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偏微分方程指含有未知函数及其偏导数的方程。描述自变量、未知函数及其偏导数之间的关系。符合这个关系的函数是方程的解。

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我们提供的偏微分方程partial difference equations及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
数学代写|偏微分方程代写partial difference equations代考|MAP4341

数学代写|偏微分方程代写partial difference equations代考|OTHER DETERMINISTIC METHODS

Outside of the three volume discretization based methods discussed in the preceding three subsections, a myriad of other methods for solving PDEs exist and continue to be used. It is probably fair to state, however, that these “other” methods are either still in their infancy – relatively speaking – or have been used only for a special class of problems for which they were designed. In terms of usage, there is no doubt that the three methods discussed in subsections 1.3.1 through 1.3.3 far outweigh any other method for solving PDEs. While construction of an exhaustive list of other methods for solving PDEs is quite daunting and, perhaps, of little use in the opinion of this author, at least a few of these other methods deserve a mention. These include the so-called meshless or meshfree method, the boundary element method, and the spectral method.

The name meshless or meshfree method is quite broad, and is applicable to any method that does not rely on a mesh. These may also include statistical or spectral methods. In the specific case of solution of PDEs, however, the name is reserved for methods where a cloud of points is used to develop an algebraic approximation to the PDE. These points have domains of influence, but do not have rigid interconnections, as in a traditional mesh. One of the notable features of meshless methods is that the points (or nodes) can be allowed to move freely, which, in the context of a traditional mesh, may end up twisting the mesh to a state where it cannot be used, and may require remeshing – a step that brings a new set of difficulties to the table. In light of this feature, meshless methods have grown in popularity for the solution of problems that involve moving boundaries, such as fluid-structure interaction [39.40]. multiphase flow $[41,42]$, moving body problems, and any problem where nodes may be created or destroyed due to material addition or removal [42]. The meshless method is a close relative of the smooth particle hydrodynamics method $[43,44]$ that was developed for fluid flow. Some researchers contend that, fundamentally, the method is closely related to either variational formulations using radial basis functions or the moving least squares method [45]. The general concept of not using a mesh has been used in a variety of methods that include the element-free Galerkin method [46], the meshless local Petrov-Galerkin method [47], the diffuse element method [48],and the immersed boundary method [49], among others. A good review of meshless methods, as it applies to the solution of advection-diffusion type PDEs, has been provided by Pepper [50].

The boundary element method (BEM) is a derivative of the finite element method. Its attractive feature is that it does not require a volumetric mesh, but only a surface mesh. This is particularly convenient for $2 \mathrm{D}$ geometries, wherein the only geometric input required is a set of coordinates of points. BEM is particularly suited to problems for which the interior solution is not of practical interest, but the quantities at the boundary are, for example, a heat transfer problem in which one is primarily interested in the heat flux and temperature at the boundaries. The method is significantly more efficient than the conventional finite element method. Unfortunately, the method has several serious limitations. BEM can only be used for PDEs with constant coefficients or in which the variation of the coefficients obey a certain format. This is because it is only applicable to problems for which Green’s functions can be computed easily. Nonlinear PDEs, PDEs with strong inhomogeneities, and 3D problems are generally outside the realm of pure BEM. The application of BEM to such scenarios usually requires a volumetric mesh and combination of BEM with a volume discretization method, which undermines its advantage of being an efficient method. On account of these limitations BEM has found limited use except for some specific problems in mechanics $[51,52]$, heat conduction $[53,54]$, and electrostatics [51].

数学代写|偏微分方程代写partial difference equations代考|OVERVIEW OF MESH TYPES

Since the focus of this text is the finite difference and finite volume methods, and both of those methods require a mesh for discretization of the governing PDE, a discussion of the mesh – both its type and attributes – is a prerequisite for understanding the discretization procedures presented in later chapters. The word, grid, usually refers to the interconnected nodes and lines (see Fig. 1.4), while the word, mesh, usually refers to the spaces enclosed by the lines. Throughout this text, this subtle distinction will be ignored since it only creates confusion in the mind of the reader. Henceforth, the words mesh and grid will be used interchangeably.

The history of discretization-based procedures for solving boundary value problems can be traced back to the works of Courant around 1928. In those days, when computers were almost nonexistent, solving an elliptic PDE on a square mesh with a few handful of node points was already considered ground breaking. The need for a complex procedure or algorithm to generate a mesh was never felt. As a matter of fact, this trend continued into the early 1960s, during which period, most reported computations were performed on rectangular domains with a perfectly orthogonal (Cartesian) mesh, and mostly for 2D geometries, as shown in Fig. 1.6(a). The seminal paper by Patankar and Spalding, published in 1972, which introduced the world to the famous SIMPLE algorithm [61], reports solution of the Navier-Stokes equation performed in a square domain with a $16 \times 16$ Cartesian mesh. With rapidly advancing computer technology, by the late 1970 s, the feasibility of performing slightly larger scale computations was realized.

Along with the desire to perform larger computations came the desire to perform computations in irregular geometries. In the earliest computations of this type, angled or curved boundaries were addressed using a stair-step grid [5,26], as shown in Fig. 1.6(b). With the growing emphasis on satisfying conservation laws, and the resulting proliferation of the finite volume method, the stair-step approach was soon replaced by the cut cell approach, which produced polygonal cells (control volumes) at curved boundaries, as depicted in Fig. 1.7. This was considered both a reasonable as well as convenient approximation – reasonable because with a fine mesh one can always approximate a curve with piecewise linear segments (it is a practice followed even today!), and convenient because the core finite volume procedure required little modification to accommodate one slanted edge.

