# 数学代写|数值分析代写numerical analysis代考|MA1020

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## 数学代写|数值分析代写numerical analysis代考|Convection—Going with the Flow

If $u(t, x)$ represents the concentration of an unreacting chemical species pulled along by a current in water with velocity $v(\boldsymbol{x})$, for example, the equation for $u(t, \boldsymbol{x})$ is
(6.3.35) $\quad \frac{\partial u}{\partial t}+(v(x) \cdot \nabla) u=\operatorname{div}(D \nabla u)+f(\boldsymbol{x}) \quad$ in $\Omega$
with various boundary conditions. The boundary conditions can describe prescribed concentrations (perhaps at the inflow to a region: $u=g$ on $\Gamma_D$ ), and zero flux conditions (that apply at a wall, for example, where $\boldsymbol{v} \cdot \boldsymbol{n} u+D \partial u / \partial n=0$ on $\Gamma_{Z F}$ ), and outflow conditions $\left(\partial u / \partial n=0\right.$ on $\left.\Gamma_o\right)$. The velocity field $v(x)$ represents the velocity of the current at $\boldsymbol{x}$.
If we look for steady-state solutions, we set $\partial u / \partial t=0$ and so
$$(\boldsymbol{v}(\boldsymbol{x}) \cdot \nabla) u-\operatorname{div}(D \nabla u)=f(\boldsymbol{x}) \quad \text { in } \Omega .$$
If we use $w(\boldsymbol{x})$ as a smooth function for creating the weak form, then the weak form is $$\int_{\Omega}\left[w \boldsymbol{v} \cdot \nabla u+D \nabla w^T \nabla u\right] d x-\int_{\partial \Omega} w D \frac{\partial u}{\partial n} d S=\int_{\Omega} w f(\boldsymbol{x}) d \boldsymbol{x} .$$
The main difference with the equation without convection is the term $\int_{\Omega} w \boldsymbol{v} \cdot \nabla u d x$. Note that
$$\operatorname{div}(w u v)=(u v) \cdot \nabla w+(w \boldsymbol{v}) \cdot \nabla u+w u \operatorname{div} \boldsymbol{v} .$$
If div $v=0$, which is the case for an incompressible flow field, then
\begin{aligned} \int_{\Omega} w \boldsymbol{v} \cdot \nabla u d x & =\int_{\Omega}[\operatorname{div}(w u v)-u v \cdot \nabla w] d x \ & =\int_{\partial \Omega} w u v \cdot n d S-\int_{\Omega} u \boldsymbol{v} \cdot \nabla w d x . \end{aligned}
If $w=0$ on $\partial \Omega$ then we get
$$\int_{\Omega} w v \cdot \nabla u d x=-\int_{\Omega} u v \cdot \nabla w d x$$

## 数学代写|数值分析代写numerical analysis代考|Higher Order Problems

Fourth order partial differential equations arise in a number of settings, such as elastic plate problems. A typical example is the biharmonic equation that can be written as
$$\Delta \Delta u=f(x) \quad \text { in } \Omega$$
where $\Delta$ is the Laplacian operation ( $\Delta u=\partial^2 u / \partial x^2+\partial^2 u / \partial y^2$ in two dimensions) and appropriate boundary conditions, such as Dirichlet conditions $u(\boldsymbol{x})=g(\boldsymbol{x})$, $\partial u / \partial n(\boldsymbol{x})=k(\boldsymbol{x})$ for $\boldsymbol{x} \in \partial \Omega$. The weak form of the equation with Dirichlet boundary conditions is that (6.3.37) $\quad \int_{\Omega}[(\Delta u)(\Delta v)-f v] d x \quad$ for all smooth $v$,
where $v=\partial v / \partial n=0$ on $\partial \Omega$. Standard conforming finite element methods have to use basis functions $\phi_i$ where $\int_{\Omega}\left(\Delta \phi_i\right)^2 d x$ is finite. This means that if $\phi_i$ is piecewise smooth, then there cannot be any jumps in $\nabla \phi_i$. The basis functions should therefore be $C^1$ (continuous first derivatives), which are harder to create. Section 4.3.2.1 shows some examples: the Argyris element (Figure 4.3.7), and the HCT macro element (Figure 4.3.8). The order of convergence of these methods is essentially given by the order of the polynomials that can be represented by the elements used. These $C^1$ finite elements are complicated to construct, so there has been a great deal of interest in other methods of solving equations like the biharmonic equation. The equation $\Delta \Delta u=f$ in $\Omega$ with Dirichlet boundary conditions is an elliptic partial differential equation on $H^2(\Omega)$. Most of the theory of this section can be extended to problems of this type, although the condition number of the system of equations $\kappa_2\left(A_h\right)=\mathcal{O}\left(h_{\min }^{-4}\right)$ rather than $\mathcal{O}\left(h_{\min }^{-2}\right)$ for the second order elliptic equations.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Convection—Going with the Flow

$$\frac{\partial u}{\partial t}+(v(x) \cdot \nabla) u=\operatorname{div}(D \nabla u)+f(\boldsymbol{x}) \text { 在 } \Omega$$

$$(\boldsymbol{v}(\boldsymbol{x}) \cdot \nabla) u-\operatorname{div}(D \nabla u)=f(\boldsymbol{x}) \quad \text { in } \Omega .$$

$$\int_{\Omega}\left[w \boldsymbol{v} \cdot \nabla u+D \nabla w^T \nabla u\right] d x-\int_{\partial \Omega} w D \frac{\partial u}{\partial n} d S=\int_{\Omega}$$

$$\int_{\Omega} w v \cdot \nabla u d x=\int_{\Omega}[\operatorname{div}(w u v)-u v \cdot \nabla w] d x$$

$$\int_{\Omega} w v \cdot \nabla u d x=-\int_{\Omega} u v \cdot \nabla w d x$$

## 数学代写|数值分析代写numerical analysis代考|Higher Order Problems

$$\Delta \Delta u=f(x) \quad \text { in } \Omega$$

$\Delta u=\partial^2 u / \partial x^2+\partial^2 u / \partial y^2$ 在二维中) 和适当的边 界条件，例如 Dirichlet 条件 $u(\boldsymbol{x})=g(\boldsymbol{x})$ ，
$\partial u / \partial n(\boldsymbol{x})=k(\boldsymbol{x})$ 为了 $\boldsymbol{x} \in \partial \Omega$. 具有 Dirichlet 边界 条件的方程的弱形式是 (6.3.37)
$$\int_{\Omega}[(\Delta u)(\Delta v)-f v] d x \quad \text { 一切顺利 } v \text { ， }$$

4.3.7) 和 HCT 宏元素 (图 4.3.8) 。这些方法的收敛顺 序基本上由可由所用元素表示的多项式的顺序给出。这 $\stackrel{\text { 些 }}{ }{ }^1$ 有限元构造起来很复杂，因此人们对其他求解方程 的方法 (如双调和方程) 很感兴趣。方程式 $\Delta \Delta u=f$ 在 $\Omega$ 具有 Dirichlet 边界条件的椭圆偏微分方程 $H^2(\Omega)$. 本节的大部分理论都可以扩展到此类问题，尽管方程组 的条件数 $\kappa_2\left(A_h\right)=\mathcal{O}\left(h_{\min }^{-4}\right)$ 而不是 $\mathcal{O}\left(h_{\min }^{-2}\right)$ 对于 二阶椭圆方程。

## 有限元方法代写

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## 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。

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