Applied Mathematics and Computation最新文献

英文中文

A finite difference method with symmetry properties for the high-dimensional Bratu equation 具有对称性的高维布拉图方程有限差分法

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-30 DOI: 10.1016/j.amc.2024.129136

Muhammad Luthfi Shahab , Hadi Susanto , Haralampos Hatzikirou

Solving the three-dimensional (3D) Bratu equation is highly challenging due to the presence of multiple and sharp solutions. Research on this equation began in the late 1990s, but there are no satisfactory results to date. To address this issue, we introduce a symmetric finite difference method (SFDM) which embeds the symmetry properties of the solutions into a finite difference method (FDM). This SFDM is primarily used to obtain more accurate solutions and bifurcation diagrams for the 3D Bratu equation. Additionally, we propose modifying the Bratu equation by incorporating a new constraint that facilitates the construction of bifurcation diagrams and simplifies handling the turning points. The proposed method, combined with the use of sparse matrix representation, successfully solves the 3D Bratu equation on grids of up to 301³ points. The results demonstrate that SFDM outperforms all previously employed methods for the 3D Bratu equation. Furthermore, we provide bifurcation diagrams for the 1D, 2D, 4D, and 5D cases, and accurately identify the first turning points in all dimensions. All simulations indicate that the bifurcation diagrams of the Bratu equation on the cube domains closely resemble the well-established behavior on the ball domains described by Joseph and Lundgren [1]. Furthermore, when SFDM is applied to linear stability analysis, it yields the same largest real eigenvalue as the standard FDM despite having fewer equations and variables in the nonlinear system.

由于存在多个尖锐解，求解三维（3D）布拉图方程极具挑战性。对该方程的研究始于 20 世纪 90 年代末，但至今仍未取得令人满意的结果。为了解决这个问题，我们引入了一种对称有限差分法（SFDM），它将解的对称性嵌入有限差分法（FDM）中。这种 SFDM 主要用于获得三维布拉图方程更精确的解和分岔图。此外，我们还建议对布拉图方程进行修改，加入一个新的约束条件，以方便构建分岔图并简化转折点的处理。所提出的方法与稀疏矩阵表示法相结合，在多达 3013 个点的网格上成功求解了三维布拉图方程。结果表明，对于三维布拉图方程，SFDM 优于之前采用的所有方法。此外，我们还提供了一维、二维、四维和五维情况下的分岔图，并准确识别了所有维度的第一个转折点。所有模拟都表明，布拉图方程在立方体域上的分岔图与约瑟夫和伦德格伦[1]在球域上描述的成熟行为非常相似。此外，当将 SFDM 应用于线性稳定性分析时，尽管非线性系统中的方程和变量较少，它却能得到与标准 FDM 相同的最大实特征值。

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引用次数: 0

Kirchhoff indices of Cayley graphs on D2n×Zm D2n×Zm 上 Cayley 图的基尔霍夫指数

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-30 DOI: 10.1016/j.amc.2024.129132

Yan Wang , Qian Zhang, Kai Yuan

In this paper, we provide closed-form formulae for the Kirchhoff indices and resistance distances between vertex pairs of Cayley graphs on groups that are the direct product of dihedral groups and cyclic groups.

本文提供了二面群和循环群的直积群上 Cayley 图顶点对之间的基尔霍夫指数和阻力距离的闭式计算公式。

引用次数: 0

Two graphs: Resolving the periodic reversibility of one-dimensional finite cellular automata 两个图形解决一维有限蜂窝自动机的周期可逆性问题

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-30 DOI: 10.1016/j.amc.2024.129151

Chen Wang, Junchi Ma, Chao Wang, Defu Lin, Weilin Chen

Finite cellular automata (FCA), as discrete dynamical systems, are widely used in simulations, coding theory, information theory, and so on. Its reversibility, related to information loss in system evolution, is one of the most important problems. In this paper, we perform calculations on two elaborate graphs — the reversibility graph and the circuit graph and discover that the reversibility of one-dimensional FCA exhibits periodicity as the number of cells increases. We provide a method to compute the reversibility sequence that encompasses the reversibility of one-dimensional FCA with any number of cells in

O (q^{m} V + q V k + V k^{2})

where

q, m, V, k

are the number of states, neighborhood, vertices, and elementary circuits in the reversibility graph of FCA. The calculations in this paper are applicable to FCA with two mainstream boundaries, null boundary and periodic boundary. This means we have an efficient method to determine the reversibility of almost all one-dimensional FCA, with a complexity independent of cell number.

