分数参数β和其他物理参数的不同值的三维和二维图来说明它们的动力学行为。利用beta导数框架考察了时间分数阶导数对引入模型的影响，该框架提供了一种更通用、更灵活的方法来提高解的准确性。通过线性稳定性理论检验了模型的稳定性，证实了所有的分析结果都是稳定的。结果清楚地表明，扩展的sinh-Gordon展开方法对于研究应用科学和工程领域的各种非线性演化方程是兼容的、可靠的和有效的。

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引用次数: 0

Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods 用分数阶欧拉和三次三角b样条方法数值逼近时间分数阶非线性偏积分微分方程

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-02 DOI: 10.1016/j.padiff.2025.101223

Mehwish Saleem , Arshed Ali , Fazal-i-Haq , Hassan Khan

Nonlinear mathematical problems arise due to existence of important complex nonlinear phenomena in engineering and science. In this article, a class of time-fractional nonlinear parabolic partial integro-differential equations is solved numerically by combination of fractional Euler and cubic trigonometric B-spline collocation methods. Backward finite difference formula is employed for time-fractional Caputo derivative to get an unconditional stable scheme. The memory(integral) term is evaluated using a second order quadrature rule. Fractional Euler method for Caputo derivative is used in computing the nonlinear memory term. At each time level, cubic trigonometric B-spline functions are applied to obtain the solution in spatial dimension which reduces the problem to a system of algebraic equations. This method has the ability to handle any kind of nonlinearity without using iterative processes. Efficiency and reliability of the current method is analyzed for the fractional-order via three highly nonlinear test problems with variable coefficients. The rate of convergence of the proposed method is also computed in temporal and spatial dimensions.

非线性数学问题是由于工程和科学中重要的复杂非线性现象的存在而产生的。本文采用分数阶欧拉与三次三角b样条配置相结合的方法，对一类时间分数阶非线性抛物型偏积分微分方程进行了数值求解。对时间分数阶卡普托导数采用后向有限差分公式，得到了一个无条件稳定格式。内存（积分）项是用二阶正交规则计算的。采用分数欧拉法求解卡普托导数的非线性记忆项。在每个时间层面上，采用三次三角b样条函数在空间维度上求解，将问题简化为代数方程组。该方法具有处理任何非线性问题的能力，无需使用迭代过程。通过三个变系数的高度非线性测试问题，分析了现有方法对分数阶的有效性和可靠性。本文还从时间和空间两个维度计算了该方法的收敛速度。

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引用次数: 0

A comparative analysis using the Laplace transform approach for some nonlinear fractional physical problems 用拉普拉斯变换方法对若干非线性分数物理问题的比较分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-06-30 DOI: 10.1016/j.padiff.2025.101253

Mohammad Alaroud

Both linear and nonlinear differential, partial equations of fractional order can be solved efficiently using the residual power series method (RPSM). Nevertheless, the process requires the residual function's (n − 1)ϱ fractional derivative(FD). We all know that figuring out the FD of a function can be difficult. A straightforward and effective analytical technique known as the Laplace transform-residual power series method (LT-RPSM) is used in this study to provide the approximate and exact solutions to nonlinear fractional partial differential equations(NFPDEs) under Caputo fractional differentiation including the nonlinear Fokker-Planck, nonlinear gas dynamics and nonlinear Klein-Gordon equations. The computations needed to find the coefficients of an expansion series are modest because the proposed method just requires the concept of an infinite limit. Three nonlinear fractional physical problems are successfully solved by the used investigation, which provides closed- form solutions and exact solutions in ordinary case, also a thorough graphical and numerical comparisons of the findings discovered. These outcomes are compared with existing solutions in the literature, especially in the meaning of absolute errors against the Laplace Adomin decompostion method LADM in light of different FD operators. Strong agreement between the results of the used method and several series solution techniques. Consequently, LT-RPSM can be considered a very successful technique and the most effective analytical algorithm to deal with numerous NFPDEs emerging in physics and engineering.

