### 数学代写|计算方法代写computational method代考|Boundary value problems

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## 数学代写|计算方法代写computational method代考|Notation

The Euclidean space in $n$ dimensions is denoted by $\mathbb{R}^{n}$. The Cartesian ${ }^{2}$ coordinate axes in $\mathbb{R}^{3}$ are labeled $x, y, z$ (in cylindrical systems $r, \theta, z$ ) and a vector in $\mathbb{R}^{n}$ is denoted by $\mathbf{u}$. For example, $\mathbf{u} \equiv\left{u_{x} u_{y} u_{z}\right}$ represents a vector in $\mathbf{R}^{3}$.

The index notation will be introduced gradually, in parallel with the familiar Cartesian notation, so that readers who are not yet acquainted with this notation can become familiar with it. The basic rules of index notation are as follows.

1. The Cartesian coordinate axes are labeled $x=x_{1}, y=x_{2}, z=x_{3}$.
2. In conventional notation the position vector in $\mathbb{R}^{3}$ is $\mathbf{x} \equiv{x y z}^{T}$. In index notation it is simply $x_{i}$. A general vector $\mathbf{a} \equiv\left{a_{x} a_{y} a_{z}\right}$ and its transpose is written simply as $a_{i^{*}}$.
3. A free index in $\mathbb{R}^{n}$ is understood to range from 1 to $n$.
4. Two free indices represent a matrix. The size of the matrix depends on the range of indices. Thus, in three dimensions $\left(\mathbb{R}^{3}\right)$ :
$$a_{i j} \equiv\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right] \equiv\left[\begin{array}{lll} a_{x x} & a_{x y} & a_{x z} \ a_{y x} & a_{y y} & a_{y z} \ a_{z x} & a_{z y} & a_{z z} \end{array}\right] .$$
The identity matrix is represented by the $\operatorname{Kronecker}^{3}$ delta $\delta_{i j}$, defined as follows:
$$\delta_{i j}= \begin{cases}1 & \text { if } \mathrm{i}=\mathrm{j} \ 0 & \text { if } \mathrm{i} \neq \mathrm{j} .\end{cases}$$
1. Repeated indices imply summation. For example, the scalar product of two vectors $a_{i}$ and $b_{j}$ is $a_{i} b_{i} \equiv a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}$. The product of two matrices $a_{i j}$ and $b_{i j}$ is written as $c_{i j}=a_{i k} b_{k j}$ *

Definition 2.1 Repeated indices are also called dummy indices. This is because summation is performed therefore the index designation is immaterial. For example, $a_{i} b_{i} \equiv a_{k} b_{k}$ *

1. In order to represent the cross product in index notation, it is necessary to introduce the permutation symbol $e_{i j k}$. The components of the permutation symbol are defined as follows:
$e_{1 y k}=0$ if the values of $i, j, k$ do not form a permutation of $1,2,3$
$e_{i j k}=1$ if the values of $i, j, k$ form an even permutation of $1,2,3$
$e_{i y k}=-1$ if the values of $i, j, k$ form an odd permutation of $1,2,3$.
The cross product of vectors $a_{j}$ and $b_{k}$ is written as
$$c_{i}=e_{i j k} a_{j} b_{k^{-}}$$
Definition $2.2$ The permutations $(1,2,3),(2,3,1)$ and $(3,1,2)$ are even permutations. The permutations $(1,3,2),(2,1,3)$ and $(3,2,1)$ are odd permutations.
2. Indices following a comma represent differentiation with respect to the variables identified by the indices. For example, if $u\left(x_{i}\right)$ is a scalar function then
$$u_{2} \equiv \frac{\partial u}{\partial x_{2}}, \quad u_{, 23} \equiv \frac{\partial^{2} u}{\partial x_{2} \partial x_{3}}$$
The gradient of $u$ is simply $u_{, i}$.
If $u_{i}=u_{i}\left(x_{k}\right)$ is a vector function in $\mathrm{R}^{3}$ then
$$u_{i, i} \equiv \frac{\partial u_{1}}{\partial x_{1}}+\frac{\partial u_{2}}{\partial x_{2}}+\frac{\partial u_{3}}{\partial x_{3}}$$
is the divergence of $u_{i}$.
3. The transformation rules for Cartesian vectors and tensors are presented in Appendix $\mathrm{K}$.

