标签： AMATH353

数学代写|偏微分方程代写partial difference equations代考|Problems with Axial Symmetry

Posted on 2023年6月17日2023年6月17日 by statistics-lab

如果你也在怎样代写偏微分方程Partial Differential Equations 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。偏微分方程Partial Differential Equations在数学中，偏微分方程（PDE）是一个方程，它规定了一个多变量函数的各种偏导数之间的关系。常微分方程构成了偏微分方程的一个子类，对应于单变量函数。截至2020年，随机偏微分方程和非局部方程是 “PDE “概念的特别广泛研究的延伸。更为经典的课题包括椭圆和抛物线偏微分方程、流体力学、玻尔兹曼方程和色散偏微分方程，目前仍有很多积极的研究。

偏微分方程Partial Differential Equations在以数学为导向的科学领域，如物理学和工程学中无处不在。例如，它们是现代科学对声音、热量、扩散、静电、电动力学、热力学、流体动力学、弹性、广义相对论和量子力学（薛定谔方程、保利方程等）的基础性认识。它们也产生于许多纯粹的数学考虑，如微分几何和变分计算；在其他值得注意的应用中，它们是几何拓扑学中证明庞加莱猜想的基本工具。部分由于这种来源的多样性，存在着广泛的不同类型的偏微分方程，并且已经开发了处理许多出现的个别方程的方法。因此，人们通常认为，偏微分方程没有 “一般理论”，专业知识在一定程度上被划分为几个基本不同的子领域。

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。

数学代写|偏微分方程代写partial difference equations代考|Problems with Axial Symmetry

The determination of a potential function $\psi$ for a system which has axial symmetry can often be considerably simplified by making use of the fact that it is sometimes a simple matter to write down the form of $\psi$ for points on the axis of symmetry. It is best in such cases to use spherical polar coordinates $r, \theta, \phi$ and to take the axis of symmetry to be the polar axis $\theta=0$. Suppose that we wish to determine the potential function $(r, \theta, \phi)$ corresponding to a given distribution of sources (such as masses, charges, etc.) and that we have been able to calculate its value $\psi(z, 0,0)$ at a point on the axis of symmetry. If we expand $\psi(z, 0,0)$ in the Laurent series
$$
\psi(z, 0,0)=\sum_{n=0}^{\infty}\left(A_n z^n+\frac{B_n}{z^{n+1}}\right)
$$
then it is readily shown that the required potential function is
$$
\psi(r, \theta, \phi)=\sum_{n=0}^{\infty}\left(A_n r^n+\frac{B_n}{r^{n+1}}\right) P_n(\cos \theta)
$$
for
(i) $\nabla^2 \psi=0$;
(ii) $\psi(r, \theta, \phi)$ takes the value (1) on the axis of symmetry, since there $P_n(\cos \theta)=1, r=z$
(iii) $\psi(r, \theta, \phi)$ is symmetrical about $O z$ as required.
The simplest example of the use of this method is the determination of the potential due to a uniform circular wire of radius $a$ charged with electricity of line density $e$. At a point on the axis of the wire it is readily seen that
$$
\psi(z, 0,0)=\frac{2 \pi e a}{\sqrt{a^2+z^2}}
$$
so that $\quad \psi(z, 0,0)=\left{\begin{array}{l}2 \pi e \sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)_n}{n !}\left(-\frac{z^2}{a^2}\right)^n \quad za\end{array}\right.$
where we have used the notation $(a)_n=a(a+1) \cdots(a+n-1)$.

数学代写|偏微分方程代写partial difference equations代考|Kelvin’s Inversion Theorem

It is a well-known result in the elementary theory of electrostatics that the solution of certain problems may be derived from that of simpler problems by means of a transformation of three-dimensional space known as inversion in a sphere. The points $P, \Pi$ with position vectors $\mathbf{r}, \rho$, respectively, are said to be inverse in a sphere $S$ of center with position vector $\mathbf{c}$ and radius $a$ if the points $P, \Pi, C$ are collinear and if $a$ is the mean proportional between the distances $C P$, $C \Pi$. We must therefore have
$$
\begin{array}{r}
\lambda \mathbf{c}+\mu \mathbf{r}+\nu \rho=\mathbf{0} \
\lambda+\mu+\nu=1
\end{array}
$$
and
$$
a^2=r \rho
$$

This transformation has the property that it carries planes or spheres into planes or spheres and carries a sphere $S^{\prime}$ into itself if and only if $S^{\prime}$ is orthogonal to $S$.

We now consider the effect of such a transformation on a harmonic function. If we write $\rho=(\xi, \eta, \zeta), \mathbf{r}=(x, y, z)$, so that
$$
\xi=\frac{a^2 x}{r^2}, \quad \eta=\frac{a^2 y}{r^2}, \quad \zeta=\frac{a^2 z}{r^2}
$$
then by the well-known rule ${ }^1$ for the transformation of the Laplacian operator it follows that
$$
\nabla^2 \psi=\frac{r^6}{a^6}\left[\frac{\partial}{\partial x}\left(\frac{a^2}{r^2} \frac{\partial \psi}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{a^2}{r^2} \frac{\partial \psi}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{a^2}{r^2} \frac{\partial \psi}{\partial z}\right)\right]
$$

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|Problems with Axial Symmetry

利用这样一个事实，有时写下对称轴上点的$\psi$的形式是一件很简单的事情，确定具有轴对称的系统的势函数$\psi$通常可以大大简化。在这种情况下，最好使用球面极坐标$r, \theta, \phi$，并将对称轴作为极轴$\theta=0$。假设我们希望确定对应于给定源(如质量、电荷等)分布的势函数$(r, \theta, \phi)$，并且我们已经能够计算其在对称轴上一点的值$\psi(z, 0,0)$。如果把$\psi(z, 0,0)$展开到洛朗级数中
$$
\psi(z, 0,0)=\sum_{n=0}^{\infty}\left(A_n z^n+\frac{B_n}{z^{n+1}}\right)
$$
那么很容易证明所需的势函数为
$$
\psi(r, \theta, \phi)=\sum_{n=0}^{\infty}\left(A_n r^n+\frac{B_n}{r^{n+1}}\right) P_n(\cos \theta)
$$
为了
(i) $\nabla^2 \psi=0$;
(ii) $\psi(r, \theta, \phi)$取对称轴上的值(1)，因为那里有$P_n(\cos \theta)=1, r=z$
(iii) $\psi(r, \theta, \phi)$按要求与$O z$对称。
使用这种方法的最简单的例子是测定由于半径为$a$的均匀圆形导线带线密度为$e$的电而产生的电势。在导线轴线上的某一点很容易看出
$$
\psi(z, 0,0)=\frac{2 \pi e a}{\sqrt{a^2+z^2}}
$$
这就是$\quad \psi(z, 0,0)=\left{\begin{array}{l}2 \pi e \sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)_n}{n !}\left(-\frac{z^2}{a^2}\right)^n \quad za\end{array}\right.$
这里我们用了$(a)_n=a(a+1) \cdots(a+n-1)$符号。

数学代写|偏微分方程代写partial difference equations代考|Kelvin’s Inversion Theorem

在静电学的基本理论中，有一个众所周知的结果，即某些问题的解可以由一些更简单的问题的解推导出来，方法是对三维空间进行变换，称为球面反演。如果点$P, \Pi, C$共线，并且$a$是距离$C P$和$C \Pi$之间的平均比例，则分别具有位置矢量$\mathbf{r}, \rho$的点$P, \Pi$在中心具有位置矢量$\mathbf{c}$和半径$a$的球体$S$中被称为逆。因此，我们必须
$$
\begin{array}{r}
\lambda \mathbf{c}+\mu \mathbf{r}+\nu \rho=\mathbf{0} \
\lambda+\mu+\nu=1
\end{array}
$$
和
$$
a^2=r \rho
$$

这个变换的性质是它将平面或球体带入平面或球体并将球体$S^{\prime}$带入自身当且仅当$S^{\prime}$与$S$正交。

现在我们考虑这种变换对调和函数的影响。如果我们写$\rho=(\xi, \eta, \zeta), \mathbf{r}=(x, y, z)$，那么
$$
\xi=\frac{a^2 x}{r^2}, \quad \eta=\frac{a^2 y}{r^2}, \quad \zeta=\frac{a^2 z}{r^2}
$$
然后根据拉普拉斯算子变换的著名规则${ }^1$，我们可以得出
$$
\nabla^2 \psi=\frac{r^6}{a^6}\left[\frac{\partial}{\partial x}\left(\frac{a^2}{r^2} \frac{\partial \psi}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{a^2}{r^2} \frac{\partial \psi}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{a^2}{r^2} \frac{\partial \psi}{\partial z}\right)\right]
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|Characteristic Curves of Second-order Equations

Posted on 2023年6月17日2023年6月17日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|Characteristic Curves of Second-order Equations

We shall now consider briefly the Cauchy problem for the secondorder partial differential equation
$$
R r+S s+T t+f(x, y, z, p, q)=0
$$
in which $R, S$, and $T$ are functions of $x$ and $y$ only. In other words, we wish to consider the problem of determining the solution of equation (1) such that on a given space curve $\Gamma$ it takes on prescribed values of $z$ and $\partial z / \partial n$, where $n$ is distance measured along the normal to the curve. This latter set of boundary conditions is equivalent to assuming that the values of $x, y, z, p, q$ are determined on the curve, but it should be noted that the values of the partial derivatives $p$ and $q$ cannot be assigned arbitrarily along the curve. For if we take the freedom equations of the curve $\Gamma$ to be
$$
x=x_0(\tau), \quad y=y_0(\tau), \quad z=z_0(\tau)
$$
then we must have at all points of $\Gamma$ the relation
$$
\dot{z}_0=p_0 \dot{x}_0+q_0 \dot{y}_0
$$
(where $\dot{z}_0$ denotes $d z_0 / d t$, etc.), showing that $p_0$ and $q_0$ are not independent. The Cauchy problem is therefore that of finding the solution of equation (1) passing through the integral strip of the first order formed by the planar elements $\left(x_0, y_0, z_0, p_0, q_0\right)$ of the curve $\Gamma$.

