## 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

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

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

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

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

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

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

## 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|Math442

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

In solving the above model, we have to specify two boundary conditions and an initial condition. The initial condition will be the distribution of temperature at time $t=0$, i.e.
$$u(x, 0)=f(x) .$$
The boundary conditions could be of several types.

1. Prescribed temperature (Dirichlet b.c.)
$$u(0, t)=p(t)$$
or
$$u(L, t)=q(t) .$$
2. Insulated boundary (Neumann b.c.)
$$\frac{\partial u(0, t)}{\partial n}=0$$
where $\frac{\partial}{\partial n}$ is the derivative in the direction of the outward normal. Thus at $x=0$
$$\frac{\partial}{\partial n}=-\frac{\partial}{\partial x}$$
and at $x=L$
$$\frac{\partial}{\partial n}=\frac{\partial}{\partial x}$$
(see Figure 2).
3. When a one dimensional wire is in contact at a boundary with a moving fluid or gas, then there is a heat exchange. This is specified by Newton’s law of cooling
4. $$5. -K(0) \frac{\partial u(0, t)}{\partial x}=-H{u(0, t)-v(t)} 6.$$
7. where $H$ is the heat transfer (convection) coefficient and $v(t)$ is the temperature of the surroundings. We may have to solve a problem with a combination of such boundary conditions. For example, one end is insulated and the other end is in a fluid to cool it.

## 数学代写|偏微分方程代写partial difference equations代考|A Vibrating String

Suppose we have a tightly stretched string of length $L$. We imagine that the ends are tied down in some way (see next section). We describe the motion of the string as a result of disturbing it from equilibrium at time $t=0$, see Figure 4 .

We assume that the slope of the string is small and thus the horizontal displacement can be neglected. Consider a small segment of the string between $x$ and $x+\Delta x$. The forces acting on this segment are along the string (tension) and vertical (gravity). Let $T(x, t$ ) be the tension at the point $x$ at time $t$, if we assume the string is flexible then the tension is in the direction tangent to the string, see Figure 5.

The slope of the string is given by
$$\tan \theta=\lim {\Delta x \rightarrow 0} \frac{u(x+\Delta x, t)-u(x, t)}{\Delta x}=\frac{\partial u}{\partial x} .$$ Thus the sum of all vertical forces is: $$T(x+\Delta x, t) \sin \theta(x+\Delta x, t)-T(x, t) \sin \theta(x, t)+\rho_0(x) \Delta x Q(x, t)$$ where $Q(x, t)$ is the vertical component of the body force per unit mass and $\rho_o(x)$ is the density. Using Newton’s law $$F=m a=\rho_0(x) \Delta x \frac{\partial^2 u}{\partial t^2} .$$ Thus $$\rho_0(x) u{t t}=\frac{\partial}{\partial x}[T(x, t) \sin \theta(x, t)]+\rho_0(x) Q(x, t)$$
For small angles $\theta$,
$$\sin \theta \approx \tan \theta$$
Combining (1.5.1) and (1.5.5) with (1.5.4) we obtain
$$\rho_0(x) u_{t t}=\left(T(x, t) u_x\right)_x+\rho_0(x) Q(x, t)$$

# 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

$$u(x, 0)=f(x) .$$

1. 规定温度 (Dirichlet bc)
$$u(0, t)=p(t)$$
或者
$$u(L, t)=q(t)$$
2. 绝缘边界 (Neumann bc)
$$\frac{\partial u(0, t)}{\partial n}=0$$
在哪里 $\frac{\partial}{\partial n}$ 是向外法线方向的导数。因此在 $x=0$
$$\frac{\partial}{\partial n}=-\frac{\partial}{\partial x}$$
在 $x=L$
$$\frac{\partial}{\partial n}=\frac{\partial}{\partial x}$$
(见图 2)。
3. 当一维导线在边界处与移动的流体或气体接触时，就会发生热交换。这是由牛顿冷却定律规定的
4. $\$ \$$5. -\mathrm{K}(0) \backslash frac {\backslash partial u(0, t)}{ partial x}=-H{u(0, t)-v(t)} 6. \ \$$
7. 在哪里 $H$ 是传热 (对流) 系数和 $v(t)$ 是周围环境的温度。我们可能不得不结合这些边界条件来解决问题。 例如，一端是绝缘的，另一端是在流体中以对其进行冷却。

