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

Exact solution of system of nonlinear fractional partial differential equations by modified semi-separation of variables method 用改进的半分离变量法精确解非线性分数阶偏微分方程组

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-12 DOI: 10.1016/j.padiff.2025.101247

Henry Kwasi Asiedu, Benedict Barnes, Isaac Kwame Dontwi, Kwaku Forkuoh Darkwah

A system of nonlinear fractional partial differential equations (FPDEs) is widely used in applied sciences, especially for modeling fluid dynamics and polymer-related problems. Given their importance, finding solutions to these systems is essential and a core property. Various methods have been developed to find a solution to a system of nonlinear FPDEs. However, these methods are difficult to implement and sometimes converge slowly. In the worst-case scenario, applying the differential transform method may produce a series that does not converge to the exact solution of a system of nonlinear FPDEs. The semi-separation of variables method (S-SVM) is a recent and reliable analytic method that has not been applied to obtain an exact solution to a system of nonlinear FPDEs. Furthermore, S-SVM has not been improved to observe faster convergence. In this paper, the S-SVM is used to obtain the exact solution to the system of nonlinear FPDEs. In addition, the S-SVM is further improved as a Modified S-SVM (MS-SVM), which is applied to find an exact solution to the system of nonlinear FPDEs. Also, numerical experiments using the S-SVM and the MS-SVM in both two and three dimensions are provided therein, along with a comparison of their solutions to those obtained from the Adomian Decomposition Method (ADM), the Laplace Variational Iteration Method (LVIM), and the Fractional Power Series Method (FPSM). The results show that the solutions obtained using S-SVM and MS-SVM converge faster than those from FPSM, ADM, and LVIM. Moreover, S-SVM and MS-SVM do not require the complex computation of Adomian polynomials.

非线性分数阶偏微分方程（FPDEs）系统广泛应用于应用科学，特别是流体动力学和聚合物相关问题的建模。考虑到它们的重要性，为这些系统寻找解决方案是必不可少的，也是核心属性。已经开发了各种方法来寻找非线性FPDEs系统的解。然而，这些方法很难实现，有时收敛速度较慢。在最坏的情况下，应用微分变换方法可能会产生一系列不收敛于非线性FPDEs系统的精确解。半分离变量法（S-SVM）是一种较新的可靠的解析方法，但尚未应用于求解非线性FPDEs系统的精确解。此外，S-SVM也没有改进到更快的收敛。本文将S-SVM用于求解非线性FPDEs系统的精确解。此外，将S-SVM进一步改进为改进的S-SVM (MS-SVM)，用于求解非线性FPDEs系统的精确解。并在二维和三维上进行了S-SVM和MS-SVM的数值实验，并与Adomian分解法（ADM）、拉普拉斯变分迭代法（LVIM）和分数阶幂级数法（FPSM）的解进行了比较。结果表明，S-SVM和MS-SVM的解收敛速度快于FPSM、ADM和LVIM的解。此外，S-SVM和MS-SVM不需要复杂的Adomian多项式计算。

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引用次数: 0

An immersed interface method for nonlinear convection–diffusion equations with interfaces 具有界面的非线性对流扩散方程的浸入界面法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-11 DOI: 10.1016/j.padiff.2025.101250

Miguel Uh Zapata , Reymundo Itza Balam , Silvia Jerez

This paper provides an initial framework for developing high-order numerical methods to solve interface problems for nonlinear elliptic partial differential equations. The proposed formulation is based on the immersed interface method dealing with a discontinuous coefficient problem. The algorithm introduces new schemes for points near the interface, whereas standard central finite difference schemes are used in smooth regions. As a consequence, a global second-order accurate solution is guaranteed. First, theoretical results on the truncation error are given for one-dimensional linear problems. Next, the algorithm is generalized to deal with nonlinear convection and diffusion cases using the using Levenberg–Marquardt algorithm. Numerical simulations for several benchmark problems show the robustness and efficiency of the proposed scheme.