残差幂级数法可以有效地求解分数阶线性和非线性微分、偏方程。然而，该过程需要残差函数的（n−1）ϱ分数导数（FD）。我们都知道计算一个函数的FD是很困难的。本文采用一种简单有效的拉普拉斯变换-剩余幂级数法（LT-RPSM），给出了Caputo分数阶微分下非线性分数阶偏微分方程（NFPDEs）的近似和精确解，包括非线性Fokker-Planck方程、非线性气体动力学方程和非线性Klein-Gordon方程。求展开式级数的系数所需的计算量不大，因为所提出的方法只需要无穷大极限的概念。本文成功地解决了三个非线性分数物理问题，给出了一般情况下的封闭解和精确解，并对所发现的结果进行了全面的图解和数值比较。将这些结果与文献中已有的解进行了比较，特别是针对不同FD算子对拉普拉斯Adomin分解方法LADM的绝对误差的含义。所用方法的结果与几种系列溶液技术的结果非常一致。因此，LT-RPSM可以被认为是一种非常成功的技术，也是处理物理和工程中出现的众多nfpde的最有效的分析算法。

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引用次数: 0

Analysis of fractional viewpoints on the Jaulent–Miodek and Whitham–Broer–Kaup coupled equations Jaulent-Miodek和Whitham-Broer-Kaup耦合方程的分数视点分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-06-27 DOI: 10.1016/j.padiff.2025.101243

Sachit Kumar , Varun Joshi , Mamta Kapoor

In this work, we use the Caputo fractional calculus to methodically examine the Coupled Jaulent–Miodek (CJM) fractional equation and the fractional Whitham–Broer–Kaup (WBK) system. The Sumudu residual power series approach and the Sumudu iteration transform method are used to analyze the nonlinear fractional differential equation systems, providing a comprehensive analytical analysis. The Sumudu iteration transform approach is used to achieve the fractional WBK system’s dynamics, as well as the Sumudu power series residual approach is utilized to investigate the CJM equation’s behavior for fractions. We thoroughly examine their interactions using known solutions, using both symbolic calculations and numerical simulations. This leads to the identification of new solutions and the clarification of the way in which certain systems of fractions behave in terms of the operator of Caputo. The outcomes demonstrate the efficacy of the strategies used to decipher the intricate dynamics of fractional nonlinear systems by demonstrating a strong convergence agreement between analytical and numerical solutions.

在这项工作中，我们使用Caputo分数微积分系统地检查了耦合Jaulent-Miodek （CJM）分数方程和分数Whitham-Broer-Kaup （WBK）系统。采用Sumudu残差幂级数法和Sumudu迭代变换法对非线性分数阶微分方程系统进行了分析，提供了全面的分析方法。采用Sumudu迭代变换方法实现分数阶WBK系统的动力学，并利用Sumudu幂级数残差方法研究分数阶CJM方程的行为。我们使用已知的解决方案，使用符号计算和数值模拟，彻底检查它们的相互作用。这导致了新的解决方案的识别，并澄清了某些分数系统在卡普托算子方面的行为方式。结果表明，通过证明在解析解和数值解之间具有很强的收敛性，用于破译分数阶非线性系统的复杂动力学的策略的有效性。

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引用次数: 0

Mathematical model of Cynodon Dactylon’s allelopathic effect on perennial ryegrass for exploring plant-plant interactions based upon ordinary differential equations 基于常微分方程的Cynodon Dactylon化感作用对多年生黑麦草的数学模型研究

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-06-27 DOI: 10.1016/j.padiff.2025.101254

Dipesh , Pankaj kumar , Anjori Sharma

In this paper, a mathematical model and analysis is proposed to study the stimulatory allelopathic impact of cynodon dactylon on perennial ryegrass using ordinary differential equations. Equilibrium points and biological interpretation is analyzed using the Routh-Hurwitz theorem. Allelopathic produces synergistic effects between two plants that can result in apparent competition for space, nutrients, water and growth of the plant or apparent organisms depending on how the life cycles of their shared exploiters and/or commensal are influenced by inducing morphological, physiological, biochemical and chemical changes in plants. Plants are competing for space, which is required for proper growth and development of roots. Based on spacing allelopathic effect is very less as space increases in respect of root length and root branches increase. These allelopathy biochemicals are used for pest management. Whenever the behaviors of exploiters and commensals respond to induce changes in comparative plant numbers, indirect -interactions among plants arise. The Mann-Kendall, MK test, Bartletts test, and Anova test is used to analyze the data and numerical simulation. Also, the main objective of this research article allelopathic impact of plant-to-plant interaction research plays an important role in climate action and life cycle on land.

