European Journal of Applied Mathematics最新文献_第7页

Two approximate symmetry frameworks for nonlinear partial differential equations with a small parameter: Comparisons, relations, approximate solutions 小参数非线性偏微分方程的两个近似对称框架:比较，关系，近似解

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-12-16 DOI: 10.1017/S0956792522000407

Mahmood R. Tarayrah, Brian Pitzel, A. Cheviakov

Abstract The frameworks of Baikov–Gazizov–Ibragimov (BGI) and Fushchich–Shtelen (FS) approximate symmetries are used to study symmetry properties of partial differential equations with a small parameter. In general, it is shown that unlike the case of ordinary differential equations (ODEs), unstable BGI point symmetries of unperturbed partial differential equations (PDEs) do not necessarily yield local approximate symmetries for the perturbed model. While some relations between the BGI and FS approaches can be established, the two methods yield different approximate symmetry classifications. Detailed classifications are presented for two nonlinear PDE families. The second family includes a one-dimensional wave equation describing the wave motion in a hyperelastic material with a single family of fibers. For this model, approximate symmetries can be used to compute approximate closed-form solutions. Wave breaking times are found numerically and using the approximate solutions, which yield comparable results.

摘要利用Baikov-Gazizov-Ibragimov (BGI)和Fushchich-Shtelen (FS)近似对称框架研究了小参数偏微分方程的对称性。一般来说，与常微分方程(ode)的情况不同，非摄动偏微分方程(PDEs)的不稳定BGI点对称性不一定产生摄动模型的局部近似对称性。虽然BGI和FS方法之间可以建立一些联系，但这两种方法产生了不同的近似对称分类。给出了两个非线性偏微分方程族的详细分类。第二类包括一维波动方程，描述具有单一纤维族的超弹性材料中的波动运动。对于该模型，近似对称性可以用来计算近似闭型解。波浪破碎时间是用数值方法和近似解求得的，得到了可比较的结果。

引用次数: 1

Constant rank factorisations of smooth maps, with applications to sonar 光滑地图的常秩因子分解及其在声纳中的应用

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-12-01 DOI: 10.1017/s0956792522000365

Michael Robinson

Sonar systems are frequently used to classify objects at a distance by using the structure of the echoes of acoustic waves as a proxy for the object’s shape and composition. Traditional synthetic aperture processing is highly effective in solving classification problems when the conditions are favourable but relies on accurate knowledge of the sensor’s trajectory relative to the object being measured. This article provides several new theoretical tools that decouple object classification performance from trajectory estimation in synthetic aperture sonar processing. The key insight is that decoupling the trajectory from classification-relevant information involves factoring a function into the composition of two functions. The article presents several new general topological invariants for smooth functions based on their factorisations over function composition. These invariants specialise to the case when a sonar platform trajectory is deformed by a non-small perturbation. The mathematical results exhibited in this article apply well beyond sonar classification problems. This article is written in a way that supports full mathematical generality.

声纳系统经常用于通过使用声波回波的结构作为物体形状和成分的代理来对远处的物体进行分类。当条件有利时，传统的合成孔径处理在解决分类问题方面非常有效，但它依赖于传感器相对于被测量对象的轨迹的准确知识。本文提供了几种新的理论工具，将合成孔径声纳处理中的目标分类性能与轨迹估计解耦。关键的见解是，将轨迹与分类相关信息脱钩涉及将一个函数分解为两个函数的组合。基于光滑函数在函数组合上的因子分解，给出了几种新的光滑函数的一般拓扑不变量。这些不变量专门用于声纳平台轨迹因非小扰动而变形的情况。本文所展示的数学结果远远超出了声纳分类问题的应用范围。这篇文章的写作方式支持数学的全面通用性。

引用次数: 0

A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery cells 参数化锂离子电池单元高效降阶模拟的建模框架

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-11-29 DOI: 10.1017/S0956792522000353

M. Landstorfer, Mario Ohlberger, S. Rave, M. Tacke

Abstract In this contribution, we present a modelling and simulation framework for parametrised lithium-ion battery cells. We first derive a continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterised non-linear system of partial differential equations, the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modelling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modelling framework.

