## 数学代写|数值分析代写numerical analysis代考|CIVL5458

2023年3月30日

couryes-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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
couryes™为您提供可以保分的包课服务

## 数学代写|数值分析代写numerical analysis代考|Handling Boundary Conditions

So far we have considered essential (or Dirichlet) boundary conditions, where $u(\boldsymbol{x})=g(\boldsymbol{x})$ for all $\boldsymbol{x} \in \partial \Omega$ with $g$ a given function. There are many other kinds of linear boundary conditions, most particularly natural (or Neumann) boundary conditions which in this case have the form $\partial u / \partial n(\boldsymbol{x})=h(\boldsymbol{x})$ on $\partial \Omega$ where $\partial / \partial n$ is the outward normal derivative, and mixed (or Robin) boundary conditions which combine the previous two types. Consider the elliptic partial differential equation (6.3.31) $-\operatorname{div}(a(\boldsymbol{x}) \nabla u)+b(\boldsymbol{x}) u=f(\boldsymbol{x}) \quad$ in $\Omega$.
We can create a weak form through multiplying by a smooth function $v(\boldsymbol{x})$ and integrating over $\Omega$. Then
\begin{aligned} \int_{\Omega} v & {[-\operatorname{div}(a(\boldsymbol{x}) \nabla u)+b(\boldsymbol{x}) u-f] d \boldsymbol{x} } \ = & \int_{\Omega}\left{-\operatorname{div}(v a(\boldsymbol{x}) \nabla u)+a(\boldsymbol{x}) \nabla v^T \nabla u+v[b(\boldsymbol{x}) u-f]\right} d \boldsymbol{x} \ = & -\int_{\partial \Omega} v a(\boldsymbol{x}) \nabla u^T \boldsymbol{n}(\boldsymbol{x}) d S(\boldsymbol{x}) \ & +\int_{\Omega}\left[a(\boldsymbol{x}) \nabla v^T \nabla u+b(\boldsymbol{x}) v u-f v\right] d \boldsymbol{x} . \end{aligned}
If we have essential boundary conditions $u(\boldsymbol{x})=g(\boldsymbol{x})$ for $\boldsymbol{x} \in \Gamma_D$ with $\Gamma_D$ a subset of $\partial \Omega$, then we need to impose the condition that $v(\boldsymbol{x})=0$ for $x \in \Gamma_D$. On the other hand, if we have natural boundary conditions $\partial u / \partial n(\boldsymbol{x})=\boldsymbol{n}(\boldsymbol{x})^T \nabla u(\boldsymbol{x})=h(\boldsymbol{x})$ for $\boldsymbol{x} \in \Gamma_N$, then we have to set
\begin{aligned} 0= & -\int_{\partial \Omega} v(\boldsymbol{x}) a(\boldsymbol{x}) \nabla u(\boldsymbol{x})^T n(\boldsymbol{x}) d S(\boldsymbol{x}) \ & +\int_{\Omega}\left[a(\boldsymbol{x}) \nabla v(\boldsymbol{x})^T \nabla u(\boldsymbol{x})+b(\boldsymbol{x}) v(\boldsymbol{x}) u(\boldsymbol{x})-f(\boldsymbol{x}) v(\boldsymbol{x})\right] d \boldsymbol{x} . \end{aligned}

## 数学代写|数值分析代写numerical analysis代考|Numerical Integration

In general, these entries will need to be computed numerically using a suitable numerical integration method, such as are described in Section 5.3.4. This will perturb the matrix entries and the right-hand side of the linear system to be solved. Using a numerical approximation of these integrals
$$a_{j k} \approx \sum_{\ell=1}^M w_{\ell}\left[a\left(z_{\ell}\right) \nabla \phi_j\left(z_{\ell}\right)^T \nabla \phi_k\left(z_{\ell}\right)+b\left(z_{\ell}\right) \phi_j\left(z_{\ell}\right) \phi_k\left(z_{\ell}\right)\right]$$
we still obtain symmetric, and provided there are sufficiently many integration points $z_{\ell}$ in each triangle, positive definite linear systems of equations.

