## 数学代写|偏微分方程代写partial difference equations代考|MATH4310

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

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

Possibly the most important scales of distribution spaces consist of the Sobolev spaces. In this text we will solely make use of the Sobolev spaces based on $L^2$, which we shall denote by $H^s\left(\mathbb{R}^n\right)$ with $s \in \mathbb{R}: H^s\left(\mathbb{R}^n\right)$ is the linear space of tempered distributions $u$ whose Fourier transform $\widehat{u}$ is a square-integrable function in $\mathbb{R}^n$ with respect to the density $\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$. The Hermitian product
$$(u, v)s=(2 \pi)^{-n} \int{\mathbb{R}^n} \widehat{u}(\xi) \overline{\widehat{v}(\xi)}\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$$ defines a Hilbert space structure on $H^s\left(\mathbb{R}^n\right)$; we use the notation $|u|_s=\sqrt{(u, u)s}$. We have $H^0\left(\mathbb{R}^n\right)=L^2\left(\mathbb{R}^n\right)$; if $s^{\prime}{s^{\prime}} \leq|u|_{s^s}$. All the Hilbert spaces $H^s\left(\mathbb{R}^n\right)$ are isomorphic: it is immediate to see that the operators
$$\left(1-\Delta_x\right)^{t / 2} \varphi(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi}\left(1+|\xi|^2\right)^{t / 2} \widehat{\varphi}(\xi) \mathrm{d} \xi, t \in \mathbb{R},$$
form a group of (continuous linear) automorphisms of $\mathcal{S}\left(\mathbb{R}^n\right) ;(2.2 .2)$ extends as an isometry of $H^s\left(\mathbb{R}^n\right)$ onto $H^{s-t}\left(\mathbb{R}^n\right)$, whatever the real numbers $s, t$.

We mention a useful inequality, valid for all $s, t \in \mathbb{R}$ such that $a=s-t>0$, all $\varepsilon>0$ and $u \in H^s\left(\mathbb{R}^n\right)$
$$|u|_t^2 \leq \varepsilon|u|_s^2+\frac{1}{4 \varepsilon}|u|_{t-a}^2,$$
a direct consequence of the inequality $A^t \leq \varepsilon A^s+\frac{1}{4 \varepsilon} A^{t-a}, A=1+|\xi|^2$.

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

We must now introduce distributions $F(x, y)$ on products $\Omega_1 \times \Omega_2$ with $\Omega_1 \subset$ $\mathbb{R}^{n_1}, \Omega_2 \subset \mathbb{R}^{n_2}$ open sets. Distributions belonging to $\mathcal{D}^{\prime}\left(\Omega_1 \times \Omega_2\right)$ are often referred to as kernels or distribution kernels. We can regard the product of two test-functions $\varphi \in C_{\mathrm{c}}^{\infty}\left(\Omega_1\right)$ and $\psi \in C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$ as an element of $C_{\mathrm{c}}^{\infty}\left(\Omega_1 \times \Omega_2\right)$, denoted by $\varphi \otimes \psi$, and evaluate $F \in \mathcal{D}^{\prime}\left(\Omega_1 \times \Omega_2\right)$ on it. Fixing $\psi$ defines a distribution in $\Omega_1$ :
$$C_{\mathrm{c}}^{\infty}\left(\Omega_1\right) \ni \varphi \mapsto\langle F, \varphi \otimes \psi\rangle \in \mathbb{C} .$$
To emphasize this partial action it is convenient to adopt the “Volterra notation”: to write $\int F(x, y) \psi(y)$ d $y$ rather than $\langle F(x, y), \psi(y)\rangle$. (Keep in mind, however, that $\int$ does not stand for a true integral!) In passing we point out that the Fubini formula is always true in distribution theory: $$\int\left(\int F(x, y) \psi(y) \mathrm{d} y\right) \varphi(x) \mathrm{d} x=\int\left(\int F(x, y) \varphi(x) \mathrm{d} x\right) \psi(y) \mathrm{d} y .$$
The map
$$C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \ni \psi \mapsto \mathfrak{I}F \psi(x)=\int F(x, y) \psi(y) \mathrm{d} y \in \mathcal{D}^{\prime}\left(\Omega_1\right)$$ is linear and continuous. The Schwartz Kernel Theorem states that, actually, every continuous linear map $C{\mathrm{c}}^{\infty}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$ is of the kind (2.3.1), and that the correspondence between continuous linear maps and distribution kernels is one-toone. This is a very special property of $\mathcal{D}^{\prime}$, obviously false for any infinite-dimensional Banach space (but true for $\mathcal{E}^{\prime}, C^{\infty}, C_{\mathrm{c}}^{\infty}$, if properly reformulated).

The composition $A_{1,2} \circ A_{2,3}$ of two linear operators $A_{1,2}: C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$, $A_{2,3}: C_{\mathrm{c}}^{\infty}\left(\Omega_3\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_2\right)$, puts requirements of regularity and support on the factors. For instance, we might require that $A_{2,3}$ maps $C_{\mathrm{c}}^{\infty}\left(\Omega_3\right)$ into $C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$, or else that $A_{1,2}$ extend as a continuous linear operator $\mathcal{D}^{\prime}\left(\Omega_2\right) \longrightarrow \mathcal{D}^{\prime}\left(\Omega_1\right)$, which is equivalent to requiring that the transpose $A_{1,2}^{\top}$ maps $C_{\mathrm{c}}^{\infty}\left(\Omega_1\right)$ into $C_{\mathrm{c}}^{\infty}\left(\Omega_2\right)$. These concerns are addressed in Definitions $2.3 .1$ and $2.3 .6$ below.

# 偏微分方程代写

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

$$(u, v) s=(2 \pi)^{-n} \int \mathbb{R}^n \widehat{u}(\xi) \overline{\hat{v}(\xi)}\left(1+|\xi|^2\right)^s \mathrm{~d} \xi$$

$$\left(1-\Delta_x\right)^{t / 2} \varphi(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi}\left(1+|\xi|^2\right)^{t / 2} \widehat{\varphi}(\xi) \mathrm{d} \xi, t \in \mathbb{R}$$

$$|u|t^2 \leq \varepsilon|u|_s^2+\frac{1}{4 \varepsilon}|u|{t-a}^2,$$

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

$$C_{\mathrm{c}}^{\infty}\left(\Omega_1\right) \ni \varphi \mapsto\langle F, \varphi \otimes \psi\rangle \in \mathbb{C} .$$

$$\int\left(\int F(x, y) \psi(y) \mathrm{d} y\right) \varphi(x) \mathrm{d} x=\int\left(\int F(x, y) \varphi(x) \mathrm{d} x\right) \psi(y) \mathrm{d} y .$$

$$C_{\mathrm{c}}^{\infty}\left(\Omega_2\right) \ni \psi \mapsto \Im F \psi(x)=\int F(x, y) \psi(y) \mathrm{d} y \in \mathcal{D}^{\prime}\left(\Omega_1\right)$$

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|Math462

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

## 数学代写|偏微分方程代写partial difference equations代考|The wave-front set of a distribution

Let $\Omega \subset \mathbb{R}^n$ be an open set and let $x^{\circ} \in \Omega, \xi^{\circ} \in \mathbb{R}^n \backslash{0}$ be arbitrary. By a cone in $\mathbb{R}^n \backslash{0}$ we shall always mean a set invariant under all dilations $\xi \mapsto \lambda \xi, \lambda>0$ (i.e., a cone with vertex at the origin).
Lemma 2.1.4 Let $u \in \mathcal{D}^{\prime}(\Omega)$ have the following property:
(NWF) There exist an open set $U \subset \subset \Omega$ containing $x^{\circ}$ and $\varphi \in C_c^{\infty}(\Omega), \varphi(x)=1$ for every $x \in U$, and an open cone $\Gamma \subset \mathbb{R}^n \backslash{0}$ containing $\xi^{\circ}$ such that
$$\forall m \in \mathbb{Z}{+}, \sup {\xi \in \Gamma}\left((1+|\xi|)^m|\overline{(\varphi u)}(\xi)|\right)<+\infty .$$
Then, if $\Gamma^{\prime} \subset \mathbb{R}^n \backslash{0}$ is an open cone such that $\Gamma^{\prime} \cap \mathbb{S}^{n-1} \subset \subset \Gamma$, we have
$$\forall m \in \mathbb{Z}{+}, \sup {\xi \in \Gamma^{\infty}}\left((1+|\xi|)^m|\widehat{(\psi u)}(\xi)|\right)<+\infty$$
for every $\psi \in C_c^{\infty}(U)$
Proof Let $\varphi$ and $\psi$ be as in the statement; we have $\psi u=\psi \varphi u$ and therefore
$$\widehat{(\psi u)}(\xi)=(2 \pi)^{-n} \int \widehat{\psi}(\xi-\eta) \widehat{(\varphi u)}(\eta) \mathrm{d} \eta .$$
Here we shall use the notation, for $k \in \mathbb{Z}{+}$, $$|\psi|_k=\sup {\xi \in \mathbb{R}^n}\left((1+|\xi|)^k|\widehat{\psi}(\xi)|\right)$$
as well as
$$|\varphi u|{k, \Gamma}=\sup {\xi \in \Gamma}\left((1+|\xi|)^k|\overline{(\varphi u)}(\xi)|\right) .$$
Using the self-evident inequality $(1+|\xi|)^m \leq(1+|\eta|)^m(1+|\xi-\eta|)^m$ we get, for $\xi \in \Gamma^{\prime}$