At every point of the integral strip $p_0=p_0(\tau), q_0=q_0(\tau)$, so that if we differentiate these equations with respect to $\tau$, we obtain the relations
$$
\dot{p}_0=r \dot{x}_0+s \dot{y}_0, \quad \dot{q}_0=s \dot{x}_0+t \dot{y}_0
$$
If we solve the three equations (1) and (4) for $r, s, t$, we find that
where
$$
\frac{r}{\Delta_1}=\frac{-s}{\Delta_2}=\frac{t}{\Delta_3}=\frac{-1}{\Delta}
$$
$$
\Delta_1=\left|\begin{array}{ccc}
S & T & f \
\dot{y}_0 & 0 & -\dot{p}_0 \
\dot{x}_0 & \dot{y}_0 & -\dot{q}_0
\end{array}\right| \text {, etc. } \quad \text { and } \quad \Delta=\left|\begin{array}{ccc}
R & S & T \
\dot{x}_0 & \dot{y}_0 & 0 \
0 & \dot{x}_0 & \dot{y}_0
\end{array}\right|
$$
If $\Delta \neq 0$, we can therefore easily calculate the expressions for the secondorder derivatives $r_0, s_0$, and $t_0$ along the curve $\Gamma$.

数学代写|偏微分方程代写partial difference equations代考|Characteristics of Equations in Three Variables

The concept of the characteristic curves of a second-order linear differential equation which was developed in the last section for equations in two independent variables may readily be extended to the case where there are $n$ independent variables. In this section we shall show how the analysis may be extended in the case $n=3$. The general result proceeds along similar lines, but the geometrical concepts are more easily visualized in the case we shall consider.

We suppose that we have three independent variables $x_1, x_2, x_3$ and one dependent variable $u$, and we write $p_{i j}$ for $\partial^2 u / \partial x_i \partial x_j, p_i$ for $\partial u / \partial x_i$. The problem we consider is that of finding a solution of the linear equation
$$
\mathrm{L}(u)=\sum_{i, j-1}^3 a_{i j} p_{i j}+\sum_{i=1}^3 b_i p_i+c u=0
$$
for which $u$ and $\partial u / \partial n$ take on prescribed values on the surface $S$ whose equation is
$$
f\left(x_1, x_2, x_3\right)=0
$$
If we suppose that the freedom equations of $S$ are
$$
x_i=\bar{x}_i\left(\tau_1, \tau_2\right) \quad i=1,2,3
$$

then we may write the boundary conditions in the form
$$
\bar{u}=F\left(\tau_1, \tau_2\right), \quad \overline{\partial u / \partial n}=G\left(\tau_1, \tau_2\right)
$$
the bar denoting that these are the values assumed by the relevant quantity on the surface $S$.
From equation (2) we have the identity
$$
\sum_{i=1}^3 \frac{\partial f}{\partial x_i}\left(\frac{\partial x_i}{\partial \tau_1} d \tau_1+\frac{\partial x_i}{\partial \tau_2} d \tau_2\right)=0
$$
so that equating to zero the coefficients of $d \tau_1$ and $d \tau_2$, we have
$$
\sum_{i=1}^3 \delta_i P_{i j}=0 \quad j=1,2
$$
where $\delta_i \equiv \partial f / \partial x_i, P_{i j}=\partial x_i / \partial \tau_j$. Solving these equations, we find that
$$
\frac{\delta_1}{\Delta_1}=\frac{\delta_2}{\Delta_2}=\frac{\delta_3}{\Delta_3}=\rho, \text { say }
$$
where $\Delta_1$ denotes the Jacobian $\partial\left(x_2, x_3\right) / \partial\left(\tau_1, \tau_2\right)$ and the others are defined similarly.

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|Compatible Systeṃs of First-order Equations

接下来我们将考虑满足这个条件，以便一阶偏微分方程的每一个解
$$
f(x, y, z, p, q)=0
$$
也是方程的解吗
$$
g(x, y, z, p, q)=0
$$
当这种情况出现时，我们说方程是相容的。如果
$$
J \equiv \frac{\partial(f, g)}{\partial(p, q)} \neq 0
$$
我们可以解式(1)和式(2)得到显式表达式
$$
p=\phi(x, y, z), \quad q=\psi(x, y, z)
$$
浏览$p$和$q$。将方程(1)和(2)对相容的条件简化为方程组(4)完全可积的条件，即方程
$$
\phi d x+\psi d y-d z=0
$$
应该是可积的。由第一章的定理5可知，这个方程可积的条件是
$$
\phi\left(-\psi_z\right)+\psi\left(\phi_z\right)-\left(\psi_x-\phi_y\right)=0
$$
它等价于
$$
\psi_x+\phi \psi_z=\phi_y+\psi \phi_z
$$
将式(4)代入式(1)，分别对$x$和$z$求导，得到式
$$
\begin{aligned}
& f_x+f_{\mathfrak{p}} \phi_x+f_a \psi_x=0 \
& f_z+f_p \phi_z+f_a \psi_z=0
\end{aligned}
$$
由此很容易推断出
$$
f_x+\phi f_z+f_p\left(\phi_x+\phi \phi_z\right)+f_a\left(\psi_x+\phi \psi_z\right)=0
$$
同样地，我们可以由式(2)推导出
$$
g_x+\phi g_z+g_p\left(\phi_x+\phi \phi_z\right)+g_a\left(\psi_x+\phi \psi_z\right)=0
$$
解这些方程，我们发现
$$
\psi_x+\phi \psi_z=\frac{1}{J}\left{\frac{\partial(f, g)}{\partial(x, p)}+\phi \frac{\partial(f, g)}{\partial(z, p)}\right}
$$
其中$J$定义为式(3)。

数学代写|偏微分方程代写partial difference equations代考|Charpit’s Method

求解偏微分方程的一种方法
$$
f(x, y, z, p, q)=0
$$
由于Charpit，是基于上一节的考虑。Charpit方法的基本思想是引入一个二阶偏微分方程
$$
g(x, y, z, p, q, a)=0
$$
它包含一个任意常数$a$，它使得:
(a)式(1)、(2)可解得
$$
p=p(x, y, z, a), \quad q=q(x, y, z, a)
$$
(b)方程可积。
$$
d z=p(x, y, z, a) d x+q(x, y, z, a) d y
$$
当找到这样的函数$g$时，式(3)的解
$$
F(x, y, z, a, b)=0
$$
包含两个任意常数$a, b$将是方程(1)的解。从第7节的考虑可以看出，方程(4)是方程(1)的完全积分。

然后主要问题是确定第二个方程(2)，但这已经在上一节中解决了，因为我们只需要寻找与给定方程$f=0$兼容的方程$g=0$。其条件用上一节的式(3)和式(8)表示。展开后一个方程，我们看到它等价于线性偏微分方程
$$
\begin{aligned}
f_p \frac{\partial g}{\partial x}+f_q \frac{\partial g}{\partial y}+\left(p f_p+q f_a\right) & \frac{\partial g}{\partial z} \
& -\left(f_x+p f_z\right) \frac{\partial g}{\partial p}-\left(f_y+q f_z\right) \frac{\partial g}{\partial q}=0
\end{aligned}
$$
用于$g$的测定。我们的问题是找到这个方程的解，尽可能简单，涉及到任意常数$a$，我们通过求子方程的积分来实现
$$
\frac{d x}{f_p}=\frac{d y}{f_\alpha}=\frac{d z}{p f_p+q f_q}=\frac{d p}{-\left(f_x+p f_z\right)}=\frac{d q}{-\left(f_y+q f_z\right)}
$$根据定理3。这些方程被称为Charpit方程，等价于第8节的特征方程(18)。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|Compatible Systeṃs of First-order Equations