## 数学代写|偏微分方程代写partial difference equations代考|A Vibrating String

$$\tan \theta=\lim \Delta x \rightarrow 0 \frac{u(x+\Delta x, t)-u(x, t)}{\Delta x}=\frac{\partial u}{\partial x} .$$

$$T(x+\Delta x, t) \sin \theta(x+\Delta x, t)-T(x, t) \sin \theta(x, t)+\rho_0(x) \Delta x Q(x, t)$$

$$F=m a=\rho_0(x) \Delta x \frac{\partial^2 u}{\partial t^2} .$$

$$\rho_0(x) u t t=\frac{\partial}{\partial x}[T(x, t) \sin \theta(x, t)]+\rho_0(x) Q(x, t)$$

$$\sin \theta \approx \tan \theta$$

$$\rho_0(x) u_{t t}=\left(T(x, t) u_x\right)_x+\rho_0(x) Q(x, t)$$

## 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|МАТH2415

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|Basic Concepts and Definitions

Definition 1. A partial differential equation (PDE) is an equation containing partial derivatives of the dependent variable.
For example, the following are PDEs
$$\begin{gathered} u_t+c u_x=0 \ u_{x x}+u_{y y}=f(x, y) \ \alpha(x, y) u_{x x}+2 u_{x y}+3 x^2 u_{y y}=4 e^x \ u_x u_{x x}+\left(u_y\right)^2=0 \ \left(u_{x x}\right)^2+u_{y y}+a(x, y) u_x+b(x, y) u=0 . \end{gathered}$$
Note: We use subscript to mean differentiation with respect to the variables given, e.g. $u_t=\frac{\partial u}{\partial t}$. In general we may write a PDE as
$$F\left(x, y, \cdots, u, u_x, u_y, \cdots, u_{x x}, u_{x y}, \cdots\right)=0$$
where $x, y, \cdots$ are the independent variables and $u$ is the unknown function of these variables. Of course, we are interested in solving the problem in a certain domain D. A solution is a function $u$ satisfying (1.1.6). From these many solutions we will select the one satisfying certain conditions on the boundary of the domain D. For example, the functions
\begin{aligned} &u(x, t)=e^{x-c t} \ &u(x, t)=\cos (x-c t) \end{aligned}
are solutions of (1.1.1), as can be easily verified. We will see later (section 3.1) that the general solution of (1.1.1) is any function of $x-c t$.

Definition 2. The order of a PDE is the order of the highest order derivative in the equation. For example (1.1.1) is of first order and (1.1.2) – (1.1.5) are of second order.

Definition 3. A PDE is linear if it is linear in the unknown function and all its derivatives with coefficients depending only on the independent variables.

## 数学代写|偏微分方程代写partial difference equations代考|Conduction of Heat in a Rod

Consider a rod of constant cross section A and length L (see Figure 1) oriented in the $x$ direction.
Let $e(x, t)$ denote the thermal energy density or the amount of thermal energy per unit volume. Suppose that the lateral surface of the rod is perfectly insulated. Then there is no thermal energy loss through the lateral surface. The thermal energy may depend on $x$ and $t$ if the bar is not uniformly heated. Consider a slice of thickness $\Delta x$ between $x$ and $x+\Delta x$.