## 数学代写|偏微分方程代写partial difference equations代考|Action of diferential operators on distributions

The action of a linear PDO on a distribution $u$ in $\Omega$ is defined by transposition:
$$\langle P(x, \mathrm{D}) u, \varphi\rangle=\left\langle u, P(x, \mathrm{D})^{\top} \varphi\right\rangle, \varphi \in \mathcal{C}{\mathrm{c}}^{\infty}(\Omega) .$$ When $u \in C^{\infty}(\Omega)$, (2.1.6) simply reflects integration by parts. Likewise, $$\langle P(x, \mathrm{D}) u, \bar{\varphi}\rangle=\left\langle u, \overline{P(x, \mathrm{D})^* \varphi}\right\rangle, \varphi \in C{\mathrm{c}}^{\infty}(\Omega) .$$
It follows directly from (2.1.6) that the inclusion (1.3.2), $\operatorname{supp} P(x, \mathrm{D}) f \subset$ supp $f$, remains valid when $f \in \mathcal{D}^{\prime}(\Omega)$. It is also obvious that
$$\text { singsupp } P(x, \text { D) } f \subset \operatorname{singsupp} f \text {, }$$
and if the coefficients of $P(x, \mathrm{D})$ are real-analytic, that
$$\text { singsupp }{\mathrm{a}} P(x, \mathrm{D}) f \subset \text { singsupp }{\mathrm{a}} f \text {. }^2$$
In other words, differential operators “decrease” the singular supports, just like they decrease the supports.

Every linear PDO maps $\mathcal{D}^{\prime}(\Omega)$ linearly and continuously into itself, and $\mathcal{E}^{\prime}(\Omega)$ into itself. In particular, $P(x, \mathrm{D}$ ) acts in the distribution sense (often called “the weak sense”) on a function $f \in L_{\text {loc }}^1(\Omega)$ :
$$\langle P(x, \mathrm{D}) f, \varphi\rangle=\int f P(x, \mathrm{D})^{\top} \varphi \mathrm{d} x, \varphi \in C_{\mathrm{c}}^{\infty}(\Omega) .$$
Actually [cf. (2.1.5)], every distribution $u \in \mathcal{D}^{\prime}(\Omega)$ can be represented locally as a finite sum of derivatives of continuous functions.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|The wave-front set of a distribution

(NWF) 存在一个开集 $U \subset \subset \Omega$ 含有 $x^0$ 和 $\varphi \in C_c^{\infty}(\Omega), \varphi(x)=1$ 每一个 $x \in U$ ，和一个开雉 $\Gamma \subset \mathbb{R}^n \backslash 0$ 含有 $\xi^{\circ}$ 这样
$$\forall m \in \mathbb{Z}+, \sup \xi \in \Gamma\left((1+|\xi|)^m|\overline{(\varphi u)}(\xi)|\right)<+\infty$$

$$\forall m \in \mathbb{Z}+, \sup \xi \in \Gamma^{\infty}\left((1+|\xi|)^m|\widehat{(\psi u)}(\xi)|\right)<+\infty$$

$$\widehat{(\psi u)}(\xi)=(2 \pi)^{-n} \int \widehat{\psi}(\xi-\eta) \widehat{(\varphi u)}(\eta) \mathrm{d} \eta .$$

$$|\psi|_k=\sup \xi \in \mathbb{R}^n\left((1+|\xi|)^k|\widehat{\psi}(\xi)|\right)$$

$$|\varphi u| k, \Gamma=\sup \xi \in \Gamma\left((1+|\xi|)^k|\overline{(\varphi u)}(\xi)|\right)$$

## 数学代写|偏微分方程代写partial difference equations代考|Action of diferential operators on distributions

$$\langle P(x, \mathrm{D}) u, \varphi\rangle=\left\langle u, P(x, \mathrm{D})^{\top} \varphi\right\rangle, \varphi \in \mathcal{C c}^{\infty}(\Omega) .$$

$$\langle P(x, \mathrm{D}) u, \bar{\varphi}\rangle=\left\langle u, \overline{P(x, \mathrm{D})^* \varphi}\right\rangle, \varphi \in C \mathrm{c}^{\infty}(\Omega) .$$

$$\text { singsupp a } P(x, \mathrm{D}) f \subset \text { singsupp a } f .{ }^2$$

$$\langle P(x, \mathrm{D}) f, \varphi\rangle=\int f P(x, \mathrm{D})^{\top} \varphi \mathrm{d} x, \varphi \in C_{\mathrm{c}}^{\infty}(\Omega) .$$

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|MATH1470

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

## 数学代写|偏微分方程代写partial difference equations代考|Basics on Distributions in Euclidean Space

Let $\Omega$ be an open subset of $\mathbb{R}^n$, as before. If $u$ is a complex-valued linear functional on the vector space $C_{\mathrm{c}}^{\infty}(\Omega)$, i.e., if $u$ is a linear map $C_{\mathrm{c}}^{\infty}(\Omega) \longrightarrow \mathbb{C}$, we denote by $\langle u, \varphi\rangle$ its evaluation at the test-function $\varphi \in C_{\mathrm{c}}^{\infty}(\Omega)$. The linear functional $u$ is a distribution in $\Omega$ if $\left\langle u, \varphi_j\right\rangle \rightarrow 0$ whenever the sequence $\left{\varphi_j\right}_{j=0,1,2, \ldots} \subset C_{\mathrm{c}}^{\infty}(\Omega)$ converges to zero in the following sense:
(•) all derivatives $\partial^\alpha \varphi_j$ converge uniformly to zero and there is a compact set $K \subset \Omega$ such that $\operatorname{supp} \varphi_j \subset K$ whatever $j$.

The space of distributions in $\Omega$ is denoted by $\mathcal{D}^{\prime}(\Omega)$. The restriction of a distribution $u \in \mathcal{D}^{\prime}(\Omega)$ to an open subset $\Omega^{\prime}$ of $\Omega$ is simply the restriction of the linear functional $u$ to the linear subspace $C_{\mathrm{c}}^{\infty}\left(\Omega^{\prime}\right)$ of $C_{\mathrm{c}}^{\infty}(\Omega)$. By using partitions of unity in $C_{\mathrm{c}}^{\infty}(\Omega)$ it is readily proved that there is a smallest closed subset of $\Omega$, called the support of $u$ and denoted by supp $u$, such that $u$ vanishes (“identically”) in $\Omega \backslash F$. The subspace of distributions in $\Omega$ that have compact support (contained in $\Omega$ ) is denoted by $\mathcal{E}^{\prime}(\Omega)$; it can be identified with the dual of $C^{\infty}(\Omega)$.

The convergence of a sequence of distributions $u_j\left(j \in \mathbb{Z}{+}\right)$is to be understood in the “weak sense”: $u_j \rightarrow 0$ if $\left\langle u_j, \varphi\right\rangle \rightarrow 0$ for each $\varphi \in C{\mathrm{c}}^{\infty}(\Omega)$. For $u_j \in \mathcal{E}^{\prime}(\Omega)$ to converge to zero in $\mathcal{E}^{\prime}(\Omega)$ it is moreover required that there be a compact set $K \subset \Omega$ such that $\operatorname{supp} u_j \subset K$ for all $j$.

Every continuous linear map of $C_{\mathrm{c}}^{\infty}(\Omega)$ into itself defines, by transposition, a continuous linear map of $\mathcal{D}^{\prime}(\Omega)$ into itself. Most important among these are multiplication by smooth functions in $\Omega$ and partial derivatives. If $P\left(x, \mathrm{D}x\right)$ is a linear partial differential operator with smooth coefficients in $\Omega$ we define, for arbitrary $u \in \mathcal{D}^{\prime}(\Omega), \varphi \in C{\mathrm{c}}^{\infty}(\Omega)$,
$$\left\langle P\left(x, \mathrm{D}_x\right) u, \varphi\right\rangle=\left\langle u, P\left(x, \mathrm{D}_x\right)^{\top} \varphi\right\rangle,$$
where $P\left(x, \mathrm{D}_x\right)^{\top}$ is the transpose of $P\left(x, \mathrm{D}_x\right)$ [cf. (1.3.3)].