Posted on 2023年6月3日2023年6月3日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|Compatible Systeṃs of First-order Equations

We shall next consider the condition to be satisfied in order that every solution of the first-order partial differential equation
$$
f(x, y, z, p, q)=0
$$
is also a solution of the equation
$$
g(x, y, z, p, q)=0
$$
When such a situation arises, the equations are said to be compatible. If
$$
J \equiv \frac{\partial(f, g)}{\partial(p, q)} \neq 0
$$
we can solve equations (1) and (2) to obtain the explicit expressions
$$
p=\phi(x, y, z), \quad q=\psi(x, y, z)
$$
for $p$ and $q$. The condition that the pair of equations (1) and (2) should be compatible reduces then to the condition that the system of equations (4) should be completely integrable, i.e., that the equation
$$
\phi d x+\psi d y-d z=0
$$
should be integrable. From Theorem 5 of Chap. 1 we see that the condition that this equation is integrable is
$$
\phi\left(-\psi_z\right)+\psi\left(\phi_z\right)-\left(\psi_x-\phi_y\right)=0
$$
which is equivalent to
$$
\psi_x+\phi \psi_z=\phi_y+\psi \phi_z
$$
Substituting from equations (4) into equation (1) and differentiating with regard to $x$ and $z$, respectively, we obtain the equations
$$
\begin{aligned}
& f_x+f_{\mathfrak{p}} \phi_x+f_a \psi_x=0 \
& f_z+f_p \phi_z+f_a \psi_z=0
\end{aligned}
$$
from which it is readily deduced that
$$
f_x+\phi f_z+f_p\left(\phi_x+\phi \phi_z\right)+f_a\left(\psi_x+\phi \psi_z\right)=0
$$
Similarly we may deduce from equation (2) that
$$
g_x+\phi g_z+g_p\left(\phi_x+\phi \phi_z\right)+g_a\left(\psi_x+\phi \psi_z\right)=0
$$
Solving these equations, we find that
$$
\psi_x+\phi \psi_z=\frac{1}{J}\left{\frac{\partial(f, g)}{\partial(x, p)}+\phi \frac{\partial(f, g)}{\partial(z, p)}\right}
$$
where $J$ is defined as equation (3).

数学代写|偏微分方程代写partial difference equations代考|Charpit’s Method

A method of solving the partial differential equation
$$
f(x, y, z, p, q)=0
$$
due to Charpit, is based on the considerations of the last section. The fundamental idea in Charpit’s method is the introduction of a second partial differential equation of the first order
$$
g(x, y, z, p, q, a)=0
$$
which contains an arbitrary constant $a$ and which is such that:
(a) Equations (1) and (2) can be solved to give
$$
p=p(x, y, z, a), \quad q=q(x, y, z, a)
$$
(b) The equation is integrable.
$$
d z=p(x, y, z, a) d x+q(x, y, z, a) d y
$$
When such a function $g$ has been found, the solution of equation (3)
$$
F(x, y, z, a, b)=0
$$
containing two arbitrary constants $a, b$ will be a solution of equation (1). From the considerations of Sec. 7 it will be seen that equation (4) is a complete integral of equation (1).

The main problem then is the determination of the second equation (2), but this has already been solved in the last section, since we need only seek an equation $g=0$ compatible with the given equation $f=0$. The conditions for this are symbolized in equations (3) and (8) of the last section. Expanding the latter equation, we see that it is equivalent to the linear partial differential equation
$$
\begin{aligned}
f_p \frac{\partial g}{\partial x}+f_q \frac{\partial g}{\partial y}+\left(p f_p+q f_a\right) & \frac{\partial g}{\partial z} \
& -\left(f_x+p f_z\right) \frac{\partial g}{\partial p}-\left(f_y+q f_z\right) \frac{\partial g}{\partial q}=0
\end{aligned}
$$
for the determination of $g$. Our problem then is to find a solution of this equation, as simple as possible, involving an arbitrary constant $a$, and this we do by finding an integral of the subsidiary equations
$$
\frac{d x}{f_p}=\frac{d y}{f_\alpha}=\frac{d z}{p f_p+q f_q}=\frac{d p}{-\left(f_x+p f_z\right)}=\frac{d q}{-\left(f_y+q f_z\right)}
$$ in accordance with Theorem 3. These equations, which are known as Charpit’s equations, are equivalent to the characteristic equations (18) of Sec. 8 .

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|Compatible Systeṃs of First-order Equations

数学代写|偏微分方程代写partial difference equations代考|Charpit’s Method

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|Partial Differential Equations

Posted on 2023年6月3日2023年6月3日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|Partial Differential Equations

We now proceed to the study of partial differential equations proper. Such equations arise in geometry and physics when the number of independent variables in the problem under discussion is two or more. When such is the case, any dependent variable is likely to be a function of more than one variable, so that it possesses not ordinary derivatives with respect to a single variable but partial derivatives with respect to several variables. For instance, in the study of thermal effects in a solid body the temperature $\theta$ may vary from point to point in the solid as well as from time to time, and, as a consequence, the derivatives
$$
\frac{\partial \theta}{\partial x}, \quad \frac{\partial \theta}{\partial y}, \quad \frac{\partial \theta}{\partial z}, \quad \frac{\partial \theta}{\partial t},
$$
will, in general, be nonzero. Furthermore in any particular problem it may happen that higher derivatives of the types
$$
\frac{\partial^2 \theta}{\partial x^2}, \quad \frac{\partial^2 \theta}{\partial x \partial t}, \quad \frac{\partial^3 \theta}{\partial x^2 \partial t} \text {, etc. }
$$
may be of physical significance.
When the laws of physics are applied to a problem of this kind, we sometimes obtain a relation between the derivatives of the kind
$$
F\left(\frac{\partial \theta}{\partial x}, \ldots, \frac{\partial^2 \theta}{\partial x^2}, \ldots, \frac{\partial^2 \theta}{\partial x \partial t}, \ldots\right)=0
$$
Such an equation relating partial derivatives is called a partial differential equation.

数学代写|偏微分方程代写partial difference equations代考|Origins of First-order Partial Differential Equations

Before discussing the solution of equations of the type (7) of the last section, we shall examine the interesting question of how they arise. Suppose that we consider the equation
$$
x^2+y^2+(z-c)^2=: a^2
$$
in which the constants $a$ and $c$ are arbitrary. Then equation (1) represents the set of all spheres whose centers lie along the $z$ axis. If we differentiate this equation with respect to $x$, we obtain the relation
$$
x+p(z-c)=0
$$
while if we differentiate it with respect to $y$, we find that
$$
y+q(z-c)=0
$$
Eliminating the arbitrary constant $c$ from these two equations, we obtain the partial differential equation
$$
y p-x q=0
$$
which is of the first order. In some sense, then, the set of all spheres with centers on the $z$ axis is characterized by the partial differential equation (2)

However, other geometrical entities can be described by the same equation. For example, the equation
$$
x^2+y^2=(z-c)^2 \tan ^2 \alpha
$$
in which both of the constants $c$ and $\alpha$ are arbitrary, represents the set of all right circular cones whose axes coincide with the line $O z$. If we differentiate equation (3) first with respect to $x$ and then with respect to $y$, we find that
$$
p(z-c) \tan ^2 \alpha=x, \quad q(z-c) \tan ^2 \alpha=y
$$
and, upon eliminating $c$ and $\alpha$ from these relations, we see that for these cones also the equation (2) is satisfied.

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|Partial Differential Equations

现在我们开始研究偏微分方程。当所讨论的问题中自变量的数目为两个或两个以上时，在几何和物理中就会出现这样的方程。在这种情况下，任何因变量都可能是一个以上变量的函数，因此它不是对单个变量具有普通导数，而是对多个变量具有偏导数。例如，在研究固体中的热效应时，温度$\theta$可能在固体中点与点之间以及时间上发生变化，因此，其导数也会发生变化
$$
\frac{\partial \theta}{\partial x}, \quad \frac{\partial \theta}{\partial y}, \quad \frac{\partial \theta}{\partial z}, \quad \frac{\partial \theta}{\partial t},
$$
通常是非零的。此外，在任何特定的问题中，可能会发生高阶导数的类型
$$
\frac{\partial^2 \theta}{\partial x^2}, \quad \frac{\partial^2 \theta}{\partial x \partial t}, \quad \frac{\partial^3 \theta}{\partial x^2 \partial t} \text {, etc. }
$$
可能具有物理意义。
当把物理定律应用于这类问题时，我们有时会得到这类导数之间的关系
$$
F\left(\frac{\partial \theta}{\partial x}, \ldots, \frac{\partial^2 \theta}{\partial x^2}, \ldots, \frac{\partial^2 \theta}{\partial x \partial t}, \ldots\right)=0
$$
这种与偏导数有关的方程叫做偏微分方程。

数学代写|偏微分方程代写partial difference equations代考|Origins of First-order Partial Differential Equations