## 数学代写|计算方法代写computational method代考|The scalar elliptic boundary value problem

The three-dimensional analogue of the model problem introduced in Section $1.1$ is the scalar elliptic boundary value problem
$$-\operatorname{div}([\kappa] \operatorname{grad} u)+c u=f(x, y, z), \quad(x, y, z) \in \Omega$$
where
$$\left[\kappa^{\prime}\right]=\left[\begin{array}{lll} \kappa_{x} & \kappa_{x y} & \kappa_{x z} \ \kappa_{y x} & \kappa_{y} & \kappa_{y z} \ \kappa_{z x} & \kappa_{z y}^{c} & \kappa_{z} \end{array}\right]$$
is a positive-definite matrix ${ }^{4}$ and $c=c(x, y, z) \geq 0$. In index notation eq. (2.3) reads:
$$-\left(\kappa_{i j} u_{, j}\right)_{, i}+c u=f$$
We will be concerned with the following linear boundary conditions:

1. Dirichlet boundary condition: $u=\hat{u}$ is prescribed on boundary region $\partial \Omega_{u}$. When $\hat{u}=0$ on $\partial \Omega_{u}$ then the Dirichlet boundary condition is said to be homogeneous.
2. Neumann boundary condition: The flux vector is defined by
$$\mathbf{q} \stackrel{\text { def }}{=}-[\kappa] \text { grad } u, \quad \text { equivalently; } \quad q_{i} \stackrel{\text { def }}{=} \kappa_{i j} u_{j^{-}}$$
The normal flux is defined by $q_{n} \stackrel{\text { def }}{=} \mathbf{q} \cdot \mathbf{n} \equiv q_{i} n_{i}$ where $\mathbf{n} \equiv n_{i}$ is the unit outward normal to the boundary. When $q_{n}=q_{n}$ is prescribed on boundary region $\partial \Omega_{q}$ then the boundary condition is called a Neumann boundary condition. When $\hat{q}{n}=0$ on $\partial \Omega{q}$ then the Neumann boundary condition is said to be homogeneous.
3. Robin boundary condition: $q_{n}=h_{R}\left(u-u_{R}\right)$ is given on boundary segment $\partial \Omega_{R}$. In this expression $h_{R}>0$ and $u_{R}$ are given functions. When $u_{R}=0$ on $\partial \Omega_{R}$ then the Robin boundary condition is said to be homogeneous.
4. Boundary conditions of convenience: In many instances the solution domain can be simplified through taking advantage of symmetry, antisymmetry and/or periodicity. These boundary conditions are called boundary conditions of convenience.

The boundary segments $\partial \Omega_{u}, \partial \Omega_{q}, \partial \Omega_{R}$ and $\partial \Omega_{p}$ are non-overlapping and collectively cover the entire boundary $\partial \Omega$. Any of the boundary segments may be empty.

Definition 2.3 The Dirichlet boundary condition is also called essential boundary condition. The Neumann and Robin conditions area called natural boundary conditions.

## 数学代写|计算方法代写computational method代考|Generalized formulation

To obtain the generalized formulation for the scalar elliptic boundary value problem we multiply eq. (2.5) by a test function $v$ and integrate over the domain $\Omega$ :
$$-\int_{\Omega}\left(\kappa_{i j} u_{j}\right){, i} v d V+\int{\Omega} c u v d V=\int_{\Omega} f v d V .$$