## 数学代写|偏微分方程代写partial difference equations代考|Tempered distributions and their Fourier transforms

As is customary, $\mathcal{S}\left(\mathbb{R}^n\right)$ stands for the (Schwartz) space of functions $\varphi \in C^{\infty}\left(\mathbb{R}^n\right)$ rapidly decaying at infinity: given arbitrary $\alpha \in \mathbb{Z}{+}^n$ and $m \in \mathbb{Z}{+}$,
$$\sup {x \in \mathbb{R}^n}\left(1+|x|^2\right)^{\frac{1}{2} m}\left|\partial_x^\alpha \varphi(x)\right|<+\infty .$$ A sequence of functions $\varphi \in \mathcal{S}\left(\mathbb{R}^n\right)$ converges to zero if the seminorms on the left in (2.1.1) converge to zero for all choices of $m$ and $\alpha ; \mathcal{S}\left(\mathbb{R}^n\right)$ is a Fréchet space and thus its topology can be defined by (equivalent) metrics that turn it into a complete metric space. The space $\mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$ of tempered distributions in $\mathbb{R}^n$ is the subspace of $\mathcal{D}^{\prime}\left(\mathbb{R}^n\right)$ consisting of the distributions $u$ which can be written as finite sums of distribution derivatives $$u=\sum{|\alpha| \leq m} \mathrm{D}^\alpha\left(P_\alpha f_\alpha\right)$$
in which the $P_\alpha$ are polynomials and the $f_\alpha$ belong, say, to $L^1\left(\mathbb{R}^n\right)$. By transposing the dense injection $C_{\mathrm{c}}^{\infty}\left(\mathbb{R}^n\right) \hookrightarrow \mathcal{S}\left(\mathbb{R}^n\right)$ the dual of $\mathcal{S}\left(\mathbb{R}^n\right)$ is identified with $\mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$. Below we often denote by $\int u(x) \varphi(x) \mathrm{d} x$ (rather than by $\langle u, \varphi\rangle$ ) the duality bracket between $u \in \mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$ and $\varphi \in \mathcal{S}\left(\mathbb{R}^n\right)$.
The Fourier transform
$$\widehat{u}(\xi)=\int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi} u(x) \mathrm{d} x$$
defines a Fréchet space isomorphism of $\mathcal{S}\left(\mathbb{R}x^n\right)$ onto $\mathcal{S}\left(\mathbb{R}{\xi}^n\right)$ whose inverse is given by
$$u(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{i x \cdot \xi} \widehat{u}(\xi) \mathrm{d} x .$$

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Basics on Distributions in Euclidean Space

(•) 所有导数 $\partial^\alpha \varphi_j$ 一致收敛于零且存在紧集 $K \subset \Omega$ 这样 $\operatorname{supp} \varphi_j \subset K$ 任何 $j$.

$$\left\langle P\left(x, \mathrm{D}_x\right) u, \varphi\right\rangle=\left\langle u, P\left(x, \mathrm{D}_x\right)^{\top} \varphi\right\rangle,$$

## 数学代写|偏微分方程代写partial difference equations代考|Tempered distributions and their Fourier transforms

$$\sup x \in \mathbb{R}^n\left(1+|x|^2\right)^{\frac{1}{2} m}\left|\partial_x^\alpha \varphi(x)\right|<+\infty .$$

$$u=\sum|\alpha| \leq m \mathrm{D}^\alpha\left(P_\alpha f_\alpha\right)$$

$$\widehat{u}(\xi)=\int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi} u(x) \mathrm{d} x$$

$$u(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{i x \cdot \xi} \widehat{u}(\xi) \mathrm{d} x$$

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|MATH4310

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

## 数学代写|偏微分方程代写partial difference equations代考|Second Order Differential Equations

A second order differential equation in two variables $x$ and $y$ is given by
$$F\left(x, y ; u ; p, q ; u_{x x}, u_{x y}, u_{y y}\right)=0, \quad \text { for } z=u(x, y) \in C^2(\Omega),$$
where function $F$ is sufficiently smooth with respect to all involved variables, and $F_p^2+F_q^2 \not \equiv 0$ over $\Omega$. In particular, a second order quasilinear equation is given by
$$a u_{x x}+b u_{x y}+c u_{y y}+F_1\left(x, y ; u ; u_x, u_y\right)=0,$$
where the coefficients $a, b, c$ are functions of the independent variables $x, y$, and also of the dependent variable $z=u(x, y)$. As said earlier, (5.1.26) is a semilinear equation when functions $a, b, c$ depend on variables $x$ and $y$ only. Also, a general second order linear equation for a function $u \in C^2(\Omega)$ is given by
$$a u_{x x}+b u_{x y}+c u_{y y}+d u_x+e u_y+f u+g=0,$$
where the coefficients $a, \ldots, g$ are functions of the independent variables $x$ and $y$ only. As in the case of a first order differential equation in two variables, we say (5.1.27) is a homogeneous equation if the $g \equiv 0$. Otherwise, it is called a nonhomogeneous equation.

The next two examples illustrate that the second order differential equations of simpler linearity types arise naturally in mathematics, and also in practical situations. The main idea is to eliminate all parameters from the given functional relation. For convenience, we may write the second order partial derivatives of a $C^2$-function $u=u(x, y)$ as
$$r=u_{x x}=\frac{\partial^2 u}{\partial x^2}, \quad s=u_{x y}=\frac{\partial^2 u}{\partial x \partial y}, \quad t=u_{y y}=\frac{\partial^2 u}{\partial y^2} .$$

## 数学代写|偏微分方程代写partial difference equations代考|Classification and Canonical Forms

Let $\Omega \subseteq \mathbb{R}^2$ be an open set, and consider the general second order linear differential equation for a function $u \in C^2(\Omega)$ given by
$$a u_{x x}+2 b u_{x y}+c u_{y y}+F_1(x, y ; u ; p, q)=0, \quad \text { for } z=u(x, y),$$
where the coefficients $a, b, c \in C^2(\Omega)$ are such that the condition $a^2+b^2+c^2 \not \equiv 0$ holds over $\Omega$. In this section, our main concern is the principle part given by
$$a u_{x x}+2 b u_{x y}+c u_{y y},$$
because only it participates in the classification procedure described below. When the coefficients $a, b, c$ are constants, the geometry type of Eq. (5.2.1) remains uniform over a domain $\Omega$. However, in the general case, the equation may be of different types across various regions of $\Omega$. We will study Eq. (5.2.1) over a domain $\Omega_1 \subseteq \Omega$ such that the discriminant given by
$$D:=b^2-a c$$
has the same sign at each point of $\Omega_1$. We show that, for $\left(x_0, y_0\right) \in \Omega_1$, there exists a neighbourhood $U_0$ of the point $\left(x_0, y_0\right)$ and sufficiently smooth functions $\varphi, \phi$ such that the transformation $(x, y) \mapsto(\xi, \eta)$ given by
$$\xi=\varphi(x, y) \quad \text { and } \quad \eta=\phi(x, y),$$
changes Eq. (5.2.1) to a differential equation that has one of the three geometry types ${ }^3$ such as given below:

1. A hyperbolic type such as the wave equation (5.1.38).
2. A parabolic type such as the heat equation (5.1.40).
3. An elliptic type such as the Laplace equation (5.1.42).

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Second Order Differential Equations

$$F\left(x, y ; u ; p, q ; u_{x x}, u_{x y}, u_{y y}\right)=0, \quad \text { for } z=u(x, y) \in C^2(\Omega),$$

$$a u_{x x}+b u_{x y}+c u_{y y}+F_1\left(x, y ; u ; u_x, u_y\right)=0,$$

$$a u_{x x}+b u_{x y}+c u_{y y}+d u_x+e u_y+f u+g=0,$$

$$r=u_{x x}=\frac{\partial^2 u}{\partial x^2}, \quad s=u_{x y}=\frac{\partial^2 u}{\partial x \partial y}, \quad t=u_{y y}=\frac{\partial^2 u}{\partial y^2} .$$

## 数学代写|偏微分方程代写partial difference equations代考|Classification and Canonical Forms

$$a u_{x x}+2 b u_{x y}+c u_{y y}+F_1(x, y ; u ; p, q)=0, \quad \text { for } z=u(x, y),$$