If the slice is small enough then the total energy in the slice is the product of thermal energy density and the volume, i.e.
$$e(x, t) A \Delta x \text {. }$$
The rate of change of heat energy is given by
$$\frac{\partial}{\partial t}[e(x, t) A \Delta x] .$$
Using the conservation law of heat energy, we have that this rate of change per unit time is equal to the sum of the heat energy generated inside per unit time and the heat energy flowing across the boundaries per unit time. Let $\varphi(x, t)$ be the heat flux (amount of thermal energy per unit time flowing to the right per unit surface area). Let $S(x, t)$ be the heat energy per unit volume generated per unit time. Then the conservation law can be written as follows
$$\frac{\partial}{\partial t}[e(x, t) A \Delta x]=\varphi(x, t) A-\varphi(x+\Delta x, t) A+S(x, t) A \Delta x .$$
This equation is only an approximation but it is exact at the limit when the thickness of the slice $\Delta x \rightarrow 0$. Divide by $A \Delta x$ and let $\Delta x \rightarrow 0$, we have
$$\frac{\partial}{\partial t} e(x, t)=-\underbrace{\lim {\Delta x \rightarrow 0} \frac{\varphi(x+\Delta x, t)-\varphi(x, t)}{\Delta x}}{=\frac{\partial \varphi(x, t)}{\partial x}}+S(x, t) .$$
We now rewrite the equation using the temperature $u(x, t)$. The thermal energy density $e(x, t)$ is given by
$$e(x, t)=c(x) \rho(x) u(x, t)$$
where $c(x)$ is the specific heat (heat energy to be supplied to a unit mass to raise its temperature by one degree) and $\rho(x)$ is the mass density. The heat flux is related to the temperature via Fourier’s law
$$\varphi(x, t)=-K \frac{\partial u(x, t)}{\partial x}$$

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Basic Concepts and Definitions

$$u_t+c u_x=0 u_{x x}+u_{y y}=f(x, y) \alpha(x, y) u_{x x}+2 u_{x y}+3 x^2 u_{y y}=4 e^x u_x u_{x x}+\left(u_y\right)^2=0\left(u_{x x}\right)^2$$

$$F\left(x, y, \cdots, u, u_x, u_y, \cdots, u_{x x}, u_{x y}, \cdots\right)=0$$

$$u(x, t)=e^{x-c t} \quad u(x, t)=\cos (x-c t)$$

## 数学代写|偏微分方程代写partial difference equations代考|Conduction of Heat in a Rod

$$e(x, t) A \Delta x .$$

$$\frac{\partial}{\partial t}[e(x, t) A \Delta x] .$$

$$\frac{\partial}{\partial t}[e(x, t) A \Delta x]=\varphi(x, t) A-\varphi(x+\Delta x, t) A+S(x, t) A \Delta x .$$

$$\frac{\partial}{\partial t} e(x, t)=-\underbrace{\lim \Delta x \rightarrow 0 \frac{\varphi(x+\Delta x, t)-\varphi(x, t)}{\Delta x}}=\frac{\partial \varphi(x, t)}{\partial x}+S(x, t) .$$

$$e(x, t)=c(x) \rho(x) u(x, t)$$

$$\varphi(x, t)=-K \frac{\partial u(x, t)}{\partial x}$$

## 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|МATH4335

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|FINITE DIFFERENCE METHOD

The finite difference method is based on the calculus of finite differences. It is a relatively straightforward method in which the governing PDE is satisfied at a set of prescribed interconnected points within the computational domain, referred to as nodes. The boundary conditions, in turn, are satisfied at a set of prescribed nodes located on the boundaries of the computational domain. The framework of interconnected nodes is referred to as a grid or mesh. Each derivative in the PDE is approximated using a difference approximation that is typically derived using Taylor series expansions. To illustrate the method, let us consider solution of the Poisson equation [Eq. (1.2)] in the computational domain shown in Fig. 1.4. Let us also assume that the value of the dependent variable, $\phi$, is prescribed at the boundary, and is known at all boundary nodes.

The objective is to determine the value of $\phi$ at all nodes. The total number of nodes is denoted by $N$. This implies that for the specific nodal arrangement shown in Fig. 1.4, $N=16$, and $\phi_{10}$ through $\phi_{16}$ needs to be determined. In order to determine these seven unknowns, seven equations are needed. In the finite difference method, these seven equations are formulated by satisfying the governing equation at the nodes 10-16, yielding
$$\left.\frac{\partial^2 \phi}{\partial x^2}\right|_i+\left.\frac{\partial^2 \phi}{\partial y^2}\right|_i=S_i \quad \forall i=10,11, \ldots, 16$$