摘要在这篇文章中，我们提出了一个参数化锂离子电池的建模和仿真框架。我们首先在非平衡热力学的基础上推导了一个相当普遍的插层电池的连续体模型。为了有效地评估由此产生的参数化非线性偏微分方程组，采用了降基方法。降阶基法是一种基于增量层次近似自正交分解方法和经验算子插值的模型降阶技术。该建模框架特别适合于研究和量化电池单元的退化影响。给出了几个数值实验来证明建模框架的范围和效率。

引用次数: 1

Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems lotka - volterra型种群系统的精确非经典对称解

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-11-25 DOI: 10.1017/S095679252200033X

P. Broadbridge, R. Cherniha, J. Goard

Abstract New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

摘要介绍了一类新的非线性反应-扩散方程的条件可积系统。它们是通过将标量偏微分方程的一个众所周知的非经典对称性扩展到向量方程而获得的。构造了具有交叉扩散的非线性捕食者-被捕食系统的新的精确解。对于这样的非线性系统，可以得到无限维类的精确解。这些解中的一些朝着消光衰减，一些围绕内部不动点振荡或螺旋。结果表明，条件可积系统与标准扩散Lotka-Volterra系统密切相关，但它们具有额外的特征。

引用次数: 2

Existence and stability of singular patterns in a fractional Ginzburg–Landau equation with a mean field 具有平均场的分数阶Ginzburg-Landau方程奇异模式的存在性和稳定性

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-11-11 DOI: 10.1017/s0956792522000286

Mingchen Gao, M. Winter, Wen Yang

In this paper, we consider the existence and stability of singular patterns in a fractional Ginzburg–Landau equation with a mean field. We prove the existence of three types of singular steady-state patterns (double fronts, single spikes, and double spikes) by solving their respective consistency conditions. In the case of single spikes, we prove the stability of single small spike solution for sufficiently large spatial period by studying an explicit non-local eigenvalue problem which is equivalent to the original eigenvalue problem. For the other solutions, we prove the instability by using the variational characterisation of eigenvalues. Finally, we present the results of some numerical computations of spike solutions based on the finite difference methods of Crank–Nicolson and Adams–Bashforth.

本文研究了一类具有平均域的分数阶Ginzburg-Landau方程奇异模式的存在性和稳定性。通过求解其一致性条件，证明了三种奇异稳态模式(双锋、单尖峰和双尖峰)的存在性。在单尖峰情况下，通过研究一个等价于原特征值问题的显式非局部特征值问题，证明了单小尖峰解在足够大的空间周期内的稳定性。对于其他解，我们利用特征值的变分刻画证明了其不稳定性。最后，我们给出了基于Crank-Nicolson和Adams-Bashforth有限差分方法的一些脉冲解的数值计算结果。

引用次数: 0

A deterministic gradient-based approach to avoid saddle points 基于确定性梯度的避鞍点方法

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-11-09 DOI: 10.1017/s0956792522000316

L. M. Kreusser, S. J. Osher, B. Wang

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $lfloor (n-1)/2rfloor$, and hence it is significantly smaller than that of GD whose dimension is $n-1$.

具有大量鞍点的损失函数是有效训练现代机器学习(ML)模型的主要障碍之一。梯度下降(GD)等一阶方法通常是训练ML模型的首选方法。然而，对于初始猜测的某些选择，这些方法收敛到鞍点。在本文中，我们对最近提出的拉普拉斯平滑梯度下降(LSGD) [Osher et al.， arXiv:1806.06317]提出了一种修正，称为修正LSGD (mLSGD)，并证明了其在不牺牲收敛速度的情况下避免鞍点的潜力。我们的分析是基于引力区域的，引力区域由所考虑的数值方案收敛于鞍点的所有起点组成。我们用解析法和数值法研究了引力区域的维数。对于一类典型的二次函数，我们证明了mLSGD的吸引域的维数为$lfloor (n-1)/2rfloor$，因此它明显小于维数为$n-1$的GD的吸引域。

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引用次数: 0

A deterministic gradient-based approach to avoid saddle points 基于确定性梯度的避鞍点方法

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-11-09 DOI: 10.1017/s0956792522000316

L. M. Kreusser, S. J. Osher, B. Wang

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $lfloor (n-1)/2rfloor$, and hence it is significantly smaller than that of GD whose dimension is $n-1$.

具有大量鞍点的损失函数是有效训练现代机器学习(ML)模型的主要障碍之一。梯度下降(GD)等一阶方法通常是训练ML模型的首选方法。然而，对于初始猜测的某些选择，这些方法收敛到鞍点。在本文中，我们对最近提出的拉普拉斯平滑梯度下降(LSGD) [Osher et al.， arXiv:1806.06317]提出了一种修正，称为修正LSGD (mLSGD)，并证明了其在不牺牲收敛速度的情况下避免鞍点的潜力。我们的分析是基于引力区域的，引力区域由所考虑的数值方案收敛于鞍点的所有起点组成。我们用解析法和数值法研究了引力区域的维数。对于一类典型的二次函数，我们证明了mLSGD的吸引域的维数为$lfloor (n-1)/2rfloor$，因此它明显小于维数为$n-1$的GD的吸引域。

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引用次数: 0

Time-dependent modelling of thin poroelastic films drying on deformable plates 多孔弹性薄膜在可变形板上干燥的时间相关模型

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-10-31 DOI: 10.1017/s0956792523000062

M. Hennessy, R. Craster, O. Matar

Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from $t$ to $t^{1/2}$ , in agreement with experiments.