有限蜂窝自动机（FCA）作为离散动力系统，被广泛应用于模拟、编码理论、信息论等领域。其可逆性与系统演化过程中的信息丢失有关，是最重要的问题之一。在本文中，我们对可逆性图和电路图这两个精心设计的图进行了计算，发现一维 FCA 的可逆性随着单元数的增加而呈现周期性。我们提供了一种方法，可以在 O(qmV+qVk+Vk2) 内计算包含任意单元数的一维 FCA 的可逆性序列，其中 q,m,V,k 是 FCA 可逆性图中的状态数、邻域数、顶点数和元电路数。本文的计算适用于具有两种主流边界（空边界和周期边界）的 FCA。这意味着我们有了一种高效的方法来确定几乎所有一维 FCA 的可逆性，其复杂性与单元数无关。

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引用次数: 0

Note on the anti-Ramsey number for matching in hypercubes 超立方体匹配的反拉姆齐数说明

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-28 DOI: 10.1016/j.amc.2024.129154

Rui Li , Yuede Ma , Zhongmei Qin , Yingping Zhao

Let Q be a host graph and

L \subseteq Q

be a subgraph. The anti-Ramsey number

a r (Q, L)

of L in Q, is defined as the largest number t that allows the existence of a t-edge-colored Q which contains no rainbow L. In this paper, the anti-Ramsey number for matchings when the host graph is hypercube is considered and some exact results are provided.

设 Q 是主图，L⊆Q 是子图。Q 中 L 的反拉姆齐数 ar(Q,L)定义为允许不包含彩虹 L 的 t 边着色 Q 存在的最大数 t。本文考虑了主图为超立方体时匹配的反拉姆齐数，并提供了一些精确结果。

引用次数: 0

A general class of constraint preconditioners for generalized saddle point linear systems 广义鞍点线性系统的一类通用约束预处理器

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-28 DOI: 10.1016/j.amc.2024.129148

Hong-Yu Wu

We propose a general class of constraint preconditioners for solving generalized saddle point linear systems, derived from a general matrix splitting of the (1,1) block of the coefficient matrix. This new constraint preconditioner can not only reduce to some existing constraint preconditioners, but also induce new constraint preconditioners under some certain matrix splitting schemes. Then we present invertibility conditions of the proposed constraint preconditioner and establish the convergence analysis of the corresponding constraint preconditioning iteration method. Numerical examples are provided to confirm that the proposed preconditioner outperforms existing ones when suitable matrix splitting schemes are chosen.

我们提出了一类用于求解广义鞍点线性系统的通用约束前提器，该约束前提器源于系数矩阵 (1,1) 块的通用矩阵拆分。这种新的约束条件器不仅能还原现有的一些约束条件器，还能在某些矩阵拆分方案下诱导出新的约束条件器。然后，我们提出了所提约束条件的可逆性条件，并建立了相应约束条件迭代方法的收敛性分析。我们提供的数值示例证实，在选择合适的矩阵拆分方案时，所提出的预条件器优于现有的预条件器。

引用次数: 0

A weighted switching sequence optimization algorithm for static output feedback control synthesis of nonlinear systems 非线性系统静态输出反馈控制合成的加权开关序列优化算法

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-28 DOI: 10.1016/j.amc.2024.129152

Jingjing Gao , Xiangpeng Xie

In this paper, the static output feedback (SOF) control synthesis of discrete-time Takagi-Sugeno (T-S) fuzzy systems is concerned upon homogeneous polynomial parameter dependent Lyapunov functions (HPPD-LFs). It is well known that SOF control always leads to inequality conditions with non-convexity, which makes the optimization problem intractable. To overcome this difficulty, a novel switching sequence convex optimization (SSCO) algorithm is proposed, which is upon the matrix decomposition concept and the inner approximation strategy to eliminate the non-convex terms formed by the controller and the slack variables. Unlike conventional methods, the controller acts as a direct optimization variable and does not require structural or multiplicative relationships between the slack variables, which opens up the possibility of obtaining improved results in terms of

l_{2}

gain performance. In particular, more relaxed design conditions are obtained for SOF controller based on the weighted switching method by effectively utilizing the membership functions information. Finally, two simulation examples demonstrate the superiority of the developed SOF control scheme.