Next, the second derivatives are approximated using the nodal values of $\phi$. For example, one may write (approximate) the second derivatives with respect to $x$ and $y$, respectively, at node 16 as
\begin{aligned} &\left.\frac{\partial^2 \phi}{\partial x^2}\right|{16} \approx \sum{k=1}^N A_{16, k}^x \phi_k, \ &\left.\frac{\partial^2 \phi}{\partial y^2}\right|{16} \approx \sum{k=1}^N A_{16, k}^y \phi_k, \end{aligned}
where $A_{16, k}^x$ and $A_{16, k}^y$ represent coefficients (or weights) for node 16 that expresses the two second derivatives as a linear combination of the nodal values of $\phi$. These coefficients may he derived in a variety of ways: nsing Taylor series expansions, interpolation functions, and splines, among others. Generally, the nodes considered in the summation shown in Eq. (1.11) include a small subset of the total number of nodes – those being the immediate neighbors and the node in question itself. Thus, for node 16 , only nodes $10-16$ may be used in the summations shown in $\mathrm{F}{\mathrm{q}}$. (1.11), implying that the coefficients wonld he zero for $k=1,2, \ldots, 10$. If the radins of influence is extended further and the neighbors of the neighbors are used, a more accurate approximation – so-called higher-order approximation – may be possible to derive. The exact mathematical procedure to derive the coefficients in the finite difference method, along with the errors incurred in the approximations, is described in Chapter 2. Upon substitution of Eq. (1.11) into Eq. (1.10) for node 16, we obtain $$\sum{k=1}^N A_{16, k}^x \phi_k+\sum_{k=1}^N A_{16, k}^y \phi_k=\sum_{k=1}^N A_{16, k} \phi_k=S_{16},$$
where $A_{16, k}=A_{16, k}^x+A_{16, k}^y$. Equation (1.12) has $N=16$ terms on the left-hand side, of which, depending on the radius of influence of each node, many are zeroes.

## 数学代写|偏微分方程代写partial difference equations代考|FINITE VOLUME METHOD

The finite volume method derives its name from the fact that in this method the governing PDE is satisfied over finite-sized control volumes, rather than at points. The first step in this method is to split the computational domain into a set of control volumes known as cells, as shown in Fig. 1.5. In general, these cells may be of arbitrary shape and size, although, traditionally, the cells are convex polygons (in 2D) or polyhedrons (in 3D), i.e., they are bounded by straight edges (in 2D) or planar surfaces (in 3D). As a result, if the bounding surface is curved, it is approximated by straight edges or planar faces, as is evident in Fig. 1.5. These bounding discrete surfaces are known as cell faces or simply, faces. The vertices of the cells, on the other hand, are called nodes, and are, in fact, the same nodes that were used in the finite difference method. All information is stored at the geometric centroids of the cells, referred to as cell centers.

The derivation of the finite volume equations commences by integrating the governing PDE over the cells constituting the computational domain. In the case of the Poisson equation, this yields
$$\int_{V_i} \nabla^2 \phi d V=\int_{V_i} S_\phi d V$$
where $V_i$ is the volume of the $i$-th cell. The volume integral on the left-hand side of Eq. (1.14) can be simplified by writing the Laplacian, as $\nabla^2 \phi=\nabla \cdot(\nabla \phi)$, and by applying the Gauss divergence theorem, to yield
$$\int_{S_i}(\nabla \phi) \cdot \hat{\mathbf{n}} d A=\int_{V_i} S_\phi d V$$
where $S_i$ is the surface area of the surface bounding the cell $i$, and $d A$ is a differential area on the surface with outward pointing unit surface normal $\hat{\mathbf{n}}$. For details on the Gauss divergence theorem, and its application to the finite volume procedure, the interested reader is referred to Chapter 7. The right-hand side of Eq. (1.15) can be simplified by applying the mean value theorem and by assuming that the mean value of $S_\phi$ over the volume $V_i$ is the same as the value of $S_\phi$ evaluated at the centroid of the cell $i$. This simplification yields
$$\int_{S_i}(\nabla \phi) \cdot \hat{\mathbf{n}} d A=S_i V_i$$

## 数学代写|偏微分方程代写partial difference equations代考|FINITE DIFFERENCE METHOD

$$\left.\frac{\partial^2 \phi}{\partial x^2}\right|i+\left.\frac{\partial^2 \phi}{\partial y^2}\right|_i=S_i \quad \forall i=10,11, \ldots, 16$$ 接下来，二阶导数使用节点值来近似 $\phi$. 例如，可以写出 (近似) 关于的二阶导数 $x$ 和 $y$ ，分别在节点 16 为 $$\frac{\partial^2 \phi}{\partial x^2}\left|16 \approx \sum=1^N A{16, k}^x \phi_k, \quad \frac{\partial^2 \phi}{\partial y^2}\right| 16 \approx \sum k=1^N A_{16, k}^y \phi_k,$$