本文为发展求解非线性椭圆型偏微分方程界面问题的高阶数值方法提供了一个初步框架。该公式基于处理不连续系数问题的浸入界面法。该算法对界面附近的点引入了新的格式，而在光滑区域则采用标准的中心有限差分格式。因此，保证了全局二阶精确解。首先，对一维线性问题给出了截断误差的理论结果。然后，利用Levenberg-Marquardt算法将该算法推广到处理非线性对流和扩散情况。若干基准问题的数值仿真结果表明了该方法的鲁棒性和有效性。

引用次数: 0

Qualitative analysis and controllability of complex tumor model with different therapies with nonsingular kernel 非奇异核不同治疗方法复杂肿瘤模型的定性分析及可控性

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-10 DOI: 10.1016/j.padiff.2025.101249

Maryam Batool , Muhammad Farman , Kottakkaran Sooppy Nisar , Evren Hincal , Shah Jahan

In this paper, consider the immune response to avascular cancer under the effect of immunotherapy, chemotherapy, and their combinations, as well as vaccination regimens, is described using a fractional order model to observe the impact of different therapies for cancer treatment. The impact of vaccination therapy is viewed as a model parameter perturbation. The effect of the global derivative, the existence, and the boundedness of the suggested system are confirmed, which are the essential characteristics of epidemic problems. The proposed system is qualitatively examined as well to determine its stable points. The Lyapunov function is used to analyze global stability, and the equilibrium states of the second derivative test are quantitatively examined. To investigate the effects of the fractional operator on the suggested model, solutions are generated using the Mittag Leffler kernel, and numerical simulations are run to demonstrate the theoretical findings. Using MATLAB, the effects of cancer treatment with various drugs and parameter values are justified. The proposed system is also treated for controllability and observability for a linear control system to monitor the close-loop design with different therapies as an input and cancer cells as an output.

本文考虑在免疫治疗、化疗及其联合治疗以及疫苗接种方案的作用下对无血管性癌症的免疫反应，使用分数阶模型来观察不同治疗方法对癌症治疗的影响。疫苗治疗的影响被视为模型参数扰动。证实了系统的全局导数效应、存在性和有界性，这是流行病问题的基本特征。对所提出的系统进行了定性检查，以确定其稳定点。利用Lyapunov函数分析了系统的全局稳定性，并对二阶导数检验的平衡态进行了定量检验。为了研究分数算子对建议模型的影响，使用Mittag Leffler核生成解决方案，并运行数值模拟来证明理论发现。利用MATLAB对不同药物和参数值对肿瘤治疗的效果进行了论证。所提出的系统还处理了线性控制系统的可控性和可观察性，以监测以不同疗法作为输入和癌细胞作为输出的闭环设计。

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引用次数: 0

Some novel properties of complex intuitionistic fuzzy ideals in classical ring 经典环中复杂直觉模糊理想的一些新性质

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-07 DOI: 10.1016/j.padiff.2024.100811

Ziad Khan , Fawad Hussain , Ikhtesham Ullah , Tariq Rahim , Madad Khan , Rashid Jan , Ibrahim Mekawy , Asma Alharbi

The complex intuitionistic fuzzy set is an extension of the intuitionistic fuzzy set where the membership and non-membership functions are expressed by a complex numbers. Ring theory is a well-known field of abstract algebra that is used in a broad area of present study in mathematics and computer science. The study of ideals is important in numerous ways in ring theory. Keeping in view the importance of complex intuitionistic fuzzy sets and ring theory, in this paper, we define the notion of complex intuitionistic fuzzy ideals in a classical ring

R

and investigate its various algebraic properties. We obtain that the intersection of any two complex intuitionistic fuzzy ideals of a classical ring

R

is again a complex intuitionistic fuzzy ideal of

R

. We also define the notion of a complex intuitionistic fuzzy level set. Furthermore, we define the concept of complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal of a classical ring and prove that the set of all complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal forms a ring under certain binary operations. Finally, we prove a complex intuitionistic fuzzy version of the fundamental theorem of a ring homomorphism.