在讨论最后一节(7)型方程的解之前，我们将研究它们是如何产生的有趣问题。假设我们考虑这个方程
$$
x^2+y^2+(z-c)^2=: a^2
$$
其中常数$a$和$c$是任意的。则式(1)表示中心沿$z$轴的所有球体的集合。如果我们对$x$求导，我们就得到了这个关系
$$
x+p(z-c)=0
$$
如果我们对$y$求导，我们会发现
$$
y+q(z-c)=0
$$
从这两个方程中消去任意常数$c$，得到偏微分方程
$$
y p-x q=0
$$
这是一阶的。因此，在某种意义上，所有以$z$为中心的球体的集合可以用偏微分方程(2)来表示。

然而，其他几何实体可以用同样的方程来描述。例如，这个方程
$$
x^2+y^2=(z-c)^2 \tan ^2 \alpha
$$
其中，常数$c$和$\alpha$都是任意的，表示轴与$O z$线重合的所有直角圆锥的集合。如果我们先对方程(3)对$x$求导然后对$y$求导，我们会发现
$$
p(z-c) \tan ^2 \alpha=x, \quad q(z-c) \tan ^2 \alpha=y
$$
并且，在从这些关系中消去$c$和$\alpha$之后，我们看到对于这些锥体，方程(2)也是满足的。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|The Mapping Theorem

Posted on 2023年6月3日2023年6月3日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|The Mapping Theorem

In this section, a generalization of Hartman’s Theorem for Maps is given. While the proof is a generalization of the one given by Pugh [62] and Hartman [35], there are subtle changes and sufficient novelty to merit its exposition. More precisely, Pugh and Hartman prove the Mapping Theorem for maps operating on the same Banach space $E: L \in \mathscr{Q}(E, E), N \in \mathscr{M}(E, E)$. The theorem below is presented for maps operating between distinct Hilbert spaces. Furthermore, due to the Hilbert space setting, a global version of the Mapping Theorem is achieved. The theorem for flows on vector fields is discussed in the next section.

The Hartman Mapping Theorem: Let $L \in \mathscr{Q}(X, Y)$ be an expansive-contractive map and $N \in$ $\operatorname{S}(X, Y)$. Then there exists a unique homeomorphism $H$ so that $H \cdot(L+N)=L \cdot H$ on sufficiently small neighborhoods $U$ of $0 \in X$ and $V$ of $0 \in Y$. That is, Diagram 8.1 below commutes.

Proof: For technical reasons, it is easier to prove the local topological conjugacy of $T \equiv L+N$ to $R \equiv L+M$. To that end, let $U=\mathscr{B}_X(0, r)$ and $V=\mathscr{B}_Y(0, s)$. Take $r$ and $s$ sufficiently small and $N$, $M \in \mathscr{S}(X, Y)$, so that the operators $T$ and $R$ map $U$ homeomorphically onto $V$. That is, $(L+N) U=$ $(L+M) U=V$. This result is a direct application of the Inverse Function (see, e.g., Schwartz [68]).
Let $H=I+h$, where $I$ is the identity operator. Choose $r$ and $s$ sufficient small so that $T$ and $R$ map $U$ onto $V$. Then, as $T$ is a homeomorphism, the estimate $\left|T(\boldsymbol{x})-T\left(\boldsymbol{x}^{\prime}\right)\right|_V \leq 2 s$ holds for any $\boldsymbol{x}, \boldsymbol{x}^{\prime}$ $\in U$. Similar estimates are obtained for the homeomorphism $R:\left|R(\boldsymbol{x})-R\left(\boldsymbol{x}^{\prime}\right)\right|_V \leq 2 s$. Moreover, for any $\boldsymbol{y}, \boldsymbol{y}^{\prime} \in V,\left|T^{-1}(\boldsymbol{y})-T^{-1}\left(\boldsymbol{y}^{\prime}\right)\right|_U \leq 2 r$, with identical results for $R^{-1}$. These estimates will be utilized later on in the proof.
The conjugacy equation
$$
H T=R H
$$
and identification $H=I+h$ are equivalent to both (8.3.2) and (8.3.3) below.
$$
\begin{aligned}
& h=(M H+L h-N) T^{-1} \
& h=L^{-1}(h T+N-M H)
\end{aligned}
$$
Indeed, (8.3.1) implies $(I+h) T=R(I+h)$ which means $T+h T=R+R h$. Now $R-T=M-N$ so that $h T=M-N+R h=M-N+(L+M) h=M(I+h)-N+L h$ or $h=(M H-N+L h) T^{-1}$. This verifies (8.3.2). By proceeding in a similar manner, (8.3.1) implies $T+h T=H T=R H=R+R h$. Subtracting $L$ from both sides of this relation produces $N+h T=M+(L+M) h=M(I+h)+L h$ so that $N+h T-M H=L h \Rightarrow L^{-1}(N+h T-M H)=h$. This is $(8.3 .3)$.

数学代写|偏微分方程代写partial difference equations代考|The Flow Theorem

Consider the evolution equation (8.1.1) as a vector field defined by the maps $L$ and $N$ on the separable Hilbert space $X$. Let $F_{L+N}^t$ and $F_L^t$ be the flows on the vector fields defined by equations (8.1.1) and (8.1.2), respectively. That is, $F_{L+N}^t[u(\boldsymbol{x}, 0)]=e^{t L}[u(\boldsymbol{x}, 0)]+\int_0^t e^{(t-s) L}[N(u(\boldsymbol{x}, s))] d s$ and $F_L^t[v(\boldsymbol{x}, 0)]=e^{t L}[v(\boldsymbol{x}, 0)]$. The Hartman-Grobman Flow Theorem states that the flows $F_{L+N}^t$ and $F_L^t$ are locally topologically conjugate. More colloquially, the theorem states that locally there is a change of variables in which a solution of the linear equation (8.1.2) can be mapped into a solution of the nonlinear equation (8.1.1).

To prove the Flow Theorem, it must first be established that $F_{L+N}^t$ is a flow on (8.1.1). This means it must be shown that there is a unique solution to $(8.1 .1)$. Key Conditions $(A),(C)$, and $(D)$ of $\$ 8.2$ guarantee the existence of a unique solution to (8.1.1) in the separable Hilbert space $H^{\mathrm{1}}([0, T]$; Z).

Existence and Uniqueness Theorem: Let $L \in \mathscr{Q}(X, Z)$ satisfy Key Condition $(A)$ and $N \in \mathscr{M}(X, Z)$ satisfy Key Conditions $(C)-(D)$. For every $f \in X$, there exists a time $T_m \in \mathbb{R}^{+}$so that (8.1.1) has a unique generalized solution $u \in H^1([0, T] ; Z)$ for every subinterval $[0, T] \subset\left[0, T_m\right)$.

The proof of this theorem will be carried out via a series of lemmas. These lemmas, in turn, are generalizations of the work of Rauch [64], Henry [37], and Haraux [32].

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|The Mapping Theorem

在这一节中，给出了哈特曼定理在映射中的推广。虽然该证明是对Pugh[62]和Hartman[35]给出的证明的概括，但仍有细微的变化和足够的新新性值得阐述。更准确地说，Pugh和Hartman证明了映射定理在同一个巴拿赫空间$E: L \in \mathscr{Q}(E, E), N \in \mathscr{M}(E, E)$上运行。下面的定理是针对不同希尔伯特空间之间的映射给出的。进一步，由于Hilbert空间的设置，得到了映射定理的一个全局版本。下一节将讨论向量场上的流动定理。

哈特曼映射定理:设$L \in \mathscr{Q}(X, Y)$是一个扩张-收缩映射，$N \in$$\operatorname{S}(X, Y)$。那么就存在一个唯一的同胚$H$，使得$H \cdot(L+N)=L \cdot H$在$0 \in X$的$U$和$0 \in Y$的$V$上有足够小的邻域。也就是说，下图8.1是通勤图。