(1.11) 包括节点总数的一小部分一一那些是直接邻居和有问题的节点本身。因此，对于节点 16 ，只有节点 $10-16$ 可用于所示的总和 $\mathrm{Fq} .(1.11)$ ，这意味着系数不会为零 $k=1,2, \ldots, 10$. 如果进一步扩大影响范围并使 用邻居的邻居，则可以推导出更准确的近似值一一所谓的高阶近似值。在第 2 章中描述了在有限差分法中推导系数 的精确数学过程，以及近似值中产生的误差。(1.11) 进入等式。(1.10) 对于节点 16，我们得到
$$\sum k=1^N A_{16, k}^x \phi_k+\sum_{k=1}^N A_{16, k}^y \phi_k=\sum_{k=1}^N A_{16, k} \phi_k=S_{16},$$

## 数学代写|偏微分方程代写partial difference equations代考|FINITE VOLUME METHOD

$$\int_{V_i} \nabla^2 \phi d V=\int_{V_i} S_\phi d V$$

$$\int_{S_i}(\nabla \phi) \cdot \hat{\mathbf{n}} d A=\int_{V_i} S_\phi d V$$

$$\int_{S_i}(\nabla \phi) \cdot \hat{\mathbf{n}} d A=S_i V_i$$

## 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|CLASSIFICATION OF PDEs

As discussed in the preceding section, at the heart of analysis that utilizes fundamental physics-based principles are differential equations. In the most general case, when the behavior of the system or device is sought as a function of both time and space, the governing equations are PDEs. The solution of a PDE – either by analytical or numerical means – is generally quite complex, and requires a deep understanding of the key attributes of the PDE. These attributes dictate the basic method of solution, how and where boundary and initial conditions must be applied, and what the general nature of the solution is expected to be. Therefore, in this section, we classify PDEs into broad canonical types, and also apply this classification to PDEs commonly encountered in engineering analysis.

We begin our classification by considering a PDE of the following general form:
$$A \frac{\partial^2 \phi}{\partial x^2}+B \frac{\partial^2 \phi}{\partial x \partial y}+C \frac{\partial^2 \phi}{\partial y^2}+\ldots=0,$$
where $x$ and $y$ are so-called independent variables, while $\phi$ is the dependent variable. The coefficients $A, B$, and $C$ are either real constants or real functions of the independent or dependent variables. If any of the three coefficients is a function of the dependent variable $\phi$, the PDE becomes nonlinear. Otherwise, it is linear. It is worth pointing out that the distinction between linear and nonlinear PDEs is not related to whether the solution to the PDE is a linear or nonlinear function of $x$ and $y$. As a matter of fact, most linear PDEs yield nonlinear solutions! What matters is whether the PDE has any nonlinearity in the dependent variable $\phi$.

Eq. (1.1) is a PDE in two independent variables. In general, of course, PDEs can be in more than two independent variables, and such scenarios will be discussed in due course. For now, we will restrict ourselves to the bare minimum number of independent variables required to deem a differential equation a PDE. Depending on the values of the coefficients $A, B$, and $C$, PDEs are classified as follows:
If $B^2-4 A C<0$, then the PDE is elliptic. If $B^2-4 A C=0$, then the PDE is parabolic. If $B^2-4 A C>0$, then the PDE is hyperbolic.
Next, we examine a variety of PDEs, commonly encountered in science and engineering disciplines, and use the preceding criteria to identify its type. We begin with the steady-state diffusion equation, written in two independent variables as [2]
$$\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=S_\phi,$$
where $S_\phi$ is the so-called source or source term that, in general, could be a function of either the dependent variables or the independent variables, or both, i.e., $S_\phi=S_\phi(x, y, \phi)$. If the source term is equal to zero, Eq. (1.2) is the so-called Laplace equation. If the source term is a function of the independent variables only, or a constant, i.e., $S_\phi=S_\phi(x, y)$, Eq. (1.2) reduces to the so-called Poisson equation. If the source term is a linear function of the dependent variable, i.e., $S_\phi=a \phi+b$, Eq. (1.2) is referred to as the Helmholtz Equation. The term “diffusion equation” stems from the fact that the differential operators shown in Eq. (1.2) usually arise out of modeling diffusion-like processes such as heat conduction, current conduction, molecular mass diffusion, and other similar phenomena, as is discussed in more detail in Chapters 6 and 7. Irrespective of the aforementioned three forms assumed by Eq. (1.2), comparing it with Eq. (1.1) yields $A=C=1$, and $B=0$, resulting in $B^2-4 A C<0$. Thus, the steady-state diffusion equation, which includes equations of the Laplace, Poisson, or Helmholtz type, is an elliptic PDE. An important characteristic of elliptic PDEs is that they require specification of boundary conditions on all surfaces that bound the domain of solution.