本文以同质多项式参数相关李亚普诺夫函数（HPPD-LFs）为基础，研究离散时间高木-菅野（Takagi-Sugeno，T-S）模糊系统的静态输出反馈（SOF）控制合成。众所周知，SOF 控制总是导致具有非凸性的不等式条件，从而使优化问题变得棘手。为了克服这一困难，我们提出了一种新颖的切换序列凸优化（SSCO）算法，该算法基于矩阵分解概念和内近似策略，以消除控制器和松弛变量形成的非凸项。与传统方法不同的是，控制器作为一个直接优化变量，不需要松弛变量之间的结构或乘法关系，这为获得更好的 l2 增益性能结果提供了可能。特别是，通过有效利用成员函数信息，基于加权切换方法的 SOF 控制器获得了更宽松的设计条件。最后，两个仿真实例证明了所开发的 SOF 控制方案的优越性。

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引用次数: 0

Improving disturbance rejection and tracking performance of interval type-2 T-S fuzzy systems with time-varying delay 改善具有时变延迟的区间型-2 T-S 模糊系统的干扰抑制和跟踪性能

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-28 DOI: 10.1016/j.amc.2024.129150

Jia-Yu Zhao, Tao Zhao

In this paper, the issue of disturbances rejection for a class of interval type-2 T-S fuzzy systems (IT-2 TSFSs) with time-varying delay is studied. First, an augmented dynamic IT-2 TSFS composed of an internal model, observer, and improved equivalent input disturbance (IEID) is constructed. The IEID estimator is inserted into the fuzzy controller to compensate for unknown external disturbances. Meanwhile, a controller structure including state feedback, internal model feedforward, error feedback, and IEID is designed to improve the disturbance rejection performance of the IT-2 TSFSs. Then, the controller design problem is transformed into a stability problem of the augmented dynamic system. In addition, a Lyapunov-Krasovskii functional (LKF) is constructed which relies on the information of integral variables and membership functions (MFs). Using the time derivative information of the MFS, the controller with switching rules is designed. To reduce the conservatism of the system design conditions and improve the degree of freedom, the slack matrix and upper and lower MFs information are introduced in the stability analysis of the control system. Finally, the availability and advantages of our proposed method are validated through a numerical simulation.

本文研究了一类具有时变延迟的区间-2 T-S 模糊系统（IT-2 TSFS）的干扰抑制问题。首先，构建了一个由内部模型、观测器和改进等效输入干扰（IEID）组成的增强动态 IT-2 TSFS。在模糊控制器中插入 IEID 估计器，以补偿未知的外部干扰。同时，设计了一种包括状态反馈、内部模型前馈、误差反馈和 IEID 的控制器结构，以提高 IT-2 TSFS 的干扰抑制性能。然后，控制器设计问题被转化为增强动态系统的稳定性问题。此外，还利用积分变量和成员函数（MFs）的信息构建了 Lyapunov-Krasovskii 函数（LKF）。利用 MFS 的时间导数信息，设计出具有切换规则的控制器。为了降低系统设计条件的保守性并提高自由度，在控制系统的稳定性分析中引入了松弛矩阵和上下 MFs 信息。最后，通过数值模拟验证了我们所提方法的可用性和优势。

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引用次数: 0

Cooperation under endogenous punishment in the spatial public goods game 空间公共产品博弈中内生惩罚下的合作

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-28 DOI: 10.1016/j.amc.2024.129156