证明:由于技术原因，更容易证明$T \equiv L+N$到$R \equiv L+M$的局部拓扑共轭性。为此，让$U=\mathscr{B}_X(0, r)$和$V=\mathscr{B}_Y(0, s)$。取足够小的$r$和$s$以及$N$、$M \in \mathscr{S}(X, Y)$，以便操作符$T$和$R$同态地将$U$映射到$V$。即$(L+N) U=$$(L+M) U=V$。这个结果是反函数的直接应用(参见Schwartz[68]等)。
设$H=I+h$，其中$I$是标识算子。选择足够小的$r$和$s$，以便$T$和$R$将$U$映射到$V$上。然后，由于$T$是一个同胚，估计$\left|T(\boldsymbol{x})-T\left(\boldsymbol{x}^{\prime}\right)\right|_V \leq 2 s$对任何$\boldsymbol{x}, \boldsymbol{x}^{\prime}$$\in U$都成立。对于同胚$R:\left|R(\boldsymbol{x})-R\left(\boldsymbol{x}^{\prime}\right)\right|_V \leq 2 s$也得到了类似的估计。此外，对于任何$\boldsymbol{y}, \boldsymbol{y}^{\prime} \in V,\left|T^{-1}(\boldsymbol{y})-T^{-1}\left(\boldsymbol{y}^{\prime}\right)\right|_U \leq 2 r$，对于$R^{-1}$都有相同的结果。这些估计将在后面的证明中使用。
共轭方程
$$
H T=R H
$$
和标识$H=I+h$等价于下面的(8.3.2)和(8.3.3)。
$$
\begin{aligned}
& h=(M H+L h-N) T^{-1} \
& h=L^{-1}(h T+N-M H)
\end{aligned}
$$
的确，(8.3.1)暗示$(I+h) T=R(I+h)$，意思是$T+h T=R+R h$。现在$R-T=M-N$那么$h T=M-N+R h=M-N+(L+M) h=M(I+h)-N+L h$或$h=(M H-N+L h) T^{-1}$。这验证了(8.3.2)。通过类似的方式进行，(8.3.1)意味着$T+h T=H T=R H=R+R h$。等式两边同时减去$L$得$N+h T=M+(L+M) h=M(I+h)+L h$，得到$N+h T-M H=L h \Rightarrow L^{-1}(N+h T-M H)=h$。这是$(8.3 .3)$。

数学代写|偏微分方程代写partial difference equations代考|The Flow Theorem

将演化方程(8.1.1)看作是可分离希尔伯特空间$X$上的映射$L$和$N$所定义的向量场。设$F_{L+N}^t$和$F_L^t$分别为式(8.1.1)和式(8.1.2)定义的矢量场上的流。即$F_{L+N}^t[u(\boldsymbol{x}, 0)]=e^{t L}[u(\boldsymbol{x}, 0)]+\int_0^t e^{(t-s) L}[N(u(\boldsymbol{x}, s))] d s$和$F_L^t[v(\boldsymbol{x}, 0)]=e^{t L}[v(\boldsymbol{x}, 0)]$。Hartman-Grobman流定理指出，流动$F_{L+N}^t$和$F_L^t$是局部拓扑共轭的。更通俗地说，该定理表明局部存在变量变化，其中线性方程(8.1.2)的解可以映射为非线性方程(8.1.1)的解。

为了证明流动定理，首先必须确定 $F_{L+N}^t$ 在(8.1.1)上是一个流。这意味着必须证明有一个唯一的解 $(8.1 .1)$．关键条件 $(A),(C)$，和 $(D)$ 的 $\$ 8.2$ 保证(8.1.1)在可分Hilbert空间中存在唯一解 $H^{\mathrm{1}}([0, T]$； z);

存在唯一性定理:设$L \in \mathscr{Q}(X, Z)$满足关键条件$(A)$和$N \in \mathscr{M}(X, Z)$满足关键条件$(C)-(D)$。对于每一个$f \in X$，存在一个时间$T_m \in \mathbb{R}^{+}$，使得(8.1.1)对每一子区间$[0, T] \subset\left[0, T_m\right)$有一个唯一的广义解$u \in H^1([0, T] ; Z)$。

这个定理的证明将通过一系列引理进行。反过来，这些引理是Rauch[64]、Henry[37]和Haraux[32]工作的概括。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|The Korteweg de Vries Equation and Inverse Scattering

Posted on 2023年5月22日2023年5月22日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|The Three–Dimensional Wave Kernels

数学代写|偏微分方程代写partial difference equations代考|The Korteweg de Vries Equation and Inverse Scattering

The Korteweg de Vries or $K d V$ equation (6.6.1) is one of the primary examples of integrable or analytically solvable nonlinear partial differential equations.
$$
\frac{\partial u}{\partial t}-6 u \frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial x^3}=0
$$

In 1895, the Dutch mathematician Diederik Johannes Korteweg and his student Gustav de Vries derived equation (6.6.1) above to model shallow water waves [47, 84]. An immediate calculation confirms that (6.6.1) may be written as
$$
\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}\left(-3 u^2+\frac{\partial^2 u}{\partial x^2}\right)=0 .
$$
Remark 6.2: The $K d V$ equation is also frequently written with a positive coefficient in the nonlinear term (i.e., +6 versus -6). A direct calculation shows that if $u$ satisfies (6.6.1), then $v=$ $-u$ satisfies $\frac{\partial v}{\partial t}+6 v \frac{\partial v}{\partial x}+\frac{\partial^3 v}{\partial x^3}=0$. Throughout this book, equation (6.6.1) will be used to represent the $K d V$ equation.

数学代写|偏微分方程代写partial difference equations代考|Traveling Wave Solution

It has been observed by Korteweg and de Vries [47], Kruskal and Zabusky [49], and many others that the $K d V$ equation (6.6.1) admits a traveling wave solution of the form $u(x, t)=\phi(x-\gamma t)$. Making this assumption and the substitution $\xi=x-\gamma t$, results in
$$
\frac{d}{d \xi}\left(3 \phi^2+\gamma \phi-\phi^{\prime \prime}\right)=0
$$
Equation (6.6.3) is the consequence of the following computations. The identification, $u(x, t)=$ $\phi(x-\gamma t)=\phi(\xi)$ implies $\frac{\partial u}{\partial t}=-\gamma \phi^{\prime}(\xi), \frac{\partial u}{\partial x}=\phi^{\prime}(\xi), \frac{\partial^2 u}{\partial x^2}=\phi^{\prime \prime}(\xi)$, and $\frac{\partial^3 u}{\partial x^3}=\phi^{\prime \prime \prime}(\xi)$ which, when substituted into (6.6.1), produces ( $6.6 .3)$.

To mimic real wave motion, the additional constraint $\phi(\xi) \rightarrow 0$ as $|\xi| \rightarrow+\infty$ is imposed upon the solution of (6.6.3). Integrating (6.6.3) with respect to $\xi$ and imposing the aforementioned asymptotic boundary condition yields $3 \phi^2+\gamma \phi-\phi^{\prime \prime}=0$. Multiplying this equation through by $\phi^{\prime}$ results in $3 \phi^2 \phi^{\prime}+\gamma \phi \phi^{\prime}-\phi^{\prime \prime} \phi^{\prime}=0$ or $\frac{d}{d \xi}\left(\phi^3+\frac{1}{2} \gamma \phi^2-\frac{1}{2}\left(\phi^{\prime}\right)^2\right)=0$. Integrating this differential equation and imposing the asymptotic boundary requirement produces $\left(\phi^{\prime}\right)^2=\phi^2(\gamma+2 \phi)$. That is, $\mathrm{d} \xi=\frac{d \phi}{\phi \sqrt{\gamma+2 \phi}}$. For $t_o=0$ and $x\left(t_o\right)=x_o$, then $\xi_o=\xi\left(t_o\right)=x_o-\gamma t_o=x_o$. Integration of the differential form above from $\xi$ to $\xi_o=x_o$ gives $\frac{1}{\sqrt{\gamma}} \ln \left(\frac{\sqrt{\gamma+2 \phi}-\sqrt{\gamma}}{\sqrt{\gamma+2 \phi}+\sqrt{\gamma}}\right)=\xi-x_o$, provided $\gamma>0$. After a considerable amount of algebra, the traveling wave solution (6.6.4) is obtained.

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|The Korteweg de Vries Equation and Inverse Scattering

Korteweg de Vries或$K d V$方程(6.6.1)是可积或解析可解非线性偏微分方程的主要例子之一。
$$
\frac{\partial u}{\partial t}-6 u \frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial x^3}=0
$$

1895年，荷兰数学家Diederik Johannes Korteweg和他的学生Gustav de Vries推导出上述式(6.6.1)来模拟浅水波浪[47,84]。直接计算证实(6.6.1)可以写成
$$
\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}\left(-3 u^2+\frac{\partial^2 u}{\partial x^2}\right)=0 .
$$
备注6.2:$K d V$方程也经常在非线性项中写成正系数(即+6对-6)。直接计算可知，如果$u$满足式(6.6.1)，则$v=$$-u$满足$\frac{\partial v}{\partial t}+6 v \frac{\partial v}{\partial x}+\frac{\partial^3 v}{\partial x^3}=0$。在本书中，方程(6.6.1)将用于表示$K d V$方程。

数学代写|偏微分方程代写partial difference equations代考|Traveling Wave Solution

Korteweg和de Vries [47]， Kruskal和Zabusky[49]以及其他许多人已经观察到$K d V$方程(6.6.1)允许形式为$u(x, t)=\phi(x-\gamma t)$的行波解。做这个假设和$\xi=x-\gamma t$替换，结果是
$$
\frac{d}{d \xi}\left(3 \phi^2+\gamma \phi-\phi^{\prime \prime}\right)=0
$$
式(6.6.3)是以下计算的结果。标识$u(x, t)=$$\phi(x-\gamma t)=\phi(\xi)$表示$\frac{\partial u}{\partial t}=-\gamma \phi^{\prime}(\xi), \frac{\partial u}{\partial x}=\phi^{\prime}(\xi), \frac{\partial^2 u}{\partial x^2}=\phi^{\prime \prime}(\xi)$，而将$\frac{\partial^3 u}{\partial x^3}=\phi^{\prime \prime \prime}(\xi)$代入(6.6.1)则生成($6.6 .3)$)。