## 数学代写|偏微分方程代写partial difference equations代考|OVERVIEW OF METHODS FOR SOLVING PDEs

The solution of PDEs is quite challenging. The number of methods available to find closed-form analytical solutions to canonical PDEs is limited. These include separation of variables, superposition, product solution methods, Fourier transforms, Laplace transforms, and perturbation methods, among a few others. Even these methods are limited by constraints such as regular geometry, linearity of the equation, constant coefficients, and others. The imposition of these constraints severely curtails the range of applicability of analytical techniques for solving PDEs, rendering them almost irrelevant for problems of practical interest. In realization of this fact, applied mathematicians and scientists have endeavored to build machines that can solve differential equations by numerical means, as outlined in the brief history of computing presented at the beginning of this chapter.

The methods for numerical solution to PDEs can broadly be classified into two types: deterministic and stochastic. A deterministic method is one in which, for a given input to an equation, the output is always the same. The output does not depend on how many times one solves the equation, at what time of the day it is solved, or what computer it is solved on (disregarding precision errors, that may be slightly different on different computers). On the other hand, a stochastic method is based on statistical principles, and the output can be slightly different for the same input depending on how many times the calculation is performed, and other factors. In this case, by “slightly different,” we mean within the statistical error bounds. The difference between these two approaches is best elucidated by a simple example. Let us consider a scenario in which a ball is released from a certain height, $h_i$, above the horizontal ground. Upon collision with the ground, the ball bounces back to a height $h_o$. Let us assume that based on experimental observations or other physical laws, we know that the ball always bounces back to half the height from which it is released. Following this information, we may construct the following deterministic equation: $h_0=(1 / 2) h_i$. If this equation is used to compute the height of a bounced ball, it would always be one-half of the height of release. In other words, the equation (or method of calculation) has one hundred percent confidence built into it. Hence, it is termed a deterministic method. The stochastic viewpoint of the same problem would be quite different. In this viewpoint, one would argue that if $n$ balls were made to bounce, by the laws of theoretical probability, $n / 2$ balls would bounce to a height slightly above half the released height, and the remaining $n / 2$ balls would bounce to a height slightly below the released height, such that in the end, when tallied, the mean height to which the balls bounce back to would be exactly half the height of release. Whether this exact result is recovered or not would depend on how many balls are bounced, i.e., the number of statistical samples used.

## 数学代写|偏微分方程代写partial difference equations代考|CLASSIFICATION OF PDEs

$$A \frac{\partial^2 \phi}{\partial x^2}+B \frac{\partial^2 \phi}{\partial x \partial y}+C \frac{\partial^2 \phi}{\partial y^2}+\ldots=0$$

$$\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=S_\phi,$$

## 数学代写|偏微分方程代写partial difference equations代考|OVERVIEW OF METHODS FOR SOLVING PDEs

PDE 的求解非常具有挑战性。可用于找到典型 PDE 的封闭式解析解的方法数量有限。其中包括变量分离、叠加、乘积求解方法、傅里叶变换、拉普拉斯变换和微扰方法等。即使这些方法也受到诸如规则几何、方程线性、常数系数等约束的限制。这些约束的强加严重限制了分析技术在求解偏微分方程中的适用范围，使得它们几乎与实际感兴趣的问题无关。为了实现这一事实，应用数学家和科学家们努力制造可以通过数值方法求解微分方程的机器，