了解机械应力的产生在干燥，颗粒负载薄膜是重要的广泛的工业过程。研究这些应力的一种方法是通过悬臂实验，将薄膜沉积在薄板的表面，薄板的一端夹在墙上。干燥过程中膜内产生的应力传递给印版，带动弯曲。数学模型使薄膜应力可以从板挠度的测量推断出来。本文的目的是利用渐近方法从连续介质力学的时间相关方程中推导出悬臂梁实验的简化模型。薄膜用非线性孔隙弹性来描述，板用非线性弹性来描述。与stone -like公式相比，简化模型考虑了薄膜厚度和应力不均匀的情况。薄膜模型可简化为可独立于平板方程求解的单一微分方程。该板块模型简化为Föppl-von Kármán (FvK)方程的扩展形式，该方程考虑了作用于板块表面的纵向牵引力梯度。通过解析板自由边缘的Saint-Venant边界层，导出了FvK方程的一致边界条件。渐近简化模型与完整控制方程的有限元解非常吻合。随着psamclet数的增加，板挠度的时间演化由$t$变为$t^{1/2}$，与实验结果一致。

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引用次数: 0

Modelling, simulation and optimisation of parabolic trough power plants 抛物线槽式电站的建模、仿真与优化

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-10-11 DOI: 10.1017/S0956792522000274

H. Bakhti, I. Gasser, Stefan Dipl.-Ing. Schuster, E. Parfenov

We present a mathematical model built to describe the fluid dynamics for the heat transfer fluid in a parabolic trough power plant. Such a power plant consists of a network of tubes for the heat transport fluid. In view of optimisation tasks in the planning and in the operational phase, it is crucial to find a compromise between a very detailed description of many possible physical phenomena and a necessary simplicity needed for a fast and robust computational approach. We present the model, a numerical approach, simulation for single tubes and also for realistic network settings. In addition, we optimise the power output with respect to the operational parameters.

本文建立了一个描述抛物线槽式电站传热流体动力学的数学模型。这样的发电厂是由输送热流体的管道网络组成的。考虑到规划和操作阶段的优化任务，在对许多可能的物理现象的非常详细的描述和快速和健壮的计算方法所需的必要的简单性之间找到折衷是至关重要的。我们提出了一个模型，一个数值方法，模拟单管和现实的网络设置。此外，我们还根据操作参数优化了功率输出。

引用次数: 0

Friedlander-Keller ray expansions in electromagnetism: Monochromatic radiation from arbitrary surfaces in three dimensions 电磁学中的Friedlander-Keller射线展开：来自三维任意表面的单色辐射

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

European Journal of Applied Mathematics

Pub Date : 2022-10-10 DOI: 10.1017/s0956792522000249

A. Radjen, R. Tew, G. Gradoni

The standard approach to applying ray theory to solving Maxwell’s equations in the large wave-number limit involves seeking solutions that have (i) an oscillatory exponential with a phase term that is linear in the wave-number and (ii) has an amplitude profile expressed in terms of inverse powers of that wave-number. The Friedlander–Keller modification includes an additional power of this wave-number in the phase of the wave structure, and this additional term is crucial when analysing certain wave phenomena such as creeping and whispering gallery wave propagation. However, other wave phenomena necessitate a generalisation of this theory. The purposes of this paper are to provide a ‘generalised’ Friedlander–Keller ray ansatz for Maxwell’s equations to obtain a new set of field equations for the various phase terms and amplitude of the wave structure; these are then solved subject to boundary data conforming to wave-fronts that are either specified or general. These examples specifically require this generalisation as they are not amenable to classic ray theory.

将射线理论应用于求解大波数极限下的麦克斯韦方程组的标准方法包括寻求具有（i）振荡指数的解，该振荡指数的相位项在波数上是线性的，并且（ii）具有用该波数的逆幂表示的振幅轮廓。Friedlander–Keller修正在波浪结构的相位中包含了该波数的附加幂，并且在分析某些波浪现象（如蠕变和回音壁波传播）时，该附加项至关重要。然而，其他波动现象需要推广这一理论。本文的目的是为麦克斯韦方程组提供一个“广义”的Friedlander–Keller-ray变换，以获得一组新的波结构不同相位项和振幅的场方程；然后根据符合特定或一般波前的边界数据来求解这些问题。这些例子特别需要这种概括，因为它们不符合经典的射线理论。

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