$$a u_{x x}+2 b u_{x y}+c u_{y y},$$

$$D:=b^2-a c$$

$$\xi=\varphi(x, y) \quad \text { and } \quad \eta=\phi(x, y),$$

1. 双曲线类型如波动方程 (5.1.38)。
2. 抛物线类型，例如热方程 (5.1.40)。
3. 椭圆类型，例如拉普拉斯方程 (5.1.42)。

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|Math462

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

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

It helps in some situations to assume that the open set $\Omega \subseteq \mathbb{R}^n$ is simply connected, ${ }^1$ and has a piecewise smooth closed boundary $\Gamma=\partial \Omega$. That is, when $\Omega \subseteq \mathbb{R}^n$ is seen as an immersed submanifold of dimension $n-1$, the boundary $\Gamma$ is assumed to be a finite union of smooth surfaces of dimension $n-1$. For example, the interior of a rectangular region $\bar{\Omega}=[a, b] \times[c, d]$ fits the above description. Further, while modelling the equilibrium state of a phenomenon such as dealing with heat conduction or potential of a conservative force field, we may also take $\Omega$ to be bounded. In most practical situations discussed earlier, the restriction of the function $u=u(\boldsymbol{x}, t): \Omega_1 \rightarrow \mathbb{R}$ to the boundary $\Gamma$ is assumed to be continuous. Each differential equation model derived in Chapter 4 can be expressed implicitly as $$F\left(x ; t ; u ; p_i ; q ; r_{i j} ; \ldots\right)=0, \quad \text { for }(x, t) \in \Omega_1 \text {, }$$
where $F$ is some nice function, and the symbols $p_i, q, r_{i j}$, etc., are the partial derivatives of the function $u$ given by
\begin{aligned} & p_i=u_{x_i}=\partial_{x_i} u=\frac{\partial u}{\partial x_i} ; \quad q=u_t=\partial_t u=\frac{\partial u}{\partial t} ; \ & r_{i j}=u_{x_i x_j}=\partial_{x_i x_j}^2 u=\frac{\partial^2 u}{\partial x_i \partial x_j}=\frac{\partial^2 u}{\partial x_j \partial x_i} ; \quad \text { etc. } \end{aligned}
Also, the del operator $\nabla$ in $n$ variables $x_1, \ldots, x_n$ is written as
$$\nabla \equiv\left(\frac{\partial}{\partial x_1}, \cdots, \frac{\partial}{\partial x_n}\right) .$$
Definition 5.1 For a function $F$ that is sufficiently smooth with respect to variables $\boldsymbol{x}, t, u, p_i, q, r_{i j}, \ldots$ such that at least one of the derivatives $F_{p_i}, F_q, F_{r_{i j}}, \ldots$, is nonzero over a suitable domain, an equation of the form (5.1.1) is called a general partial differential equation for a function $u=u(\boldsymbol{x}, t)$, with $(\boldsymbol{x}, t) \in \Omega_1$. The order of (5.1.1) is the order of the highest derivative of $u$ appearing in the equation. The sum of highest order terms is called the principle part of a differential equation.

## 数学代写|偏微分方程代写partial difference equations代考|Linearity Type of a Differential Equation

Other than the order of a differential equation, as defined above, the linearity type of the function $F$ with respect to the variables $u, p_i, q, r_{i j}, \ldots$ provides a first stage classification of equations of the form (5.1.1), determines the complexity level of the geometry involved with any related problem, and hence also of the solution method to be applied in a particular situation.
Definition $5.2$ A differential equation (5.1.1) is said to be a

1. Linear equation if the function $F$ is linear with respect to variables $u, p_i, q, r_{i j}$, $\ldots$, whichever is present in the equation, and the coefficient functions depend on the independent variables $x_1, \ldots, x_n$ and $t$ only;
2. Semilinear equation (or almost linear) if the principle part of the equation is linear, and the coefficients appearing in the principle part are functions of $(n+1)$ independent variables $x_1, \ldots, x_n, t$ only;
3. Quasilinear equation if the principle part of the equation is linear, the coefficients appearing in the principle part are functions of $(n+1)$ independent variables and also of the dependent variable $u(\boldsymbol{x}, t)$, and at least one coefficient function appearing in lower order terms depends on $(n+1)$ independent variables and also on some lower order derivative of the function $u$.

A differential equation for a function $u=u(\boldsymbol{x}, t)$ is called fully nonlinear if it is not a quasilinear equation.

As in the case of an ordinary differential equation, the linear differential equations are simplest to deal with. Notice that, however, though each linear ordinary differential equation admits a global solutions, but the same is not true for latter types of differential equations. A general $k$ th order linear differential equation for a function $u=u(\boldsymbol{x}, t)$ can be written in operator notation as given below. For $\boldsymbol{x}=\left(x_1, \ldots, x_n\right)$ and an $n$-tuple $a=\left(a_1, \ldots, a_n\right)$ of non-negative integers, we write
$$\boldsymbol{x}^a=x_1^{a_1} \cdots x_n^{a_n} \quad \text { and } \quad D^a=D_1^{a_1} \cdots D_n^{a_n},$$
where $D_j$ denote the partial differential operator $\partial / \partial x_j$, for $j=1, \ldots, n$.

# 偏微分方程代写

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

$$F\left(x ; t ; u ; p_i ; q ; r_{i j} ; \ldots\right)=0, \quad \text { for }(x, t) \in \Omega_1 \text {, }$$

$$p_i=u_{x_i}=\partial_{x_i} u=\frac{\partial u}{\partial x_i} ; \quad q=u_t=\partial_t u=\frac{\partial u}{\partial t} ; \quad r_{i j}=u_{x_i x_j}=\partial_{x_i x_j}^2 u=\frac{\partial^2 u}{\partial x_i \partial x_j}=\frac{\partial^2 u}{\partial x_j \partial x}$$

$$\nabla \equiv\left(\frac{\partial}{\partial x_1}, \cdots, \frac{\partial}{\partial x_n}\right)$$

(5.1.1) 的阶数是的最高导数的阶数 $u$ 出现在等式中。最高阶项的总和称为微分方程的主要部分。

## 数学代写|偏微分方程代写partial difference equations代考|Linearity Type of a Differential Equation

1. 函数的线性方程 $F$ 相对于变量是线性的 $u, p_i, q, r_{i j} ， \ldots$ ，以方程中出现的那个为准，系数函数取决 于自变量 $x_1, \ldots, x_n$ 和 $t$ 只要;
2. 半线性方程 (或几乎线性) 如果方程的主部分是线性的，并且出现在主部分中的系数是函数 $(n+1)$ 自变量 $x_1, \ldots, x_n, t$ 只要;
3. 拟线性方程 如果方程的主项是线性的，则主项中出现的系数是 $(n+1)$ 自变量和因变量 $u(\boldsymbol{x}, t)$ ，并 且至少一个以低阶项出现的系数函数取决于 $(n+1)$ 自变量以及函数的一些低阶导数 $u$.
函数的微分方程 $u=u(\boldsymbol{x}, t)$ 如果它不是拟线性方程，则称为完全非线性。
与常微分方程的情况一样，线性微分方程最容易处理。然而，请注意，虽然每个线性常微分方程都承认全 局解，但对于后一类微分方程则不然。一个将军 $k$ 函数的 th 阶线性溦分方程 $u=u(\boldsymbol{x}, t)$ 可以用下面给出 的运算符符号来写。为了 $\boldsymbol{x}=\left(x_1, \ldots, x_n\right)$ 和 $n$-元组 $a=\left(a_1, \ldots, a_n\right)$ 的非负整数，我们写
$$\boldsymbol{x}^a=x_1^{a_1} \cdots x_n^{a_n} \text { and } D^a=D_1^{a_1} \cdots D_n^{a_n},$$
在哪里 $D_j$ 表示偏微分算子 $\partial / \partial x_j$ ，为了 $j=1, \ldots, n$.

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|MATH1470

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

## 数学代写|偏微分方程代写partial difference equations代考|Momentum Transfer: Equations of Motion

We derive here system of partial differential equations as a model to study practical problems related to the dynamics of real fluids, which are called equations of motion. The related initial-boundary value problems help study physical processes dealing with momentum transfer.

In this case, in addition to three components of velocity vector $\mathbf{u}=\mathbf{u}(\boldsymbol{x}, t)$, the pressure $p=p(\boldsymbol{x}, t)>0$ (as global interaction among all particles), and the mass density $\rho=\rho(\boldsymbol{x}, t)$, we also take into consideration amount of the energy dissipation in terms of viscosity and heat exchange among different parts of it. Therefore, the proposed model must also include internal energy density function $e=e(\boldsymbol{x}, t)$ and the heat flux density function $q=q(\boldsymbol{x}, t)$. Notice that the frictional forces inside the fluid enhance the local coherence of the flow leading to a lower velocity if the fluid particles move faster than the averages of their neighbourhoods. We consider next two forces acting on a control volume:

1. The body forces due to gravity $\mathbf{g}$ given by
$$\mathbf{F}_v:=\int_V \rho \mathbf{g d} V .$$
2. The surface forces given by
$$\mathbf{F}_S:=\int_S(-p) \mathbf{n d} S .$$
Such types of forces are caused due to collisions between fluid molecules on either side of the surface $S$, which produce a flux of momentum across the boundary in the direction of the normal $\mathbf{n}$. We have considered only the effect of the pressure $p>0$ on the volume $V$ bounded by the surface $S$.

By Newton’s momentum law, the sum total of forces acting on a control volume $\mathrm{d} V$ must equals the rate of change of momentum. Therefore, by using the divergence theorem, we have
$$\int_V \rho \frac{D \mathbf{u}}{D t} \mathrm{~d} V=\int_V[-\nabla p+\rho \mathbf{g}] \mathrm{d} V .$$

## 数学代写|偏微分方程代写partial difference equations代考|Energy Transfer: Diffusion Equations

Finally, as an important case of energy transfer, we consider diffusion of a substance like a chemical through a homogeneous liquid (or a gas in air) contained in a bounded volume. Recall that diffusion process takes place due to the collisions of neighbouring molecules by which the kinetic energy of molecules is transferred from one to its nearest neighbour. It is assumed that the fluid is nearly motionless to avoid convection. Over a period of time, the substance diffuses throughout the fluid randomly moving from regions of higher concentration to the lower concentration. Under these conditions, we derive partial differential equations that are used to study time rate change in concentration levels of the diffusing substance at a position. These are called diffusion equations.