## 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。

## 数学代写|偏微分方程代写partial difference equations代考|Math442

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|Introduction to Numerical Methods for Solving Differential Equations

The immense power of mathematics is, arguably, best divulged by “crunching” numbers. While an equation or a formula can provide significant insight into a physical phenomenon, its depth, as written on paper, can only be appreciated by a limited few – ones that already have a fairly rigorous understanding of the phenomenon to begin with. The same equation or formula, however, when put to use to generate numbers, reveals significantly more. For example, the Navier-Stokes equations, which govern fluid flow, are not particularly appealing on paper except, perhaps, to a select few. However, their solution, when appropriately postprocessed and depicted in the form of line plots, field plots, and animations, can be eye-opening even to a middle-school student! In realization of the fact that the numbers generated out of sophisticated equations are far more revealing than the equations themselves, for more than a century, applied mathematicians have endeavored to find ways to rapidly generate numbers from equations. The desire to generate numbers has also been partly prompted by the fact that closed-form analytical solutions exist only for a limited few scenarios, and even those require number crunching or computing to some degree.
Although the history of computing can be traced back to Babylon, where the abacus was believed to have been invented around $2400 \mathrm{BC}$, it was not until the nineteenth century that the development of devices that could, according to the modern sense of the word, compute, came to be realized. While the industrial revolution created machines that made our everyday life easier, the nineteenth and twentieth century witnessed strong interest among mathematicians and scientists in building a machine that could crunch numbers or compute repeatedly and rapidly. The so-called Analytical Engine, proposed by Charles Babbage around 1835, is believed to be the first computer design capable of logic-based computing. Unfortunately, it was never built due to political and economic turn of events. In 1872, Sir William Thomson built an analog tide-predicting machine that could integrate differential equations. The Russian naval architect and mathematician, Alexei Krylov (1863-1945), also built a machine capable of integrating an ordinary differential equation in 1904. These early analog machines were based on mechanical principles and built using mechanical parts. As a result, they were slow. The Second World War stimulated renewed interest in computing both on the German and British sides. The Zuse Z3, designed by be the world’s first programmable electromechanical computer. It was also around this time that the British cryptanalyst Alan Turing, known as the father of computer science and artificial intelligence, and brought to the limelight recently by The Imitation Game, built an electromechanical machine to decode the Enigma machine that was being used by the German military for their internal communication. Shortly after the war, Turing laid the theoretical foundation for the modern stored-program programmable computer – a machine that does not require any rewiring to execute a different set of instructions. This so-called Turing Machine later became the theoretical standard for computer design, and modern computer designs, upon satisfying a set of mandatory design requirements, are referred to as “Turing complete.”

## 数学代写|偏微分方程代写partial difference equations代考|ROLE OF ANALYSIS

During the industrial revolution, and for decades afterwards, the need for analysis of man-made devices was not critical. The so-called factor of safety in-built into the design of most devices was so large that they rarely failed. Most devices were judged by their ability or inability to perform a certain task, not necessarily by how efficiently the task was performed. One marveled at an automobile because of its ability to transport passengers from point A to point B. Metrics, such as miles per gallon, was not even remotely in the picture. With an exponential rise in the world’s population, and dwindling natural resources, building efficient devices and conserving natural resources is now a critical need rather than a luxury. Improvement in the efficiency requires understanding of the functioning of a device or system at a deeper level.
Analysis refers to the use of certain physical and mathematical principles to establish the relationship between cause and effect as applied to the device or system in question. The causal relationship, often referred to as the mathematical model, may be in the form of a simple explicit equation or in the form of a complex set of partial differential equations (PDEs). It may be based on empirical correlations or fundamental laws such as the conservation of mass, momentum, energy or other relevant quantities. Irrespective of whether the mathematical model is fundamental physics based, or empirical, it enables us to ask “what if?” questions. What if the blade angle was changed by 2 degrees? What if the blade speed was altered by $2 \%$ ? Such analysis enables us to explore the entire design space in a relatively short period of time and, hopefully, with little use of resources. It also helps eliminate designs that are not promising from a scientific standpoint, thereby narrowing down the potential designs that warrant further experimental study.

## 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。