数学代写|偏微分方程代写partial difference equations代考|MAP4341

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|OTHER DETERMINISTIC METHODS

除了前面三个小节中讨论的基于体积离散化的三个方法之外,还有无数其他求解 PDE 的方法存在并继续使用。然而,公平地说,这些“其他”方法要么仍处于起步阶段(相对而言),要么仅用于解决它们所设计的特殊类别的问题。在使用方面,毫无疑问,1.3.1 到 1.3.3 小节中讨论的三种方法远远超过求解 PDE 的任何其他方法。虽然构建用于求解 PDE 的其他方法的详尽列表非常令人生畏,并且在作者看来可能没什么用处,但至少有一些其他方法值得一提。这些包括所谓的无网格或无网格法、边界元法和谱法。

无网格或无网格方法的名称相当广泛,适用于任何不依赖网格的方法。这些也可能包括统计或光谱方法。然而,在 PDE 求解的特定情况下,该名称保留用于使用点云来开发 PDE 的代数近似的方法。这些点具有影响域,但没有像传统网格中那样的刚性互连。无网格方法的显着特征之一是可以允许点(或节点)自由移动,这在传统网格的上下文中可能最终将网格扭曲到无法使用的状态,并且可能需要重新划分网格——这一步骤带来了一系列新的困难。鉴于这一特点,无网格方法在解决涉及移动边界的问题方面越来越受欢迎,例如流固耦合[39.40]。多相流[41,42],移动体问题,以及由于材料添加或移除而可能创建或破坏节点的任何问题[42]。无网格法是光滑粒子流体动力学方法的近亲[43,44]这是为流体流动而开发的。一些研究人员认为,从根本上说,该方法与使用径向基函数的变分公式或移动最小二乘法 [45] 密切相关。不使用网格的一般概念已用于多种方法,包括无单元 Galerkin 方法 [46]、无网格局部 Petrov-Galerkin 方法 [47]、扩散单元方法 [48] 和浸入式边界方法[49]等。Pepper [50] 对无网格方法进行了很好的回顾,因为它适用于对流扩散型 PDE 的解决方案。

边界元法(BEM)是有限元法的衍生。它的吸引人的特点是它不需要体积网格,而只需要一个表面网格。这特别方便2D几何,其中唯一需要的几何输入是一组点的坐标。BEM 特别适用于内部解不具有实际意义的问题,但边界处的量是例如传热问题,其中人们主要对边界处的热通量和温度感兴趣。该方法明显比传统的有限元方法更有效。不幸的是,该方法有几个严重的局限性。BEM 只能用于具有恒定系数或系数变化服从某种格式的偏微分方程。这是因为它仅适用于可以轻松计算格林函数的问题。非线性偏微分方程、具有强不均匀性的偏微分方程和 3D 问题通常不在纯 BEM 的范围内。边界元法在此类场景中的应用通常需要体积网格以及边界元法与体积离散化方法的结合,这削弱了其作为一种高效方法的优势。由于这些限制,BEM 的使用受到了限制,除了一些特定的力学问题[51,52], 热传导[53,54],和静电[51]。

数学代写|偏微分方程代写partial difference equations代考|OVERVIEW OF MESH TYPES

由于本文的重点是有限差分法和有限体积法,并且这两种方法都需要网格来对控制 PDE 进行离散化,因此对网格(包括其类型和属性)的讨论是理解离散化过程的先决条件在后面的章节中介绍。网格一词通常是指相互连接的节点和线(见图 1.4),而网格一词通常是指由线包围的空间。在整本书中,这种微妙的区别将被忽略,因为它只会在读者心中造成混乱。此后,网格和网格这两个词将互换使用。

求解边值问题的基于离散化的程序的历史可以追溯到 1928 年左右 Courant 的工作。在那些日子里,计算机几乎不存在,在具有少量节点的方形网格上求解椭圆 PDE 是已经被认为是开创性的。从未感觉到需要复杂的程序或算法来生成网格。事实上,这种趋势一直持续到 1960 年代初期,在此期间,大多数报告的计算都是在具有完全正交(笛卡尔)网格的矩形域上进行的,并且主要针对 2D 几何,如图 1.6(a) 所示. Patankar 和 Spalding 于 1972 年发表的开创性论文向世界介绍了著名的 SIMPLE 算法 [61],报告了在平方域中执行的 Navier-Stokes 方程的解16×16笛卡尔网格。随着计算机技术的飞速发展,到 1970 年代后期,实现了进行稍大规模计算的可行性。

随着执行更大计算的愿望而来的是对不规则几何图形执行计算的愿望。在这种类型的最早计算中,有角度的或弯曲的边界是使用阶梯网格[5,26]来处理的,如图 1.6(b) 所示。随着对满足守恒定律的日益重视,以及由此产生的有限体积方法的普及,阶梯方法很快被切割单元方法所取代,切割单元方法在弯曲边界处产生多边形单元(控制体积),如图 1 所示。 1.7. 这被认为是一种既合理又方便的近似——合理,因为使用精细网格,总是可以用分段线性段近似曲线(即使在今天也是一种做法!),

数学代写|偏微分方程代写partial difference equations代考

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题,以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法,其中不假设数据来自于由少数参数决定的规定模型;这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型(GLM)归属统计学领域,是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。

有限元方法代写

有限元方法(FEM)是一种流行的方法,用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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随机分析代写


随机微积分是数学的一个分支,对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。

回归分析代写

多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

R语言代写问卷设计与分析代写
PYTHON代写回归分析与线性模型代写
MATLAB代写方差分析与试验设计代写
STATA代写机器学习/统计学习代写
SPSS代写计量经济学代写
EVIEWS代写时间序列分析代写
EXCEL代写深度学习代写
SQL代写各种数据建模与可视化代写