复直觉模糊集是直觉模糊集的扩展，其隶属函数和非隶属函数用复数表示。环理论是一个著名的抽象代数领域，在数学和计算机科学的广泛研究中得到应用。在环理论中，理想的研究在很多方面都很重要。考虑到复直觉模糊集和环理论的重要性，在经典环R中定义了复直觉模糊理想的概念，并研究了它的各种代数性质。我们得到了经典环R的任意两个复直觉模糊理想的交点仍然是R的复直觉模糊理想，并定义了复直觉模糊水平集的概念。进一步，我们定义了经典环上复直觉模糊理想的复直觉模糊协集的概念，并证明了在一定的二元运算下，复直觉模糊理想的所有复直觉模糊协集的集合构成环。最后，我们证明了环同态基本定理的一个复杂直觉模糊版本。

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引用次数: 0

The Modified Homogeneous Balance Method for solving fractional Cahn–Allen and equal width equations 求解分数阶Cahn-Allen方程及等宽方程的修正齐次平衡法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-05 DOI: 10.1016/j.padiff.2025.101246

Francis Tuffour, Benedict Barnes, Isaac Kwame Dontwi, Kwaku Forkuoh Darkwah

This paper presents exact solutions to the Fractional Cahn–Allen (FC–A) and the Fractional Equal Width (FEW) equations using the Modified Homogeneous Balance Method (MHBM). The MHBM transforms the FC–A and FEW equations into fractional ordinary differential equations via a wave transformation. By balancing the highest-order derivative with the leading nonlinear term, the method determines the appropriate polynomial degree. A fractional Riccati equation with a quadratic nonlinearity facilitates the construction of exact solutions without resorting to infinite series expansions. Compared to existing methods, the MHBM offers a finite and well-defined solution structure, avoiding the rigidity of the

tan (\frac{ξ (η)}{2})

-expansion method and the complexities associated with the Riemann–Hilbert and algebro–geometric methods. It also provides clearer criteria for convergence analysis than the

ϕ^{6}

-expansion method. The MHBM accommodates various solution types, including trigonometric, hyperbolic, rational, and elliptic functions, with fewer parameter restrictions and potential for multi-wave structures. Numerical simulations shows that as the spatial variable

x

increases, the solitons tend to stabilize, and the plots for different values of the fractional order

α

closely aligned, indicating minor sensitivity to

α

. Furthermore, the FEW soliton exhibits a dense tiling structure along the time axis in its surface plot, while the FC–A soliton demonstrates a smooth kink-like transition along

ξ

, characteristic of solutions connecting two stable equilibrium states. These findings underscore the robustness and versatility of the MHBM in analyzing fractional nonlinear evolution equations.

本文用改进齐次平衡法给出了分数阶Cahn-Allen方程（FC-A）和分数阶等宽方程（FEW）的精确解。MHBM通过波变换将FC-A和FEW方程转化为分数阶常微分方程。该方法通过平衡最高阶导数与前导非线性项，确定合适的多项式次。具有二次非线性的分数阶里卡蒂方程便于构造精确解而不需要无穷级数展开。与现有方法相比，MHBM提供了有限且定义良好的解结构，避免了tanξ(η)2展开方法的刚性和黎曼-希尔伯特和代数-几何方法的复杂性。它还为收敛分析提供了比ϕ6展开法更清晰的准则。MHBM适用于各种解类型，包括三角函数、双曲函数、有理函数和椭圆函数，具有更少的参数限制和多波结构的潜力。数值模拟表明，随着空间变量x的增大，孤子趋于稳定，不同分数阶α值的图排列紧密，表明孤子对α的敏感性较小。此外，在其表面图中，FEW孤子沿时间轴呈现致密的平铺结构，而FC-A孤子沿ξ呈现光滑的扭结状跃迁，这是连接两个稳定平衡态的解的特征。这些发现强调了MHBM在分析分数阶非线性演化方程中的鲁棒性和通用性。

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引用次数: 0

Non-uniqueness of the solution of the Cauchy problem for one higher-order equation with a fractional derivative 具有分数阶导数的高阶方程Cauchy问题解的非唯一性

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-07-04 DOI: 10.1016/j.padiff.2025.101252

B. Yu. Irgashev , H.H. Pulatova

In the article a non-trivial solution of the homogeneous Cauchy problem for a homogeneous high-order equation with a fractional Caputo derivative is constructed.

本文构造了具有分数阶Caputo导数的齐次高阶方程的齐次Cauchy问题的非平凡解。

引用次数: 0

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