为了模拟真实的波动，在(6.6.3)的解上附加约束$\phi(\xi) \rightarrow 0$为$|\xi| \rightarrow+\infty$。对$\xi$积分(6.6.3)并施加上述渐近边界条件，得到$3 \phi^2+\gamma \phi-\phi^{\prime \prime}=0$。将这个方程乘以$\phi^{\prime}$得到$3 \phi^2 \phi^{\prime}+\gamma \phi \phi^{\prime}-\phi^{\prime \prime} \phi^{\prime}=0$或$\frac{d}{d \xi}\left(\phi^3+\frac{1}{2} \gamma \phi^2-\frac{1}{2}\left(\phi^{\prime}\right)^2\right)=0$。对微分方程积分并施加渐近边界要求，得到$\left(\phi^{\prime}\right)^2=\phi^2(\gamma+2 \phi)$。也就是$\mathrm{d} \xi=\frac{d \phi}{\phi \sqrt{\gamma+2 \phi}}$。对于$t_o=0$和$x\left(t_o\right)=x_o$，然后是$\xi_o=\xi\left(t_o\right)=x_o-\gamma t_o=x_o$。从$\xi$到$\xi_o=x_o$的微分形式的积分得到$\frac{1}{\sqrt{\gamma}} \ln \left(\frac{\sqrt{\gamma+2 \phi}-\sqrt{\gamma}}{\sqrt{\gamma+2 \phi}+\sqrt{\gamma}}\right)=\xi-x_o$，提供$\gamma>0$。经过大量的代数运算，得到行波解(6.6.4)。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|The Three–Dimensional Wave Kernels

Posted on 2023年5月22日2023年5月22日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|The Three–Dimensional Wave Kernels

To construct the wave kernels $K_1$ and $K_2$ of (5.1.3), define the radial indicator function on the sphere of radius $t$ and “thickness” $\varepsilon$. That is, $\mathscr{B}_{\mathbb{B}(\boldsymbol{0}, t+\varepsilon) \mathbb{B}(\boldsymbol{\theta}, t)}(\boldsymbol{x})= \begin{cases}1 & \text { for } x \in \mathbb{B}(\boldsymbol{0}, t+\varepsilon) \backslash \mathbb{B}(\boldsymbol{0}, t) \ 0 & \text { otherwise }\end{cases}$ where $\mathbb{B}(\boldsymbol{0}, t)=\left{\boldsymbol{x} \in \mathbb{R}^3:|\boldsymbol{x}|^2=x^2+y^2+z^2 \leq t^2\right}$ is the three-dimensional ball of radius $t$ centered at the origin $\boldsymbol{0}$. The solid $\mathbb{B}(\boldsymbol{0}, t+\varepsilon) \mathbb{B}(\boldsymbol{0}, t)=\left{\boldsymbol{x} \in \mathbb{R}^3: t^2 \leq|\boldsymbol{x}|^2 \leq(t+\varepsilon)^2\right}$ is the three-dimensional ball whose thickness is $\varepsilon$. Figure 5.1 illustrates a cross-section of this “spherical shell” as per Champeney [16] and Dettman [22].

Define $f_{\varepsilon}(x, t)=\frac{1}{c t \varepsilon} \Phi_{B(\theta, t+\varepsilon) \mathbb{B}(\theta, t)}(\boldsymbol{x})$. The challenge is to determine the Fourier transform of $f_{\varepsilon}(x$, t). It is evident from the geometry that spherical coordinates $x=r \cdot \sin (\beta) \cdot \cos (\theta), y=r \cdot \sin (\beta) \cdot \sin (\theta)$, $z=r \cdot \sin (\beta), 0 \leq \beta \leq \pi, 0 \leq \theta \leq 2 \pi, t \leq r \leq t+\varepsilon$ should be utilized. Before attempting to compute the Fourier transform, the Jacobian matrix reflecting the change of variables from Euclidean $(x, y$, $z)$ to spherical $(r, \beta, \theta)$ coordinates must be calculated. That is,
$$
J_{r \beta \theta}^{x y z}=\left[\begin{array}{ccc}
\frac{d x}{d r} & \frac{d x}{d \beta} & \frac{d x}{d \theta} \
\frac{d y}{d r} & \frac{d y}{d \beta} & \frac{d y}{d \theta} \
\frac{d z}{d r} & \frac{d z}{d \beta} & \frac{d z}{d \theta}
\end{array}\right]=\left[\begin{array}{ccc}
\sin (\beta) \cos (\theta) & r \cos (\beta) \cos (\theta) & -r \sin (\beta) \sin (\theta) \
\sin (\beta) \sin (\theta) & r \cos (\beta) \sin (\theta) & r \sin (\beta) \cos (\theta) \
\cos (\beta) & -r \sin (\beta) & 0
\end{array}\right] .
$$
A direct calculation shows that $\operatorname{det}\left[J_{r \beta \theta}^{x y z}\right]=r^2 \sin (\beta)$.

数学代写|偏微分方程代写partial difference equations代考|Huygen’s Principle and Duhamel’s Principle

This section relies heavily upon the work of Zachmanoglou and Thoe [87]. Since convolution is symmetric with respect to the function in translation, then (5.1.3) can be written as
$$
\begin{aligned}
A(\boldsymbol{x}, t) & =\frac{1}{(\sqrt{2 \pi})^3} \int_{\mathbb{R}^3}\left(\phi(\boldsymbol{x}-\boldsymbol{\xi}) \cdot K_1(\boldsymbol{\xi}, t)+\psi(\boldsymbol{x}-\boldsymbol{\xi}) \cdot K_2(\boldsymbol{\xi}, t)\right) d \boldsymbol{\xi} \
& +\frac{1}{(\sqrt{2 \pi})^3} \int_0^t \int_{\mathbb{R}^3} p(\boldsymbol{x}-\boldsymbol{\xi}, \tau) \cdot K_2(\boldsymbol{\xi}, t-\tau) d \xi d \tau .
\end{aligned}
$$
Utilizing the results of $\S 5.2$ and Table 5.1, shifting once again to spherical coordinates over the sphere of radius $t / \alpha$, selecting $v=(\sin (\beta) \cdot \cos (\theta), \sin (\beta) \cdot \sin (\theta), \cos (\beta))$ as the position vector on the unit sphere, and setting $\boldsymbol{\xi}=(t / \alpha) \cdot \boldsymbol{v}$ to be the position vector on $\mathbb{S}(\boldsymbol{\theta}, t / \alpha)$, results in
$$
\begin{aligned}
\int_{\mathbb{R}^3} \psi(\boldsymbol{x}-\xi) \cdot K_2(\xi, t) d \xi & =\int_0^{2 \pi} \int_0^\pi \psi(\boldsymbol{x}-c t \cdot v) \cdot K_2(\boldsymbol{v}, t) \sin (\beta) d \beta d \theta \
& =\int_{\mathbb{S}1} \sqrt{\frac{\pi}{2}} \frac{1}{c t} \cdot \psi(\boldsymbol{x}-c t \cdot \boldsymbol{v}) d \boldsymbol{v} \end{aligned} $$ where $d \boldsymbol{v}=\sin (\beta) d \beta d \theta$ is the element of surface area on the unit sphere $\mathbb{S}_1=\left{\boldsymbol{x} \in \mathbb{R}^3:|\boldsymbol{x}|^2=1\right}$. Using the relationship between $K_1(\boldsymbol{x}, t)$ and $K_2(\boldsymbol{x}, t)$, it is similarly seen that $$ \begin{aligned} \int{\mathbb{R}^3} \phi(\boldsymbol{x}-\boldsymbol{\xi}) \cdot K_1(\boldsymbol{\xi}, t) d \xi & =\int_0^{2 \pi} \int_0^\pi \phi(\boldsymbol{x}-c t \cdot \boldsymbol{v}) \cdot \frac{\partial}{\partial t} K_2(\boldsymbol{v}, t) \sin (\beta) d \beta d \theta \
& =\frac{\partial}{\partial t} \int_0^{2 \pi} \int_0^\pi \phi(\boldsymbol{x}-c t \cdot v) \cdot K_2(\boldsymbol{v}, t) \sin (\beta) d \beta d \theta \
& =\frac{\partial}{\partial t} \int_{\mathrm{S}1} \sqrt{\frac{\pi}{2}} \frac{1}{c t} \cdot \phi(\boldsymbol{x}-c t \cdot \boldsymbol{v}) d \boldsymbol{v} \end{aligned} $$ and $$ \begin{aligned} \int{\mathbb{R}^3} p(\boldsymbol{x}-\boldsymbol{\xi}, \tau) \cdot K_2(\boldsymbol{\xi}, t-\tau) d \xi & =\int_0^{2 \pi} \int_0^\pi p(\boldsymbol{x}-c t \cdot \boldsymbol{v}, \tau) \cdot K_2(\boldsymbol{v}, t-\tau) \sin (\beta) d \beta d \theta \
& =\int_{\mathrm{S}_1} \sqrt{\frac{\pi}{2}} \frac{1}{c(t-\tau)} \cdot p(\boldsymbol{x}-c t \cdot \boldsymbol{v}, c(t-\tau)) d \boldsymbol{v}
\end{aligned}
$$