Let the mass density of the diffusing substance at position $x \in \Omega \subset \mathbb{R}^n$ and at time $t \geq 0$ be given by a sufficiently smooth function $u=u(\boldsymbol{x}, t)$. The initial concentration may be assumed to be given by a function $f(\boldsymbol{x})=u(\boldsymbol{x}, 0)$. Then, the total mass $M=M(t)$ inside the volume $V$ of the region $\Omega$ at time $t$ is given by
$$M(t)=\int_V u(\boldsymbol{x}, t) \mathrm{d} V, \quad \text { for } t \geq 0 .$$
If the flux vector $\mathbf{Q}=\mathbf{Q}(\boldsymbol{x})$ changes only when it goes through the surface element $\mathrm{d} S$ of a control volume $\mathrm{d} V$, then we have
$$\frac{\mathrm{d}}{\mathrm{d} t} \int_V u(\boldsymbol{x}, t) \mathrm{d} V=-\int_S \mathbf{Q} \cdot \mathbf{n} \mathrm{d} S,$$
where $\mathbf{n}$ is the (outward) normal to the surface $S$. Also, by Theorem 2.25, we have $$\int_S \mathbf{Q} \cdot \mathbf{n} \mathrm{d} S=\int_V \nabla \cdot \mathbf{Q} \mathrm{d} V .$$
Putting together the previous two equations, we obtain
$$\int_V\left[u_t(\boldsymbol{x}, t)+\nabla \cdot \mathbf{Q}\right] \mathrm{d} V=0 .$$
Assuming the necessary smoothness of the vector $\mathbf{Q}$, and using the fact that control volume $\mathrm{d} V$ is arbitrary, we obtain
$$u_t+\nabla \cdot \mathbf{Q}=0,$$
which is the differential form of an $n$-dimensional balance equation. Many physical phenomena in science and engineering such as heat conduction, Brownian motion, population dynamics are best described by Eq. (4.3.23).

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Momentum Transfer: Equations of Motion

1. 由于重力而产生的体力 $\mathbf{g}$ 由
$$\mathbf{F}_v:=\int_V \rho \mathbf{g d} V$$
2. 由下式给出的表面力
$$\mathbf{F}_S:=\int_S(-p) \mathbf{n d} S$$
这种类型的力是由于表面两侧的流体分子之间的碰撞引起的 $S$ ，这会产生沿法线方向穿过边界的动 量通量 $\mathbf{n}$. 我们只考虑了压力的影响 $p>0$ 在音量上 $V$ 以表面为界 $S$.
根据牛顿动量定律，作用在控制体积上的力的总和 $\mathrm{d} V$ 必须等于动量的变化率。因此，利用散度定理，我 们有
$$\int_V \rho \frac{D \mathbf{u}}{D t} \mathrm{~d} V=\int_V[-\nabla p+\rho \mathbf{g}] \mathrm{d} V$$

## 数学代写|偏微分方程代写partial difference equations代考|Energy Transfer: Diffusion Equations

$$M(t)=\int_V u(\boldsymbol{x}, t) \mathrm{d} V, \quad \text { for } t \geq 0 .$$

$$\frac{\mathrm{d}}{\mathrm{d} t} \int_V u(\boldsymbol{x}, t) \mathrm{d} V=-\int_S \mathbf{Q} \cdot \mathbf{n} \mathrm{d} S,$$

$$\int_S \mathbf{Q} \cdot \mathbf{n} \mathrm{d} S=\int_V \nabla \cdot \mathbf{Q} \mathrm{d} V .$$

$$\int_V\left[u_t(\boldsymbol{x}, t)+\nabla \cdot \mathbf{Q}\right] \mathrm{d} V=0 .$$

$$u_t+\nabla \cdot \mathbf{Q}=0$$

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|MATH1470

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

## 数学代写|偏微分方程代写partial difference equations代考|Circulations and Stokes’ Theorem

Here, we prove Stokes’ curl theorem. Notice that, since Theorem $2.26$ holds when $d \mathbf{v}$ is replaced by $d \mathbf{a}$, while making respective changes in the statement, the integral definition of curl $f=\nabla \times f$ is obtained as given in the next definition.

Definition 2.36 Let $f \in C^1(\Omega)$ be a 3-dimensional field. The component of $\nabla \times \boldsymbol{f}$ in the direction of the normal $\mathbf{n}$ is the integral given by
$$\mathbf{n} \cdot(\nabla \times \boldsymbol{f}):=\lim {\delta S} \frac{1}{\delta S} \oint{\delta C} \boldsymbol{f} \cdot \mathbf{r},$$ where $\delta S$ is a surface element orthogonal to normal $\mathbf{n}$, and $\delta C$ is the positively oriented ${ }^4$ boundary of the surface element $\delta S$.

In this case, $\nabla \times f$ is the ratio of the work done by the field $f$ while moving around the loop $\delta C$ to the area of the surface element $\delta S$, which explains why curl measures how much the field $\boldsymbol{f}$ swirls locally. So, $\operatorname{curl}(\boldsymbol{f})(\boldsymbol{x}) \neq 0$ gives a region of whirlpool of positive or negative curvature, and $\operatorname{curl}(f)(\boldsymbol{x})=0$ correspond to a the point of circulation-free motions. Expressing component functions of $\boldsymbol{f}$ in terms of Cartesian coordinates and using standard basis $\mathbf{e}{\mathbf{i}}$ for $\mathbf{n}$, this definition gives earlier definition of curl as $$\text { curl } f=\left(\mathbf{c}_1 \cdot \nabla \times f, \mathbf{c}_2 \cdot \nabla \times f, \mathbf{c}_3 \cdot \nabla \times f\right) \text {. }$$ Theorem $2.27$ (Stokes Theorem) Let $\boldsymbol{f}$ be a continuously differentiable vector field defined over a surface $S$, with a closed boundary curve $C$. Then $$\int_S \nabla \times \boldsymbol{f} \cdot \boldsymbol{n} d \boldsymbol{a}=\oint_C \boldsymbol{f} \cdot d \boldsymbol{r}$$ where $C$ is positively oriented with respect to the normal $\boldsymbol{n}$ in the sense as described earlier in a footnote remark. Proof Notice that, at infinitesimal level, (2.3.36) gives $$\nabla \times \boldsymbol{f} \cdot \mathbf{n} \approx \frac{1}{\delta S_i} \oint{\delta C_i} \boldsymbol{f} \cdot \ell,$$
where the surface element $\delta S_i$ is orthogonal to $\mathbf{n}$, and has $\delta C_i$ as the posively oriented closed boundary curve. Adding contributions over all infinitesimal surface elements, we obtain
$$\sum_i[\nabla \times \boldsymbol{f} \cdot \mathbf{n}] \delta S_i \approx \sum_i \oint_{\delta C_i} \boldsymbol{f} \cdot \mathbf{r}$$

## 数学代写|偏微分方程代写partial difference equations代考|Helmholtz Decomposition Theorem

We conclude the chapter with a discussion about the fundamental theorem of vector calculus due to Hermann von Helmholtz(1821-1894): Every sufficiently wellbehaved vector field $\boldsymbol{f}$ defined over a simply connected domain $\Omega \rightarrow \mathbb{R}^3$, with a piecewise smooth boundary, can be expressed as the sum of two suitably chosen vector fields, where the one is curl-free and the other divergence-free. It is also known as the Helmholtz’s decomposition theorem, which has numerous applications in physics and engineering, especially to problems related to electromagnetism. The theorem was known to Stokes since 1849, who published the related work in 1856.
Recall that a conservative field $\boldsymbol{f}=\left(f_1, f_2, f_3\right)$ can be written as
$$\boldsymbol{f}=-\nabla \varphi \quad \Longleftrightarrow \quad f_1=\varphi_x, \quad f_2=\varphi_y, \quad f_3=\varphi_z,$$
where $\varphi \in C^1(\Omega)$ is called a scalar potential of the field $f$ (Theorem 2.23). If $f$ represents the velocity field of a conservative fluid flow, then the level curves of $\varphi$ are known as the potential lines of the flow. Therefore, to solve a system of differential equations for the function $\boldsymbol{f}$, it suffices to solve the relate differential equations for the function $\varphi$. In most such cases, we are led to solve a Laplace equation of the form
$$u_{x x}+u_{y y}+u_{z z}=0, \quad \text { for some } u=u(x, y, z) \in C^2(\Omega) .$$

Also, since it is known by Maxwell law that a magnetic field $\mathbf{B}$ do not diverge from anything, it only curls around, i.e.,

