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# 数学代写|偏微分方程代考Partial Differential Equations代写|Math442

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Helmholtz’s Second Theorem

If $\Psi(\mathbf{r})$ is a solution of the space form of the wave equation whose partial derivatives of the first and second orders are continuous outside the volume $V$ and on the closed surface $S$ bounding $V$, if $r \Psi(\mathbf{r})$ is bounded, and if
$$r\left(\frac{\partial \Psi}{\partial r}-i k \Psi\right) \rightarrow 0$$
uniformly with respect to the angle variables as $r \rightarrow \infty$, then
\begin{aligned} \frac{1}{4 \pi} \int_S\left{\Psi\left(\mathbf{r}^{\prime}\right) \frac{\partial}{\partial n}\left(\frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\right)-\frac{\partial \Psi\left(\mathbf{r}^{\prime}\right)}{\partial n} \frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\right} d S^{\prime} \ =\left{\begin{aligned} \Psi(\mathbf{r}) & \text { if } \mathbf{r} \notin V \ 0 & \text { if } \mathbf{r} \in V \end{aligned}\right. \end{aligned}
where $\mathbf{n}$ is the outward normal to $S$.
It would appear from Helmholtz’s formulas that the values taken by $\Psi$ and $\partial \Psi / \partial n$ on the surface $S$ can be assigned arbitrarily and independently of each other. By use of a Green’s function $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ with singularity at $P$ (see Sec. 7 below) we can express $\Psi(\mathbf{r})$ in terms of $\Psi\left(\mathbf{r}^{\prime}\right)$ alone through the equation
$$\Psi(\mathbf{r})=-\frac{1}{4 \pi} \int_S \Psi\left(\mathbf{r}^{\prime}\right) \frac{\partial G}{\partial n} d S^{\prime}$$
so that knowing the value of $\Psi$ on the surface $S$, we can, in general, determine $\Psi(\mathbf{r})$ uniquely and, in particular, calculate the value of $\partial \Psi / \partial n$ on $S$. It can also be shown that if $\partial \Psi / \partial n$ is prescribed on $S$, $\Psi(\mathbf{r})$ is in general determined uniquely so that its value on $S$ can be determined. The values of $\Psi$ and $\partial \Psi / \partial n$ on $S$ are therefore related. If the functions $f(\mathbf{r})$ and $g(\mathbf{r})$ are defined in an arbitrary way, then the function
satisfies the space form of the wave equation, but it does not necessarily follow that $\Psi\left(\mathbf{r}^{\prime}\right)=g\left(\mathbf{r}^{\prime}\right), \partial \Psi / \partial n=f\left(\mathbf{r}^{\prime}\right)$ on $S$.

## 数学代写|偏微分方程代考Partial Differential Equations代写|Weber’s Theorem

Weber’s Theorem. If $\Psi(\rho)$ is a solution of the space form of the two-dimensional wave equation $\nabla_1^2 \Psi+k^2 \Psi=0$ whose partial derivatives of the first and second orders are continuous within the area $S$ and on the closed curve $\Gamma$ bounding $S$, then
\begin{aligned} \frac{1}{4 i} \int_{\Gamma}\left{\Psi\left(\rho^{\prime}\right) \frac{\partial}{\partial n} H_0^{(1)}\left(k\left|\rho-\rho^{\prime}\right|\right)-H_0^{(1)}\left(k\left|\rho-\rho^{\prime}\right|\right)\right. & \left.\frac{\partial \Psi\left(\rho^{\prime}\right)}{\partial n}\right} d s^{\prime} \ & = \begin{cases}\Psi(\rho) & \text { if } \rho \in S \ 0 & \text { if } \rho \notin S\end{cases} \end{aligned}
where $\mathbf{n}$ is the outward normal to $\Gamma$. The proof is left as an exercise to the reader. ${ }^1$