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|The Three–Dimensional Wave Kernels

为构造式(5.1.3)中的波核$K_1$和$K_2$，在半径为$t$和“厚度”为$\varepsilon$的球上定义径向指示函数。即$\mathscr{B}_{\mathbb{B}(\boldsymbol{0}, t+\varepsilon) \mathbb{B}(\boldsymbol{\theta}, t)}(\boldsymbol{x})= \begin{cases}1 & \text { for } x \in \mathbb{B}(\boldsymbol{0}, t+\varepsilon) \backslash \mathbb{B}(\boldsymbol{0}, t) \ 0 & \text { otherwise }\end{cases}$，其中$\mathbb{B}(\boldsymbol{0}, t)=\left{\boldsymbol{x} \in \mathbb{R}^3:|\boldsymbol{x}|^2=x^2+y^2+z^2 \leq t^2\right}$是以原点$\boldsymbol{0}$为中心的半径为$t$的三维球。实体$\mathbb{B}(\boldsymbol{0}, t+\varepsilon) \mathbb{B}(\boldsymbol{0}, t)=\left{\boldsymbol{x} \in \mathbb{R}^3: t^2 \leq|\boldsymbol{x}|^2 \leq(t+\varepsilon)^2\right}$为三维球体，其厚度为$\varepsilon$。图5.1显示了根据Champeney[16]和Dettman[22]设计的“球壳”的横截面。

定义$f_{\varepsilon}(x, t)=\frac{1}{c t \varepsilon} \Phi_{B(\theta, t+\varepsilon) \mathbb{B}(\theta, t)}(\boldsymbol{x})$。挑战在于确定$f_{\varepsilon}(x$, t)的傅里叶变换。从几何上可以明显看出，应该利用球坐标$x=r \cdot \sin (\beta) \cdot \cos (\theta), y=r \cdot \sin (\beta) \cdot \sin (\theta)$, $z=r \cdot \sin (\beta), 0 \leq \beta \leq \pi, 0 \leq \theta \leq 2 \pi, t \leq r \leq t+\varepsilon$。在试图计算傅里叶变换之前，必须计算反映从欧几里得$(x, y$, $z)$到球面$(r, \beta, \theta)$坐标的变量变化的雅可比矩阵。也就是说，
$$
J_{r \beta \theta}^{x y z}=\left[\begin{array}{ccc}
\frac{d x}{d r} & \frac{d x}{d \beta} & \frac{d x}{d \theta} \
\frac{d y}{d r} & \frac{d y}{d \beta} & \frac{d y}{d \theta} \
\frac{d z}{d r} & \frac{d z}{d \beta} & \frac{d z}{d \theta}
\end{array}\right]=\left[\begin{array}{ccc}
\sin (\beta) \cos (\theta) & r \cos (\beta) \cos (\theta) & -r \sin (\beta) \sin (\theta) \
\sin (\beta) \sin (\theta) & r \cos (\beta) \sin (\theta) & r \sin (\beta) \cos (\theta) \
\cos (\beta) & -r \sin (\beta) & 0
\end{array}\right] .
$$
直接计算表明$\operatorname{det}\left[J_{r \beta \theta}^{x y z}\right]=r^2 \sin (\beta)$。

数学代写|偏微分方程代写partial difference equations代考|Huygen’s Principle and Duhamel’s Principle

这一部分很大程度上依赖于Zachmanoglou和Thoe[87]的工作。由于卷积相对于平移中的函数是对称的，那么(5.1.3)可以写成
$$
\begin{aligned}
A(\boldsymbol{x}, t) & =\frac{1}{(\sqrt{2 \pi})^3} \int_{\mathbb{R}^3}\left(\phi(\boldsymbol{x}-\boldsymbol{\xi}) \cdot K_1(\boldsymbol{\xi}, t)+\psi(\boldsymbol{x}-\boldsymbol{\xi}) \cdot K_2(\boldsymbol{\xi}, t)\right) d \boldsymbol{\xi} \
& +\frac{1}{(\sqrt{2 \pi})^3} \int_0^t \int_{\mathbb{R}^3} p(\boldsymbol{x}-\boldsymbol{\xi}, \tau) \cdot K_2(\boldsymbol{\xi}, t-\tau) d \xi d \tau .
\end{aligned}
$$
利用$\S 5.2$和表5.1的结果，再次在半径为$t / \alpha$的球体上移动到球坐标，选择$v=(\sin (\beta) \cdot \cos (\theta), \sin (\beta) \cdot \sin (\theta), \cos (\beta))$作为单位球体上的位置向量，并设置$\boldsymbol{\xi}=(t / \alpha) \cdot \boldsymbol{v}$为$\mathbb{S}(\boldsymbol{\theta}, t / \alpha)$上的位置向量，得到
$$
\begin{aligned}
\int_{\mathbb{R}^3} \psi(\boldsymbol{x}-\xi) \cdot K_2(\xi, t) d \xi & =\int_0^{2 \pi} \int_0^\pi \psi(\boldsymbol{x}-c t \cdot v) \cdot K_2(\boldsymbol{v}, t) \sin (\beta) d \beta d \theta \
& =\int_{\mathbb{S}1} \sqrt{\frac{\pi}{2}} \frac{1}{c t} \cdot \psi(\boldsymbol{x}-c t \cdot \boldsymbol{v}) d \boldsymbol{v} \end{aligned} $$其中$d \boldsymbol{v}=\sin (\beta) d \beta d \theta$是单位球面上的表面积元$\mathbb{S}1=\left{\boldsymbol{x} \in \mathbb{R}^3:|\boldsymbol{x}|^2=1\right}$。使用$K_1(\boldsymbol{x}, t)$和$K_2(\boldsymbol{x}, t)$之间的关系，可以类似地看到$$ \begin{aligned} \int{\mathbb{R}^3} \phi(\boldsymbol{x}-\boldsymbol{\xi}) \cdot K_1(\boldsymbol{\xi}, t) d \xi & =\int_0^{2 \pi} \int_0^\pi \phi(\boldsymbol{x}-c t \cdot \boldsymbol{v}) \cdot \frac{\partial}{\partial t} K_2(\boldsymbol{v}, t) \sin (\beta) d \beta d \theta \ & =\frac{\partial}{\partial t} \int_0^{2 \pi} \int_0^\pi \phi(\boldsymbol{x}-c t \cdot v) \cdot K_2(\boldsymbol{v}, t) \sin (\beta) d \beta d \theta \ & =\frac{\partial}{\partial t} \int{\mathrm{S}1} \sqrt{\frac{\pi}{2}} \frac{1}{c t} \cdot \phi(\boldsymbol{x}-c t \cdot \boldsymbol{v}) d \boldsymbol{v} \end{aligned} $$和 $$ \begin{aligned} \int{\mathbb{R}^3} p(\boldsymbol{x}-\boldsymbol{\xi}, \tau) \cdot K_2(\boldsymbol{\xi}, t-\tau) d \xi & =\int_0^{2 \pi} \int_0^\pi p(\boldsymbol{x}-c t \cdot \boldsymbol{v}, \tau) \cdot K_2(\boldsymbol{v}, t-\tau) \sin (\beta) d \beta d \theta \
& =\int_{\mathrm{S}_1} \sqrt{\frac{\pi}{2}} \frac{1}{c(t-\tau)} \cdot p(\boldsymbol{x}-c t \cdot \boldsymbol{v}, c(t-\tau)) d \boldsymbol{v}
\end{aligned}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|Shifting/Translation

Posted on 2023年5月22日2023年5月22日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|Shifting/Translation

For any $\boldsymbol{a} \in \mathbb{R}^n$,
$$
\mathscr{F}\left\mathrm{e}^{i(\boldsymbol{a} \boldsymbol{x})} f(\boldsymbol{x})\right=\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} e^{i(\boldsymbol{a} \cdot \boldsymbol{x})} f(\boldsymbol{x}) e^{i(x \cdot \boldsymbol{\omega})} d \boldsymbol{x}=\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{x}) e^{i(x \cdot[\boldsymbol{\omega}+a])} d \boldsymbol{x}=\mathscr{F}f(\boldsymbol{x})
$$
and
$$
\begin{aligned}
\mathscr{F}f(\boldsymbol{x}+\boldsymbol{a}) & =\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{x}-\boldsymbol{a}) e^{i(\boldsymbol{x} \cdot \boldsymbol{\omega})} d \boldsymbol{x} \stackrel{\boldsymbol{x}-\boldsymbol{a} \mapsto z}{=} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(z) e^{i((z+a] \cdot \omega)} d z \
& =e^{i(a \cdot \omega)} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{x}) e^{i(\boldsymbol{x} \cdot \omega)} d \boldsymbol{x}=e^{i(\boldsymbol{a} \cdot \omega)} \mathcal{F}f(\boldsymbol{x}) .
\end{aligned}
$$
\$4.1.5. Differentiation (with respect to the spatial variable)
$$
\mathscr{J}\left\frac{\partial f}{\partial x_j}(\boldsymbol{x})\right=\left(-i \cdot \omega_j\right) \mathscr{f}f(\boldsymbol{x}) \text { and more generally, }
$$
$$
\mathscr{F}\left\frac{\partial^k f}{\partial x_j^k}(\boldsymbol{x})\right=\left(-i \cdot \omega_j\right)^k \mathscr{F}f(\boldsymbol{x}) .
$$