B is the curl of some vector field $\boldsymbol{f}$, called a vector potential, and so it is always solenoidal (see Appendix A.2 for details). Also, for the Newton’s vector field
$$f(x)=-c \frac{x-x_1}{\left|x-x_1\right|^3},$$
defined over the star-shaped region
$$\Omega=\mathbb{R}^3 \backslash\left{\boldsymbol{x}_1+u\left(\boldsymbol{x}_1-\boldsymbol{x}_0\right): u \geq 0\right}, \text { for } \boldsymbol{x}_0 \neq \boldsymbol{x}_1,$$
with respect to the point $\boldsymbol{x}_0$, the vector potential $\mathbf{w}=\mathbf{w}(\boldsymbol{x})$ is given by
$$-c \frac{\left(x_0-\boldsymbol{x}_1\right) \times\left(\boldsymbol{x}-\boldsymbol{x}_1\right)}{\left|\boldsymbol{x}_0-\boldsymbol{x}_1\right|\left|\boldsymbol{x}-\boldsymbol{x}_1\right|^2+\left(\left(\boldsymbol{x}_0-\boldsymbol{x}_1\right) \times\left(\boldsymbol{x}-\boldsymbol{x}_1\right)\right)\left|\boldsymbol{x}-\boldsymbol{x}_1\right|} .$$
In general, when a vector field $\boldsymbol{f}$ is solenoidal, i.e., $\nabla \cdot \boldsymbol{f}=0$, we can write $\boldsymbol{f}=$ $\nabla \times \mathbf{w}$, for some vector field $\mathbf{w}$ (Theorem 2.24). The vector field $\mathbf{w}$ is called a vector potential for the field $\boldsymbol{f}$.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Circulations and Stokes’ Theorem

$$\mathbf{n} \cdot(\nabla \times \boldsymbol{f}):=\lim \delta S \frac{1}{\delta S} \oint \delta C \boldsymbol{f} \cdot \mathbf{r}$$

$$\operatorname{curl} f=\left(\mathbf{c}1 \cdot \nabla \times f, \mathbf{c}_2 \cdot \nabla \times f, \mathbf{c}_3 \cdot \nabla \times f\right) .$$ 定理 $2.27$ (斯托克斯定理) 让 $\boldsymbol{f}$ 是在曲面上定义的连续可微矢量场 $S$ ，具有封闭的边界曲线 $C$. 然后 $$\int_S \nabla \times \boldsymbol{f} \cdot \boldsymbol{n} d \boldsymbol{a}=\oint_C \boldsymbol{f} \cdot d \boldsymbol{r}$$ 在哪里 $C$ 相对于法线是正向的 $n$ 在前面脚注中描述的意义上。证明 注意，在无穷小的水平上，(2.3.36) 给 出 $$\nabla \times \boldsymbol{f} \cdot \mathbf{n} \approx \frac{1}{\delta S_i} \oint \delta C_i \boldsymbol{f} \cdot \ell,$$ 其中表面元素 $\delta S_i$ 正交于 $\mathbf{n}$ ，并且有 $\delta C_i$ 作为正向封闭边界曲线。添加对所有无穷小表面元素的贡献，我们 得到 $$\sum_i[\nabla \times \boldsymbol{f} \cdot \mathbf{n}] \delta S_i \approx \sum_i \oint{\delta C_i} \boldsymbol{f} \cdot \mathbf{r}$$

## 数学代写|偏微分方程代写partial difference equations代考|Helmholtz Decomposition Theorem

$$\boldsymbol{f}=-\nabla \varphi \quad f_1=\varphi_x, \quad f_2=\varphi_y, \quad f_3=\varphi_z,$$

$$u_{x x}+u_{y y}+u_{z z}=0, \quad \text { for some } u=u(x, y, z) \in C^2(\Omega) .$$

$B$ 是某个矢量场的旋度 $\boldsymbol{f}$ ，称为矢量势，因此它始终是螺线管（有关详细信息，请参见附录 A.2)。此外， 对于牛顿矢量场
$$f(x)=-c \frac{x-x_1}{\left|x-x_1\right|^3},$$

$$-c \frac{\left(x_0-\boldsymbol{x}_1\right) \times\left(\boldsymbol{x}-\boldsymbol{x}_1\right)}{\left|\boldsymbol{x}_0-\boldsymbol{x}_1\right|\left|\boldsymbol{x}-\boldsymbol{x}_1\right|^2+\left(\left(\boldsymbol{x}_0-\boldsymbol{x}_1\right) \times\left(\boldsymbol{x}-\boldsymbol{x}_1\right)\right)\left|\boldsymbol{x}-\boldsymbol{x}_1\right|} .$$

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|MATH4310

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

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

The development of vector analysis is primarily due to English mathematician Oliver Heaviside (1850-1925), and independently by American mathematician Josiah Gibbs (18391903). Heaviside published his work in 1893 as part of the book “The Elements of Vectorial Algebra and Analysis”, whereas Gibbs’ work first appeared as the book “Elements of Vector Analysis”, published in 1901 as the compilation of the his lectures delivered in 1881 at Yale University. Heaviside applied vector analysis tools to reformulate the twelve of twenty equations related to electromagnetic radiations in vector form, which were originally proposed by Scottish mathematician and scientist James Maxwell (1831-1879) during 1861-62. The twelve equations are recognised in modern physics as the Maxwell’s four fundamental equations (see Appendix A.2 for details). Most notations and terminology introduced in this section are due to Gibbs.

In this section, we discuss the three fundamental theorems due to Gauss, Stokes, and Helmholtz that are applied in the next two chapters to derive differential equation models for some important practical problems related to physical phenomena such as fluid flow, heat conduction, mechanical vibrations, and electromagnetic waves. In all that follows, the 3-dimensional del operator as introduced earlier plays the lead role. We shall use shorthand operator notations as given below:
$$\partial_x \equiv \frac{\partial}{\partial x}, \quad \partial_y \equiv \frac{\partial}{\partial y}, \quad \partial_{x x} \equiv \frac{\partial^2}{\partial x^2}, \quad \partial_{y x} \equiv \frac{\partial^2}{\partial y \partial x}, \quad \text { etc. }$$

Let $\Omega \subseteq \mathbb{R}^n$ be a domain. Recall that a $C^1$-function $\varphi: \Omega \rightarrow \mathbb{R}$ is called a scalar field, and, a vector field is a $C^1$-function $f: \Omega \rightarrow \mathbb{R}^n$. Clearly, each coordinate function $f_i: \Omega \rightarrow \mathbb{R}$ of a vector field $f$ is a $C^1$-scalar field, for $i=1, \ldots, n$. More generally, for $k \geq 1$, a vector field $f$ is a $C^k$-function if and only if each coordinate function $f_i \in C^k(\Omega)$. That is, for $k \geq 1$,
$$f=\left(f_1, \ldots, f_n\right) \in C^k(\Omega) \Leftrightarrow f_i \in C^k(\Omega), \text { for all } 1 \leq i \leq n .$$
Therefore, $f$ is a $C^{\infty}$-function if and only if each coordinate function $f_i \in C^{\infty}(\Omega)$. In latter case, we also say that $f$ is a smooth vector field. A similar modification holds for other notations and terminology applicable to the vector fields. We are mainly dealing with the case when $n=2$ or $n=3$.

Definition 2.29 For $I \subseteq \mathbb{R}$, and a domain $U \subseteq \mathbb{R}^n$, let $\Omega=I \times U \subseteq \mathbb{R}^{n+1}$, and $\boldsymbol{f}: U \rightarrow \mathbb{R}^n$ be a $C^1$ vector field. For any $\left(t_0, \boldsymbol{x}_0\right) \in \Omega$, let $\delta>0, \varepsilon>0$ be such that $J=\left[t_0-\delta, t_0+\varepsilon\right] \subset I$. A $C^1$-function $\boldsymbol{x}:\left[t_0-\delta, t_0+\varepsilon\right] \rightarrow \Omega$ is called an integral curve of the field $f$ if
$$x^{\prime}(t)=f(x(t)) \text {, for all } t \in J \text {, with } x\left(t_0\right)=x_0 \text {. }$$