Given an integration method on a reference triangle $\widehat{K}$ and using an affine transformation $\boldsymbol{T}K: K \rightarrow K$ we have a corresponding integration method on $K$ : $$\int{\widehat{K}} \widehat{\psi}(\widehat{\boldsymbol{x}}) d \widehat{\boldsymbol{x}} \approx \sum_{\ell=1}^{\hat{M}} \widehat{w}{\ell} \widehat{\psi}\left(\widehat{\boldsymbol{z}}{\ell}\right)$$
If $\psi(\boldsymbol{x})=\widehat{\psi}(\widehat{\boldsymbol{x}})$ where $\boldsymbol{x}=\boldsymbol{T}K(\widehat{\boldsymbol{x}})=A_K \widehat{\boldsymbol{x}}+\boldsymbol{b}_K$, we have the approximation \begin{aligned} \int_K \psi(\boldsymbol{x}) & =\left|\operatorname{det} A_K\right| \int{\widehat{K}} \widehat{\psi}(\widehat{\boldsymbol{x}}) d \widehat{\boldsymbol{x}} \ & \approx\left|\operatorname{det} A_K\right| \sum_{\ell=1}^{\hat{M}} \widehat{w}{\ell} \widehat{\psi}\left(\widehat{z}{\ell}\right) \ & =\left|\operatorname{det} A_K\right| \sum_{\ell=1}^{\hat{M}} \widehat{w}{\ell} \psi\left(\boldsymbol{T}_K\left(\widehat{z}{\ell}\right)\right) . \end{aligned}
For $\phi_j(\boldsymbol{x})=\widehat{\phi}_r(\widehat{\boldsymbol{x}})$ when $\boldsymbol{x} \in K$, note that $\nabla \phi_j(\boldsymbol{x})=A_K^{-T} \nabla \widehat{\phi}_r(\widehat{\boldsymbol{x}})$, and so
$$\nabla \phi_j(\boldsymbol{x})^T \nabla \phi_k(\boldsymbol{x})=\nabla \widehat{\phi}_r(\widehat{\boldsymbol{x}})^T A_K^{-1} A_K^{-T} \nabla \widehat{\phi}_s(\widehat{\boldsymbol{x}}),$$
where $\phi_k(\boldsymbol{x})=\widehat{\phi}_s(\widehat{\boldsymbol{x}})$ for $\boldsymbol{x} \in K$.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Handling Boundary Conditions

$$0=-\int_{\partial \Omega} v(\boldsymbol{x}) a(\boldsymbol{x}) \nabla u(\boldsymbol{x})^T n(\boldsymbol{x}) d S(\boldsymbol{x}) \quad+\int_{\Omega}$$

## 数学代写|数值分析代写numerical analysis代考|Numerical Integration

$$a_{j k} \approx \sum_{\ell=1}^M w_{\ell}\left[a\left(z_{\ell}\right) \nabla \phi_j\left(z_{\ell}\right)^T \nabla \phi_k\left(z_{\ell}\right)+b\left(z_{\ell}\right) \phi_j\right.$$

$$\int \widehat{K} \widehat{\psi}(\widehat{\boldsymbol{x}}) d \widehat{\boldsymbol{x}} \approx \sum_{\ell=1}^{\hat{M}} \widehat{w} \ell \widehat{\psi}(\widehat{\boldsymbol{z}} \ell)$$

$\boldsymbol{x}=\boldsymbol{T} K(\widehat{\boldsymbol{x}})=A_K \widehat{\boldsymbol{x}}+\boldsymbol{b}_K$ ，我们有近似值
$$\int_K \psi(\boldsymbol{x})=\left|\operatorname{det} A_K\right| \int \widehat{K} \widehat{\psi}(\widehat{\boldsymbol{x}}) d \widehat{\boldsymbol{x}} \quad \approx \mid \operatorname{det} A_K$$如果 $\psi(\boldsymbol{x})=\widehat{\psi}(\widehat{\boldsymbol{x}})$ 在哪里
$\boldsymbol{x}=\boldsymbol{T} K(\widehat{\boldsymbol{x}})=A_K \widehat{\boldsymbol{x}}+\boldsymbol{b}_K$ ，我们有近似值
$$\int_K \psi(\boldsymbol{x})=\left|\operatorname{det} A_K\right| \int \widehat{K} \widehat{\psi}(\widehat{\boldsymbol{x}}) d \widehat{\boldsymbol{x}} \quad \approx \mid \operatorname{det} A_K$$

$$\nabla \phi_j(\boldsymbol{x})^T \nabla \phi_k(\boldsymbol{x})=\nabla \widehat{\phi}_r(\widehat{\boldsymbol{x}})^T A_K^{-1} A_K^{-T} \nabla \widehat{\phi}_s(\widehat{\boldsymbol{x}})$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。