本文利用常微分方程，建立了犬齿草对多年生黑麦草刺激化感作用的数学模型并进行了分析。利用劳斯-赫维茨定理分析了平衡点和生物学解释。化感作用在两种植物之间产生协同效应，根据植物形态、生理、生化和化学变化对共同利用者和/或共生者的生命周期的影响，可以导致植物或表观生物对空间、养分、水分和生长的明显竞争。植物在争夺空间，这是根系正常生长和发育所必需的。基于间距的化感效应随根长和根枝间距的增加而减小。这些化感生物化学物质用于害虫管理。每当利用者和共生者的行为引起比较植物数量的变化时，植物间就会产生间接相互作用。采用Mann-Kendall检验、MK检验、Bartletts检验和Anova检验对数据进行分析和数值模拟。植物间化感作用的研究在气候作用和陆地生命周期中发挥着重要作用。

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引用次数: 0

The impact of carbon nanotubes (CNT) on heat generation and absorption, the behaviour of water and blood suspensions in an inclined channel with a porous matrix 碳纳米管（CNT）对热产生和吸收的影响，水和血液悬浮液在多孔基质倾斜通道中的行为

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-06-26 DOI: 10.1016/j.padiff.2025.101241

Mangala Kandagal , Ramesh Kempepatil , Jagadish V. Tawade , Nodira Nazarova , Manish Gupta , M. Khan

The study investigates the heat generation and absorption of engine oil, human blood and single wall carbon Nano tube (SWCNT) in an inclined channel filled with a porous matrix. Two regions are considered, both regions are of porous medium. Due to their enhanced thermal conductivity, are utilized to improve heat transfer efficiency in various applications. Formulation of the problem is framed using conservation of mass, energy and momentum in both regions. The flow of oil, human blood through a porous medium is analysed, considering the effects of both heat generation and absorption within the system. Key parameters such as the inclination angle of the channel, the porosity and the type of fluids are examined to understand their impact on the overall heat transfer process and velocity. To solve the problem regular perturbation method is applied for non-dimensional quantities; The presence of CNTs significantly improves the thermal conductivity of both engine oil and blood suspensions, leading to improved heat dissipation or absorption capabilities, which are influenced by the inclination and the porous structure. This study offers valuable insights into fluid flow processes in the human body using Nanotubes. Influence of CNTs on the fluid flow of human body and heat generation/absorption with porous matrix in both regions is unsolved problem.

研究了发动机机油、人体血液和单壁碳纳米管（SWCNT）在多孔基质填充的倾斜通道中的产热和吸热特性。考虑两个区域，两个区域均为多孔介质。由于其增强的导热性，在各种应用中被用来提高传热效率。问题的表述是利用两个区域的质量、能量和动量守恒来构建的。考虑到系统内热的产生和吸收的影响，分析了油、人的血液在多孔介质中的流动。研究了通道倾角、孔隙度和流体类型等关键参数，以了解它们对整体传热过程和速度的影响。为解决这一问题，对无量纲量采用正则摄动法；CNTs的存在显著提高了机油悬浮液和血液悬浮液的导热性，从而提高了其散热或吸收能力，而这些能力受倾斜和多孔结构的影响。这项研究为利用纳米管研究人体流体流动过程提供了有价值的见解。CNTs对人体流体流动以及多孔基质在这两个区域产生/吸收热量的影响是尚未解决的问题。

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引用次数: 0

Detailed analysis under fully constrained and relaxed boundary conditions of linear fields in the vicinity of a corner 角附近线性场在完全约束和松弛边界条件下的详细分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-06-26 DOI: 10.1016/j.padiff.2025.101244

Ayelet Goldstein, Ofer Eyal, Jorge Berger

This work examines the behavior of fields near corners under various boundary conditions (BCs), focusing on singularities arising from fully constrained and relaxed BCs. We analyze this behavior across diverse physical systems governed by similar equations, including electromagnetism, superconductivity, and two-phase fluid flow. The corner geometry presents a challenge due to potentially diverging field solutions as the corner is approached (r

\to

0). This motivates the investigation of relaxed BCs, which regularize the field by introducing a characteristic length (Ls) that relates the field’s value to its normal derivative at the boundary.

We explore both single-medium (single-phase) and double-medium (two-phase) systems. While prior research has addressed relaxed BCs in specific contexts, their application to corners, particularly in diverse physical systems, remains under-explored. We develop a series solution method to analyze the field behavior near the corner under different BCs. Concrete examples illustrate the theoretical framework, examining both fully constrained and relaxed scenarios. The implications of this work extend to fields such as fluid mechanics, electromagnetism, and heat transfer.