## 数学代写|偏微分方程代写partial difference equations代考|Flux and Divergence Theorem

As it has been for the line integral of a vector field along a smooth curve, the surface integral (or the volume integral) of a vector field over a regular surface $S$ also depends on the orientation of the surface. To introduce the concept, let $\mathbf{r}=\mathbf{r}(u, v): \Omega \rightarrow$ $\mathbb{R}^3$ be a parametrisation of $S$. In general, we study the geometry of $S$ at a point $a=\left(u_0, v_0\right) \in \Omega$ by using the two (orthogonal) curves given by
$$\mathbf{r}_1(u)=\mathbf{r}\left(u, v_0\right) \quad \text { and } \quad \mathbf{r}_2(v)=\mathbf{r}\left(u_0, v\right),$$
respectively, called the $u$-curve and $v$-curve. Notice that the derivatives $\mathbf{r}_u=\mathbf{r}^{\prime}(u)$ and $\mathbf{r}_v=\mathbf{r}^{\prime}(v)$ are, respectively, the tangent vectors to the two curves $\mathbf{r}_1$ and $\mathbf{r}_2$ on $S$. Also, by the vector identity
$$\left|\mathbf{r}_u \times \mathbf{r}_v\right|^2=\left(\mathbf{r}_u \times \mathbf{r}_v\right) \cdot\left(\mathbf{r}_u \times \mathbf{r}_v\right)=\operatorname{det}\left(\begin{array}{l} \mathbf{r}_u \cdot \mathbf{r}_u \mathbf{r}_u \cdot \mathbf{r}_v \ \mathbf{r}_v \cdot \mathbf{r}_u \mathbf{r}_v \cdot \mathbf{r}_v \end{array}\right)$$
it follows that the regularity condition as given in Definition $2.27$ is equivalent to the condition that the vectors $\mathbf{r}_u, \mathbf{r}_v$ are linearly independent. Therefore, there is a unique (shifted) tangent plane $\Pi(a)$ at $\mathbf{r}(\boldsymbol{a}) \in S$ spanned by the tangent vectors $\mathbf{r}_u$ and $\mathbf{r}_v$. In fact, the two vectors form a natural basis for the tangent plane $\Pi(\boldsymbol{x})$ in the sense we explain shortly. In particular, if $\varphi \in C^1(\Omega)$, then for the regular surface $$\Gamma_{\varphi}(x, y, z): \quad \varphi(x, y)-z=0,$$
we have $\mathbf{r}x=\left(1,0, \varphi_x\right)$ and $\mathbf{r}_y=\left(0,1, \varphi_y\right)$, which implies that $$\mathbf{r}_x \times \mathbf{r}_y=\left(-\varphi_x,-\varphi_y, 1\right),$$ and the equation of the tangent plane at a point $\boldsymbol{a}=\left(x_0, y_0, z_0\right)$ is given by $$\varphi_x\left(x-x_0\right)+\varphi_y\left(y-y_0\right)-\left(z-z_0\right)=0, \quad \text { where } z_0=\varphi\left(x_0, y_0\right) .$$ Therefore, for any $x \in 1{\varphi}^{\prime}$, we have
$$\mathbf{n}(\boldsymbol{x})=\pm \frac{\mathbf{r}_u \times \mathbf{r}_v}{\left|\mathbf{r}_u \times \mathbf{r}_v\right|}=\frac{\left(-\varphi_x,-\varphi_y, 1\right)}{\sqrt{\varphi_x^2+\varphi_y^2+1}}$$

# 偏微分方程代写

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

Gibbs (18391903)。Heaviside 于 1893 年发表了他的作品，作为“矢量代数和分析的要素”一书的一部分，

$$\partial_x \equiv \frac{\partial}{\partial x}, \quad \partial_y \equiv \frac{\partial}{\partial y}, \quad \partial_{x x} \equiv \frac{\partial^2}{\partial x^2}, \quad \partial_{y x} \equiv \frac{\partial^2}{\partial y \partial x}, \quad \text { etc. }$$

$$f=\left(f_1, \ldots, f_n\right) \in C^k(\Omega) \Leftrightarrow f_i \in C^k(\Omega), \text { for all } 1 \leq i \leq n .$$

$x^{\prime}(t)=f(x(t))$, for all $t \in J$, with $x\left(t_0\right)=x_0$.

## 数学代写|偏微分方程代写partial difference equations代考|Flux and Divergence Theorem

$$\mathbf{r}1(u)=\mathbf{r}\left(u, v_0\right) \quad \text { and } \quad \mathbf{r}_2(v)=\mathbf{r}\left(u_0, v\right),$$ 分别称为 $u$-曲线和 $v$-曲线。注意导数 $\mathbf{r}_u=\mathbf{r}^{\prime}(u)$ 和 $\mathbf{r}_v=\mathbf{r}^{\prime}(v)$ 分别是两条曲线的切向量 $\mathbf{r}_1$ 和 $\mathbf{r}_2$ 上 $S$. 此 外，通过矢量标识 $$\left|\mathbf{r}_u \times \mathbf{r}_v\right|^2=\left(\mathbf{r}_u \times \mathbf{r}_v\right) \cdot\left(\mathbf{r}_u \times \mathbf{r}_v\right)=\operatorname{det}\left(\mathbf{r}_u \cdot \mathbf{r}_u \mathbf{r}_u \cdot \mathbf{r}_v \mathbf{r}_v \cdot \mathbf{r}_u \mathbf{r}_v \cdot \mathbf{r}_v\right)$$ 由此得出定义中给出的规律性条件 $2.27$ 等同于向量的条件 $\mathbf{r}_u, \mathbf{r}_v$ 是线性独立的。因此，存在唯一的（移动 的）切平面 $\Pi(a)$ 在 $\mathbf{r}(\boldsymbol{a}) \in S$ 由切向量跨越 $\mathbf{r}_u$ 和 $\mathbf{r}_v$. 事实上，这两个向量构成了切平面的自然基础 $\Pi(\boldsymbol{x})$ 从某种意义上说，我们很快就会解释。特别是，如果 $\varphi \in C^1(\Omega)$ ，那么对于规则曲面 $$\Gamma{\varphi}(x, y, z): \quad \varphi(x, y)-z=0,$$

$$\mathbf{r}_x \times \mathbf{r}_y=\left(-\varphi_x,-\varphi_y, 1\right),$$

$$\varphi_x\left(x-x_0\right)+\varphi_y\left(y-y_0\right)-\left(z-z_0\right)=0, \quad \text { where } z_0=\varphi\left(x_0, y_0\right) .$$

$$\mathbf{n}(\boldsymbol{x})=\pm \frac{\mathbf{r}_u \times \mathbf{r}_v}{\left|\mathbf{r}_u \times \mathbf{r}_v\right|}=\frac{\left(-\varphi_x,-\varphi_y, 1\right)}{\sqrt{\varphi_x^2+\varphi_y^2+1}}$$

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|Math462

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

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

The set $\mathbb{R}^n$ of $n$-tuples of real numbers is a linear space over the field $\mathbb{R}$, where the addition and scalar multiplication are defined, respectively, as \begin{aligned} \boldsymbol{x}+\boldsymbol{y} & =\left(x_1+y_1, \ldots, x_n+y_n\right) \ a \cdot \boldsymbol{x} & =\left(a x_1, \ldots, a x_n\right), \end{aligned}
for $\boldsymbol{x}=\left(x_1, \ldots, x_n\right), \boldsymbol{y}=\left(y_1, \ldots, y_n\right) \in \mathbb{R}^n$ and $a \in \mathbb{R}$. The usual dot product of vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ defines an inner product on $\mathbb{R}^n$, which is written as $\langle\boldsymbol{x}, \boldsymbol{y}\rangle$. That is,
$$\langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{x} \cdot \boldsymbol{y}=x_1 y_1+\cdots+x_n y_n .$$
It can be shown that the function $\langle\rangle:, \mathbb{R}^n \rightarrow \mathbb{R}$ given by (2.1.1) is a positive definite, symmetric, bilinearfunctional (Exercise 2.1). Therefore, $\mathbb{R}^n$ is an inner product space. In general, a linear space $X$ over the field $\mathbb{R}$ is called an inner product space if there exists a positive definite, symmetric, bilinear functional $b: X \times X \rightarrow \mathbb{R}$.

Further, a linear space $X$ over the field $\mathbb{R}$ is called a normed space if there exists a positive definite, absolute homogeneous, subadditive function $p: X \rightarrow \mathbb{R}$, where $p(x)$ is called the norm of $x \in X$. In particular, it can be shown that the function ||$: \mathbb{R}^n \rightarrow \mathbb{R}$ given by
$$|\boldsymbol{x}|:=\sqrt{\langle\boldsymbol{x}, \boldsymbol{y}\rangle}=\sqrt{x_1^2+\cdots+x_n^2} .$$
gives a norm on $\mathbb{R}^n$ (Exercise 2.2). Therefore, $\mathbb{R}^n$ is a normed space, where $|\boldsymbol{x}|$ and their norms is called the norm of a vector $\boldsymbol{x}$ induced by the inner product (2.1.1). An interesting relation between the inner product of two points $\boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^n$ is the Cauchy-Schwartz inequality given by
$$|\langle\boldsymbol{x}, \boldsymbol{y}\rangle| \leq|\boldsymbol{x}||\boldsymbol{y}|, \quad \text { for } \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^n .$$
A yet another interesting relation for the case when $n=3$ is the identity given by
$$|\boldsymbol{a} \times \boldsymbol{b}|^2=|\boldsymbol{a}|^2|\boldsymbol{b}|^2-\langle\boldsymbol{a}, \boldsymbol{b}\rangle^2, \quad \text { for } \boldsymbol{a}, \boldsymbol{b} \in \mathbb{R}^3,$$
which finds many important applications, where the cross product $\boldsymbol{a} \times \boldsymbol{b}$ of vectors $\boldsymbol{a}=\left(a_1, a_2, a_3\right)$ and $\boldsymbol{b}=\left(b_1, b_2, b_3\right)$ is a vector in $\mathbb{R}^3$ given by
$$\boldsymbol{a} \times \boldsymbol{b}:=\left(a_2 b_3-a_3 b_2, a_3 b_1-a_1 b_3, a_1 b_2-a_2 b_1\right) .$$