Shiping Gao, Jinghui Suo, Nan Li

Punishment and network reciprocity have profound implications for the evolution of cooperation. However, existing research on the consequences of cooperation under punishment in social networks has largely relied on agent-based models and laboratory experiments. Moreover, different from the majority of existing studies where punishment is always believed to be deterministic, the individuals' preferences for certain behaviors are always stochastic and vary with the environment. There is an urgent need to explore how cooperation evolves when punishment is stochastic and endogenous in social networks. In this paper, we propose a theoretical model of endogenous punishment in spatial public goods games. Cooperators each can stochastically choose whether to participate in the punishment for defectors. The choice to penalize defectors comes with a price. Whether and how defectors are punished is endogenously determined by the cooperators' preferences for executing the costly punishment. We analyze how cooperation evolves under endogenous punishment based on a regular network in the mean-field limit and outline the conditions under which endogenous punishment can support cooperation. When network reciprocity is unfavorable for cooperation, endogenous punishment can be effective in supporting cooperation. On the contrary, endogenous punishment no longer supports or even hinders the promoting effect of network reciprocity on cooperation. These findings illustrate that the effectiveness of endogenous punishment in fostering cooperation is dependent on the cooperators' willingness to pay for punishment as well as the topology of social networks.

惩罚和网络互惠对合作的演化有着深远的影响。然而，现有关于社会网络中惩罚下的合作后果的研究大多依赖于基于代理的模型和实验室实验。此外，与大多数现有研究总是认为惩罚是确定性的不同，个体对某些行为的偏好总是随机的，并随着环境的变化而变化。当社会网络中的惩罚具有随机性和内生性时，合作是如何演变的亟待探索。本文提出了空间公共物品博弈中内生惩罚的理论模型。每个合作者都可以随机选择是否参与对叛逃者的惩罚。选择惩罚叛逃者是有代价的。是否惩罚叛逃者以及如何惩罚叛逃者由合作者对执行代价高昂的惩罚的偏好内生决定。我们以均值场极限中的规则网络为基础，分析了内生惩罚下的合作如何演化，并概述了内生惩罚支持合作的条件。当网络互惠不利于合作时，内生惩罚可以有效地支持合作。相反，内生惩罚不再支持甚至阻碍网络互惠对合作的促进作用。这些发现说明，内生惩罚在促进合作方面的有效性取决于合作者是否愿意为惩罚付出代价以及社会网络的拓扑结构。

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引用次数: 0

Convergence analysis of collocation solutions for delay Volterra integral equations with weakly singular kernels 具有弱奇异核的延迟 Volterra 积分方程配位解的收敛性分析

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-24 DOI: 10.1016/j.amc.2024.129122

P. Peyrovan , A. Tari , H. Brunner

Convergence analysis of the collocation solutions for second-kind Volterra integral equations (VIEs) with weakly singular kernels (WSKs) in continuous piecewise polynomial space (PPS) under certain conditions on the collocation parameters has been established previously. In this paper, we study the analogue convergence analysis for delay Volterra integral equations (DVIEs) with WSKs and vanishing delay. We also investigate the existence, uniqueness and regularity of solution. Finally, we present some illustrative numerical examples to confirm the theoretical results.

对于连续片断多项式空间（PPS）中具有弱奇异内核（WSKs）的第二类伏特拉积分方程（VIEs），在一定的配位参数条件下，其配位解的收敛性分析已经建立。本文研究了具有 WSK 和消失延迟的延迟 Volterra 积分方程 (DVIE) 的类比收敛分析。我们还研究了解的存在性、唯一性和正则性。最后，我们给出了一些数值示例来证实理论结果。

引用次数: 0

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation 二维时间分数反应-次扩散方程的完全离散 GL-ADI 方案

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation

Pub Date : 2024-10-24 DOI: 10.1016/j.amc.2024.129147

Yubing Jiang , Hu Chen , Chaobao Huang , Jian Wang

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at

t = 0

is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with

0 < α < 1

) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in

L^{2}

norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is

O (τ t_{s}^{α - 1} + τ^{2 α} + h_{1}^{2} + h_{2}^{2})

. Numerical experiments are given to verify the theoretical analysis.

本文研究了交替方向隐式（ADI）差分法求解二维反应-次扩散方程，该方程的解在 t=0 处表现为弱奇点。在均匀网格上对 Caputo 分数导数（阶数 α，0<α<1）的离散化采用了 Grünwald-Letnikov (GL) 近似方法。严格确定了完全离散 ADI 方案的稳定性和收敛性。在离散分式 Gronwall 不等式的帮助下，我们得到了尖锐的误差估计。严格证明了 GL-ADI 方案在 L2 规范下的稳定性和收敛性，其中收敛阶数为 O(τtsα-1+τ2α+h12+h22)。数值实验验证了理论分析。

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