Helmholtz’s first theorem can be expressed in another way by introducing the idea of a retarded value. If $\psi\left(\mathbf{r}^{\prime}, t\right)$ is a function of the coordinates of a variable point with position vector $\mathbf{r}^{\prime}$, then we define the retarded value $[\psi]$ of $\psi$ by the equation
$$[\psi]=\psi\left(\mathbf{r}^{\prime}, t-\frac{\lambda}{c}\right), \quad \lambda=\left|\mathbf{r}^{\prime}-\mathbf{r}\right|$$
where $\mathbf{r}$ is the position vector of some fixed point. If
$$\psi\left(\mathbf{r}^{\prime}, t\right)=\Psi\left(\mathbf{r}^{\prime}\right) e^{-i k c t}$$
then it is obvious that
$$[\psi]=\psi\left(\mathbf{r}^{\prime}, t\right) e^{i k \lambda}, \quad\left[\frac{\partial \psi}{\partial t}\right]=-i k c[\psi]$$
If, now, we multiply both sides of the equation which occurs in Helmholtz’s first theorem by $e^{i k c t}$, we find that if the point with position vector $\mathbf{r}$ is inside the surface $S$, then that equation can be written in the form
$$\psi(\mathbf{r}, t)=\frac{1}{4 \pi} \int_S\left{-[\psi] \frac{\partial \lambda}{\partial n}\left(\frac{i k}{\lambda}+\frac{d}{d \lambda}\left(\frac{1}{\lambda}\right)\right}+\frac{1}{\lambda}\left[\frac{\partial \psi}{\partial n}\right]\right} d S^{\prime}$$
which, because of the second of equations (6), can be written in the form
$$\psi(\mathbf{r}, t)=\frac{1}{4 \pi} \int_S\left{-[\psi] \frac{\partial}{\partial n}\left(\frac{1}{\lambda}\right)+\frac{1}{c \lambda} \frac{\partial \lambda}{\partial n}\left[\frac{\partial \psi}{\partial t}\right]+\frac{1}{\lambda}\left[\frac{\partial \psi}{\partial n}\right]\right} d S^{\prime}$$

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Helmholtz’s Second Theorem

$$r\left(\frac{\partial \Psi}{\partial r}-i k \Psi\right) \rightarrow 0$$

\begin{aligned} \frac{1}{4 \pi} \int_S\left{\Psi\left(\mathbf{r}^{\prime}\right) \frac{\partial}{\partial n}\left(\frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\right)-\frac{\partial \Psi\left(\mathbf{r}^{\prime}\right)}{\partial n} \frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\right} d S^{\prime} \ =\left{\begin{aligned} \Psi(\mathbf{r}) & \text { if } \mathbf{r} \notin V \ 0 & \text { if } \mathbf{r} \in V \end{aligned}\right. \end{aligned}

$$\Psi(\mathbf{r})=-\frac{1}{4 \pi} \int_S \Psi\left(\mathbf{r}^{\prime}\right) \frac{\partial G}{\partial n} d S^{\prime}$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Weber’s Theorem

\begin{aligned} \frac{1}{4 i} \int_{\Gamma}\left{\Psi\left(\rho^{\prime}\right) \frac{\partial}{\partial n} H_0^{(1)}\left(k\left|\rho-\rho^{\prime}\right|\right)-H_0^{(1)}\left(k\left|\rho-\rho^{\prime}\right|\right)\right. & \left.\frac{\partial \Psi\left(\rho^{\prime}\right)}{\partial n}\right} d s^{\prime} \ & = \begin{cases}\Psi(\rho) & \text { if } \rho \in S \ 0 & \text { if } \rho \notin S\end{cases} \end{aligned}

$$[\psi]=\psi\left(\mathbf{r}^{\prime}, t-\frac{\lambda}{c}\right), \quad \lambda=\left|\mathbf{r}^{\prime}-\mathbf{r}\right|$$

$$\psi\left(\mathbf{r}^{\prime}, t\right)=\Psi\left(\mathbf{r}^{\prime}\right) e^{-i k c t}$$

$$[\psi]=\psi\left(\mathbf{r}^{\prime}, t\right) e^{i k \lambda}, \quad\left[\frac{\partial \psi}{\partial t}\right]=-i k c[\psi]$$

$$\psi(\mathbf{r}, t)=\frac{1}{4 \pi} \int_S\left{-[\psi] \frac{\partial \lambda}{\partial n}\left(\frac{i k}{\lambda}+\frac{d}{d \lambda}\left(\frac{1}{\lambda}\right)\right}+\frac{1}{\lambda}\left[\frac{\partial \psi}{\partial n}\right]\right} d S^{\prime}$$

$$\psi(\mathbf{r}, t)=\frac{1}{4 \pi} \int_S\left{-[\psi] \frac{\partial}{\partial n}\left(\frac{1}{\lambda}\right)+\frac{1}{c \lambda} \frac{\partial \lambda}{\partial n}\left[\frac{\partial \psi}{\partial t}\right]+\frac{1}{\lambda}\left[\frac{\partial \psi}{\partial n}\right]\right} d S^{\prime}$$

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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。