This equation must hold for arbitrary $v$, provided that the indicated operations are defined. The first integral can be written as:
$$\int_{\Omega}\left(\kappa_{i j} u_{j}\right){, i} v d V=\int{\Omega}\left(\kappa_{i j} u_{j} v\right){, j} d V-\int{\Omega} \kappa_{i j} u_{, j} v_{i} d V .$$
Applying the divergence theorem (eq. (2.2)) we have:
$$\int_{\Omega}\left(\kappa_{i j} u_{, j} v\right){, i} d V=\int{\partial \Omega} \kappa_{i j} u_{j} n_{i} v d S$$
where $n_{i}$ is the unit normal vector to the boundary surface. Therefore eq. (2.7) can be written in the following form:
$$-\int_{\partial \Omega} \kappa_{i j} u_{j} n_{i} v d S+\int_{\Omega} \kappa_{i j} u_{j} v_{, i} d V+\int_{\Omega} c u v d V=\int_{\Omega} f v d V$$
It is customary to write
$$q_{i}=-\kappa_{i j} u_{j} \quad \text { and } \quad q_{n}=q_{i} n_{i} .$$
With this notation we have:
$$\int_{\Omega} \kappa_{i j} u_{j} v_{j i} d V+\int_{\Omega} c u v d V=\int_{\Omega} f v d V-\int_{\partial \Omega} q_{n} v d S$$
This is the generalization of eq. (1.18) to two and three dimensions. As we have seen in Section $1.2$, the specific statement of a generalized formulation depends on the boundary conditions. In the general case $u=\hat{u}$ is prescribed on $\partial \Omega_{u}$ (Dirichlet boundary condition); $q_{n}=\hat{q}{n}$ is prescribed on $\partial \Omega{q}$ (Neumann boundary condition) and $q_{n}=h_{R}\left(u-u_{R}\right.$ ) is prescribed on $\Omega_{R}$ (Robin boundary condition), see Section 2.2. We now define the bilinear form as follows:
$$B(u, v)=\int_{\Omega} \kappa_{i j} u_{j} v_{\dot{i}} d V+\int_{\Omega} c u v d V+\int_{d \Omega_{R}} h_{R} u v d S$$
and the linear functional:
$$F(v)=\int_{\Omega} f v d V-\int_{\partial \Omega_{q}} q_{n} v d S+\int_{\partial \Omega_{R}} h_{R} u_{R} v d S .$$
When $\partial \Omega_{R}$ is empty then the last terms in equations (2.10) and (2.11) are omitted. When Neumann condition is prescribed on the entire boundary and $c=0$ then the data must satisfy the following condition:
$$\int_{\Omega} f d V=\int_{\partial \Omega} q_{n} d S$$
The space $E(\Omega)$ is defined by
$$E(\Omega) \stackrel{\text { def }}{=}{u \mid B(u, u)<\infty}$$
and the energy norm
$$|u|_{E} \stackrel{\text { def }}{=} \sqrt{\frac{1}{2} B(u, u)}$$
is associated with $E(\Omega)$. The space of admissible functions is defined by:
$$\tilde{E}(\Omega) \stackrel{\text { def }}{=}\left{u \mid u \in E(\Omega), u=\hat{u} \text { on } \partial \Omega_{u}\right}$$

## 数学代写|计算方法代写computational method代考|Notation

1. 笛卡尔坐标轴被标记X=X1,是=X2,和=X3.
2. 在传统表示法中，位置向量R3是X≡X是和吨. 在索引符号中，它很简单X一世. 一般向量\mathbf{a} \equiv\left{a_{x} a_{y} a_{z}\right}\mathbf{a} \equiv\left{a_{x} a_{y} a_{z}\right}它的转置简单地写成一种一世∗.
3. 中的免费索引Rn被理解为范围从 1 到n.
4. 两个自由索引代表一个矩阵。矩阵的大小取决于索引的范围。因此，在三个维度(R3) :
一种一世j≡[一种11一种12一种13 一种21一种22一种23 一种31一种32一种33]≡[一种XX一种X是一种X和 一种是X一种是是一种是和 一种和X一种和是一种和和].
单位矩阵由克罗内克3三角洲d一世j，定义如下：
d一世j={1 如果 一世=j 0 如果 一世≠j.
5. 重复索引意味着求和。例如，两个向量的标量积一种一世和bj是一种一世b一世≡一种1b1+一种2b2+一种3b3. 两个矩阵的乘积一种一世j和b一世j写成C一世j=一种一世ķbķj *