本研究考察了不同边界条件下拐角附近场的行为，重点研究了完全约束和松弛边界条件下产生的奇点。我们分析了由类似方程控制的不同物理系统的这种行为，包括电磁学，超导性和两相流体流动。当拐角接近（r→0）时，由于潜在的发散场解，拐角的几何形状带来了挑战。这激发了对松弛bc的研究，它通过引入一个特征长度（Ls）来使场正则化，该特征长度将场的值与其在边界处的法向导数联系起来。我们探索单介质（单相）和双介质（两相）系统。虽然之前的研究已经在特定的环境中解决了松弛的bc，但它们在角落的应用，特别是在不同的物理系统中，仍然没有得到充分的探索。我们提出了一种级数解的方法来分析不同bc下的拐角附近的场行为。具体的例子说明了理论框架，检查了完全约束和放松的场景。这项工作的意义延伸到流体力学、电磁学和传热等领域。

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引用次数: 0

An iterative method for solving sparse linear algebraic systems with continuum solution dependent right-hand side for elliptic partial differential equations 求解椭圆型偏微分方程右侧连续解相关稀疏线性代数系统的迭代方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-06-21 DOI: 10.1016/j.padiff.2025.101236

Sudipta Lal Basu , Kirk M. Soodhalter , Breiffni Fitzgerald , Biswajit Basu

Krylov subspace iterative methods such as bi-conjugate gradients stabilized (BiCGStab) to approximately solve sparse linear algebraic systems are well known. However, there are certain instances in real-world engineering applications with underlying governing partial differential equation where the discretized right-hand side can only be exactly determined using the unavailable continuum solution. In such cases, an iterative method such as BiCGStab may not converge to a physically correct solution or may diverge completely. Such a method must be modified to accommodate inexact knowledge of the discrete right-hand side, using an updating scheme as the iteration proceeds. In this paper, we present such an updating strategy for physical problems governed by elliptic partial differential equations. This strategy must be performed in a numerically stable manner, which we also discuss. We present this as a modified BiCGStab iteration and investigate its effectiveness on both test problems, wherein it is shown to perform well and agrees with the analytical solutions, and on some more realistic problems arising in the study of Hele-Shaw flow, composite materials and power generation from wind farms.

Krylov子空间迭代方法如双共轭梯度稳定（BiCGStab）近似求解稀疏线性代数系统是众所周知的。然而，在现实世界的工程应用中，存在某些具有潜在控制偏微分方程的实例，其中离散的右侧只能使用不可用的连续介质解来精确确定。在这种情况下，像bicstab这样的迭代方法可能不会收敛到物理上正确的解决方案，或者可能完全发散。这种方法必须进行修改，以适应离散右边的不精确知识，并在迭代进行时使用更新方案。在本文中，我们提出了一种求解椭圆型偏微分方程物理问题的更新策略。此策略必须以数值稳定的方式执行，我们也将讨论这一点。我们将其作为改进的BiCGStab迭代，并研究其在两个测试问题上的有效性，其中它显示出良好的性能并与解析解一致，以及在研究Hele-Shaw流，复合材料和风力发电场发电中出现的一些更现实的问题。

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引用次数: 0

An enhanced Artificial Neural Network approach for solving nonlinear fractional-order differential equations 求解非线性分数阶微分方程的增强人工神经网络方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-06-18 DOI: 10.1016/j.padiff.2025.101230

Nikhil Sharma , Sunil Joshi , Pranay Goswami

This paper introduces a hybrid Chebyshev Collocation Method (CCM) and Artificial Neural Network (ANN) approach to address the computational challenges of nonlinear Caputo fractional differential equations. The purpose is to improve accuracy for static solutions by approximating the fractional derivative spatially. The methodology leverages CCM for spatial discretization and ANN for residual minimization, achieving low MSEs (e.g.,

1 0^{- 5}

) in three examples. The findings confirm improved convergence with increasing node count, with implications for efficient fractional PDE solvers. The novelty lies in the static CCM+ANN integration, offering a practical alternative to dynamic methods.

本文介绍了一种混合Chebyshev配置法（CCM）和人工神经网络（ANN）方法来解决非线性Caputo分数阶微分方程的计算难题。目的是通过在空间上近似分数阶导数来提高静态解的精度。该方法利用CCM进行空间离散化，利用ANN进行残差最小化，在三个示例中实现了低mse（例如10−5）。研究结果证实，随着节点数的增加，收敛性得到改善，这对有效的分数阶PDE求解器具有重要意义。新颖之处在于静态CCM+ANN集成，为动态方法提供了一种实用的替代方案。

引用次数: 0