## 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。