## 数学代写|偏微分方程代写partial difference equations代考|Classical Theory of Surfaces and Curves

Let $\Omega \subseteq \mathbb{R}^n$ be a domain, and consider a $C^1$-function $\varphi: \Omega \rightarrow \mathbb{R}$, with $\nabla \varphi \not \equiv 0$ over $\Omega$. We may write $\Gamma_{\varphi}$ for the graph of $\varphi$ as given by
$$\Gamma_{\varphi}={(\boldsymbol{x}, \varphi(\boldsymbol{x})) \in \Omega \times \varphi(\Omega): \boldsymbol{x} \in \Omega} \subset \mathbb{R}^{n+1} .$$
In the particular case, when $\Omega=\mathbb{R}^n$ and $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ is a linear map, the graph $\Gamma_{\varphi} \subset \mathbb{R}^{n+1}$ is a linear subspace of dimension $n$ with a basis given by the vectors
$$\left(e_1, \varphi\left(e_1\right)\right), \ldots,\left(e_n, \varphi\left(e_n\right)\right),$$
where $\boldsymbol{e}_1, \ldots, \boldsymbol{e}_n$ is the standard basis of the space $\mathbb{R}^n$. In the general case, when $n=1,2$, a function $\varphi$ can be visualised easily in terms of its graph. For example, if $D$ is a domain in $\mathbb{R}^2$ and $f \in C^1(D)$, then the geometry of the graph surface $\Gamma_f$ can be identified as a family of curves obtained as the intersection of $\Gamma_f$ with planes parallel to coordinates planes. A more interesting situation corresponds to the case when such types of curves are sections of $\Gamma_f$ formed by using the planes $z=c$, for $c \in f(D)$. We call these as the level curves of surface $\Gamma_f$, provided $\nabla f(c) \neq 0$.
Example 2.7 For the function $f_1: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by
$$f_1(x, y)=4 x^2+9 y^2, \quad \text { with }(x, y) \in \mathbb{R}^2 \backslash{(0,0)},$$
the level curves are ellipses given by the equation $4 x^2+9 y^2=f_1(a, b)$, for some $(a, b) \neq(0,0)$. Similarly, for the function $f_2(x, y)=x y,(x, y) \in \mathbb{R}^2 \backslash{(0,0)}$, the level curves are hyperbolas of the type $x y=f_2(a, b)$, for some $(a, b) \neq(0,0)$. Also, for the function $f_3(x, y)=x^2+y^2-1,(x, y) \in \mathbb{R}^2 \backslash{(0,0)}$, the level curves are circles given by $x^2+y^2-1=f_3(a, b)$, for some $(a, b) \neq(0,0)$.

When $\Omega \subseteq \mathbb{R}^3$ is a domain and $F \in C^1(\Omega)$, with $\nabla F \not \equiv 0$ over $\Omega$, a projected surface curve given by $\Gamma_F \bigcap \mathbb{R}^3$ is called a contour map if the function $F$ remains constant along the curve. In general, contour map in lower dimensions is obtained by keeping fixed one of the independent variables, which provides a visually intuitive way to see the level surfaces of the graph $\Gamma_F$. For example, taking $x=a$, the contour map is the intersection of the graph $\Gamma_F$ with the plane $x=a$. A level surface of the function $F$ is the contour map obtained from the intersection of $\Gamma_F$ with the plane $z=c$. In what follows, we reserve the term level set for the surface obtained from $\Gamma_{\varphi}$ by slicing it with the plane $x_{n+1}=\varphi\left(x_0\right)$, for some $x_0 \in \Omega$.

# 偏微分方程代写

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

$$\boldsymbol{x}+\boldsymbol{y}=\left(x_1+y_1, \ldots, x_n+y_n\right) a \cdot \boldsymbol{x}=\left(a x_1, \ldots, a x_n\right),$$

$$\langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{x} \cdot \boldsymbol{y}=x_1 y_1+\cdots+x_n y_n .$$

$$|\boldsymbol{x}|:=\sqrt{\langle\boldsymbol{x}, \boldsymbol{y}\rangle}=\sqrt{x_1^2+\cdots+x_n^2} .$$

$$|\langle\boldsymbol{x}, \boldsymbol{y}\rangle| \leq|\boldsymbol{x} | \boldsymbol{y}|, \quad \text { for } \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^n .$$

$$|\boldsymbol{a} \times \boldsymbol{b}|^2=|\boldsymbol{a}|^2|\boldsymbol{b}|^2-\langle\boldsymbol{a}, \boldsymbol{b}\rangle^2, \quad \text { for } \boldsymbol{a}, \boldsymbol{b} \in \mathbb{R}^3,$$

$$\boldsymbol{a} \times \boldsymbol{b}:=\left(a_2 b_3-a_3 b_2, a_3 b_1-a_1 b_3, a_1 b_2-a_2 b_1\right) .$$

## 数学代写|偏微分方程代写partial difference equations代考|Classical Theory of Surfaces and Curves

$$\Gamma_{\varphi}=(\boldsymbol{x}, \varphi(\boldsymbol{x})) \in \Omega \times \varphi(\Omega): \boldsymbol{x} \in \Omega \subset \mathbb{R}^{n+1}$$

$$\left(e_1, \varphi\left(e_1\right)\right), \ldots,\left(e_n, \varphi\left(e_n\right)\right),$$

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|AMATH353

statistics-lab™ 为您的留学生涯保驾护航 在代写偏微分方程partial difference equations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。我们的专家在代写偏微分方程partial difference equations代写方面经验极为丰富，各种代写偏微分方程partial difference equations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

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

If an endpoint of the string is fixed, then the displacement is zero and this can be written as
$$u(0, t)=0$$
or
$$u(L, t)=0 .$$
We may vary an endpoint in a prescribed way, e.g.
$$u(0, t)=b(t) .$$
A more interesting condition occurs if the end is attached to a dynamical system (see e.g. Haberman [4])
$$T_0 \frac{\partial u(0, t)}{\partial x}=k\left(u(0, t)-u_E(t)\right) .$$
This is known as an elastic boundary condition. If $u_E(t)=0$, i.e. the equilibrium position of the system coincides with that of the string, then the condition is homogeneous.
As a special case, the free end boundary condition is
$$\frac{\partial u}{\partial x}=0 .$$
Since the problem is second order in time, we need two initial conditions. One usually has
$$\begin{gathered} u(x, 0)=f(x) \ u_t(x, 0)=g(x) \end{gathered}$$
i.e. given the displacement and velocity of each segment of the string.

## 数学代写|偏微分方程代写partial difference equations代考|Diffusion in Three Dimensions

Diffusion problems lead to partial differential equations that are similar to those of heat conduction. Suppose $C(x, y, z, t)$ denotes the concentration of a substance, i.e. the mass per unit volume, which is dissolving into a liquid or a gas. For example, pollution in a lake. The amount of a substance (pollutant) in the given domain $V$ with boundary $\Gamma$ is given by
$$\int_V C(x, y, z, t) d V .$$
The law of conservation of mass states that the time rate of change of mass in $V$ is equal to the rate at which mass flows into $V$ minus the rate at which mass flows out of $V$ plus the rate at which mass is produced due to sources in $V$. Let’s assume that there are no internal sources. Let $\vec{q}$ be the mass flux vector, then $\vec{q} \cdot \vec{n}$ gives the mass per unit area per unit time crossing a surface element with outward unit normal vector $\vec{n}$.
$$\frac{d}{d t} \int_V C d V=\int_V \frac{\partial C}{\partial t} d V=-\int_{\Gamma} \vec{q} \cdot \vec{n} d S .$$
Use Gauss divergence theorem to replace the integral on the boundary
$$\int_{\Gamma} \vec{q} \cdot \vec{n} d S=\int_V \operatorname{div} \vec{q} d V .$$
Therefore
$$\frac{\partial C}{\partial t}=-\operatorname{div} \vec{q} .$$
Fick’s law of diffusion relates the flux vector $\vec{q}$ to the concentration $C$ by
$$\vec{q}=-D \operatorname{grad} C+C \vec{v}$$
where $\vec{v}$ is the velocity of the liquid or gas, and $D$ is the diffusion coefficient which may depend on $C$. Combining (1.7.4) and (1.7.5) yields
$$\frac{\partial C}{\partial t}=\operatorname{div}(D \operatorname{grad} C)-\operatorname{div}(C \vec{v})$$

# 偏微分方程代写

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

$$u(0, t)=0$$

$$u(L, t)=0 .$$

$$u(0, t)=b(t)$$

$$T_0 \frac{\partial u(0, t)}{\partial x}=k\left(u(0, t)-u_E(t)\right) .$$

$$\frac{\partial u}{\partial x}=0 .$$

$$u(x, 0)=f(x) u_t(x, 0)=g(x)$$

## 数学代写|偏微分方程代写partial difference equations代考|Diffusion in Three Dimensions

$$\int_V C(x, y, z, t) d V .$$

$$\frac{d}{d t} \int_V C d V=\int_V \frac{\partial C}{\partial t} d V=-\int_{\Gamma} \vec{q} \cdot \vec{n} d S .$$

$$\int_{\Gamma} \vec{q} \cdot \vec{n} d S=\int_V \operatorname{div} \vec{q} d V$$

$$\frac{\partial C}{\partial t}=-\operatorname{div} \vec{q}$$

$$\vec{q}=-D \operatorname{grad} C+C \vec{v}$$

$$\frac{\partial C}{\partial t}=\operatorname{div}(D \operatorname{grad} C)-\operatorname{div}(C \vec{v})$$

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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。