1. 为了用索引符号表示叉积，需要引入置换符号和一世jķ. 置换符号的组成部分定义如下：
和1是ķ=0如果的值一世,j,ķ不形成排列1,2,3
和一世jķ=1如果的值一世,j,ķ形成一个偶数排列1,2,3
和一世是ķ=−1如果的值一世,j,ķ形成奇排列1,2,3.
向量的叉积一种j和bķ写成
C一世=和一世jķ一种jbķ−
定义2.2排列组合(1,2,3),(2,3,1)和(3,1,2)甚至是排列。排列组合(1,3,2),(2,1,3)和(3,2,1)是奇数排列。
2. 逗号后面的索引表示相对于由索引标识的变量的区分。例如，如果在(X一世)那么是标量函数
在2≡∂在∂X2,在,23≡∂2在∂X2∂X3
的梯度在简直就是在,一世.
如果在一世=在一世(Xķ)是向量函数R3然后
在一世,一世≡∂在1∂X1+∂在2∂X2+∂在3∂X3
是的分歧在一世.
3. 笛卡尔向量和张量的变换规则见附录ķ.

## 数学代写|计算方法代写computational method代考|The scalar elliptic boundary value problem

−div⁡([ķ]毕业⁡在)+C在=F(X,是,和),(X,是,和)∈Ω

[ķ′]=[ķXķX是ķX和 ķ是Xķ是ķ是和 ķ和Xķ和是Cķ和]

−(ķ一世j在,j),一世+C在=F

1. 狄利克雷边界条件：在=在^规定在边界区域∂Ω在. 什么时候在^=0在∂Ω在则称狄利克雷边界条件是齐次的。
2. Neumann 边界条件：通量矢量定义为
q= 定义 −[ķ] 毕业 在, 等效地； q一世= 定义 ķ一世j在j−
法向通量定义为qn= 定义 q⋅n≡q一世n一世在哪里n≡n一世是边界外法线的单位。什么时候qn=qn规定在边界区域∂Ωq则该边界条件称为 Neumann 边界条件。什么时候q^n=0在∂Ωq则称 Neumann 边界条件是齐次的。
3. 罗宾边界条件：qn=HR(在−在R)在边界段上给出∂ΩR. 在这个表达式中HR>0和在R被赋予功能。什么时候在R=0在∂ΩR则称 Robin 边界条件是齐次的。
4. 方便的边界条件：在许多情况下，可以通过利用对称性、反对称性和/或周期性来简化解域。这些边界条件称为方便边界条件。

## 数学代写|计算方法代写computational method代考|Generalized formulation

−∫Ω(ķ一世j在j),一世在d在+∫ΩC在在d在=∫ΩF在d在.

∫Ω(ķ一世j在j),一世在d在=∫Ω(ķ一世j在j在),jd在−∫Ωķ一世j在,j在一世d在.

∫Ω(ķ一世j在,j在),一世d在=∫∂Ωķ一世j在jn一世在d小号

−∫∂Ωķ一世j在jn一世在d小号+∫Ωķ一世j在j在,一世d在+∫ΩC在在d在=∫ΩF在d在

q一世=−ķ一世j在j 和 qn=q一世n一世.

∫Ωķ一世j在j在j一世d在+∫ΩC在在d在=∫ΩF在d在−∫∂Ωqn在d小号

B(u, v)=\int_{\Omega} \kappa_{ij} u_{j} v_{\dot{i}} d V+\int_{\Omega} cuvd V+\int_{d \Omega_{ R } } h_{R} 紫外线 S

F(v)=\int_{\Omega} fvd V-\int_{\partial \Omega_{q}} q_{n} vd S+\int_{\partial \Omega_{ R }} h_{R} u_{R} vd S 。

\int_{\Omega} fd V=\int_{\partial \Omega} q_{n} d S

E(\Omega) \stackrel{\text { def }}{=}{u \mid B(u, u)<\infty}

|u|_{E} \stackrel{\text { def }}{=} \sqrt{\frac{1}{2} B(u, u)}

\tilde{E}(\Omega) \stackrel{\text { def }}{=}\left{u \mid u \in E(\Omega), u=\hat{u} \text { on } \partial \Omega_{u}\right}


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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。