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

$$
\mathscr{J}f(\boldsymbol{x}) \cdot \mathscr{f}g(\boldsymbol{x})=\mathscr{F}(f \odot g)(\boldsymbol{x}) .
$$
Proof: Proceeding directly, it is seen that
$$
\begin{aligned}
\mathscr{F}(f \odot g)(\boldsymbol{x}) & =\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n}\left[\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{\xi}) g(\boldsymbol{x}-\boldsymbol{\xi}) d \boldsymbol{\xi}\right] e^{i(x \cdot \boldsymbol{\omega})} d \boldsymbol{\omega} \
& =\frac{1}{(\sqrt{2 \pi})^n} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(\boldsymbol{\xi}) g(\boldsymbol{x}-\boldsymbol{\xi}) d \boldsymbol{\xi} e^{i(x, \omega)} d \boldsymbol{\omega} .
\end{aligned}
$$
Let $\boldsymbol{u}=\boldsymbol{x}-\boldsymbol{\xi}$ so that
$$
\begin{aligned}
\mathscr{F}(f \odot g)(\boldsymbol{x}) & =\frac{1}{(\sqrt{2 \pi})^n} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(\xi) g(\boldsymbol{u}) e^{i(\xi \cdot \omega)} d \xi e^{i(\boldsymbol{u} \cdot \omega)} d \boldsymbol{u} \
& =\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\xi) e^{i(\xi \cdot \omega)} d \xi \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} g(\boldsymbol{u}) e^{i(\boldsymbol{u} \cdot \omega)} d \boldsymbol{u} \text { (by Fubini’s Theorem) } \
& =\mathscr{J}f(\boldsymbol{x}) \cdot \mathscr{J}g(\boldsymbol{x}) .
\end{aligned}
$$
The final item in the list of Fourier transform properties is the invariance of the transform of functions of spatial and temporal variables under differentiation with respect to time. That is, if $f$ is a function of $\boldsymbol{x} \in \mathbb{R}^n$ and $t$, then as operators the derivative $\partial / \partial t$ and $\mathscr{F}$ commute. This is a crucial property of the Fourier transform as it can turn a linear PDE into a linear $O D E$.

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|Shifting/Translation

对于任何$\boldsymbol{a} \in \mathbb{R}^n$，
$$
\mathscr{F}\left\mathrm{e}^{i(\boldsymbol{a} \boldsymbol{x})} f(\boldsymbol{x})\right=\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} e^{i(\boldsymbol{a} \cdot \boldsymbol{x})} f(\boldsymbol{x}) e^{i(x \cdot \boldsymbol{\omega})} d \boldsymbol{x}=\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{x}) e^{i(x \cdot[\boldsymbol{\omega}+a])} d \boldsymbol{x}=\mathscr{F}f(\boldsymbol{x})
$$
和
$$
\begin{aligned}
\mathscr{F}f(\boldsymbol{x}+\boldsymbol{a}) & =\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{x}-\boldsymbol{a}) e^{i(\boldsymbol{x} \cdot \boldsymbol{\omega})} d \boldsymbol{x} \stackrel{\boldsymbol{x}-\boldsymbol{a} \mapsto z}{=} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(z) e^{i((z+a] \cdot \omega)} d z \
& =e^{i(a \cdot \omega)} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{x}) e^{i(\boldsymbol{x} \cdot \omega)} d \boldsymbol{x}=e^{i(\boldsymbol{a} \cdot \omega)} \mathcal{F}f(\boldsymbol{x}) .
\end{aligned}
$$
4.1.5美元。微分(关于空间变量)
$$
\mathscr{J}\left\frac{\partial f}{\partial x_j}(\boldsymbol{x})\right=\left(-i \cdot \omega_j\right) \mathscr{f}f(\boldsymbol{x}) \text { and more generally, }
$$
$$
\mathscr{F}\left\frac{\partial^k f}{\partial x_j^k}(\boldsymbol{x})\right=\left(-i \cdot \omega_j\right)^k \mathscr{F}f(\boldsymbol{x}) .
$$

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

$$
\mathscr{J}f(\boldsymbol{x}) \cdot \mathscr{f}g(\boldsymbol{x})=\mathscr{F}(f \odot g)(\boldsymbol{x}) .
$$
证明:直接进行，可以看出
$$
\begin{aligned}
\mathscr{F}(f \odot g)(\boldsymbol{x}) & =\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n}\left[\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\boldsymbol{\xi}) g(\boldsymbol{x}-\boldsymbol{\xi}) d \boldsymbol{\xi}\right] e^{i(x \cdot \boldsymbol{\omega})} d \boldsymbol{\omega} \
& =\frac{1}{(\sqrt{2 \pi})^n} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(\boldsymbol{\xi}) g(\boldsymbol{x}-\boldsymbol{\xi}) d \boldsymbol{\xi} e^{i(x, \omega)} d \boldsymbol{\omega} .
\end{aligned}
$$
让$\boldsymbol{u}=\boldsymbol{x}-\boldsymbol{\xi}$
$$
\begin{aligned}
\mathscr{F}(f \odot g)(\boldsymbol{x}) & =\frac{1}{(\sqrt{2 \pi})^n} \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f(\xi) g(\boldsymbol{u}) e^{i(\xi \cdot \omega)} d \xi e^{i(\boldsymbol{u} \cdot \omega)} d \boldsymbol{u} \
& =\frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} f(\xi) e^{i(\xi \cdot \omega)} d \xi \frac{1}{(\sqrt{2 \pi})^n} \int_{\mathbb{R}^n} g(\boldsymbol{u}) e^{i(\boldsymbol{u} \cdot \omega)} d \boldsymbol{u} \text { (by Fubini’s Theorem) } \
& =\mathscr{J}f(\boldsymbol{x}) \cdot \mathscr{J}g(\boldsymbol{x}) .
\end{aligned}
$$
傅里叶变换性质列表中的最后一项是空间和时间变量的函数在微分下对时间的变换的不变性。也就是说，如果$f$是$\boldsymbol{x} \in \mathbb{R}^n$和$t$的函数，那么作为运算符，导数$\partial / \partial t$和$\mathscr{F}$可以交换。这是傅里叶变换的一个重要性质因为它可以把线性偏微分方程变成线性$O D E$。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|Laplace’s Equation and the Classical Approach

Posted on 2023年5月11日2023年5月11日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|Laplace’s Equation and the Classical Approach

Let $\Omega \subset \mathbb{R}^n$ be a domain (that is, an open, connected set) with smooth boundary $\partial \Omega$. The closure $\bar{\Omega}$ of a domain $\Omega$ is the closed set which is a union of the set with its boundary $\bar{\Omega}=\Omega \cup \partial \Omega$. If $\Omega$ is bounded, then the boundary $\partial \Omega$ is a curve (in $n=2$ spatial dimensions) or surface ${ }^8(n \geq 3)$. As mentioned above, a function $u \in C^2(\Omega)$ that satisfies Laplace’s equation (2.1.1) is called harmonic
$$
\Delta u=\nabla^2 u=\sum_{j=1}^n \frac{\partial^2 u}{\partial x_j^2}=0 .
$$
How can harmonic functions be constructed? That is, how can (2.1.1) be solved? For the onedimensional problem (i.e., $n=1$ ), it is evident that $u(x)=x+1$ satisfies $\frac{d^2 u}{d x^2}=0$. The same is true for any scalar multiple of $u(x)$. What happens when $n>1$ ? The development of harmonic functions by John [43] and Evans [23] leads to the consideration of the general case. Since the one-dimensional harmonic function is along the line $y=x+1$, it appears reasonable to guess that the $n$-dimensions harmonic functions are along the radial length $r=|\boldsymbol{x}|=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$. Observing that $\partial r / \partial x_j=x_j / r=x_j \cdot r^{-1}$ and $\partial^2 r / \partial x_j^2=1 / r-x_j^2 / r^3$, the Laplacian of $r$ takes the simple form $\Delta r=(n-1) / r$. Plainly, the Laplacian of the radial length is not equal to 0 and therefore $r=$ $|\boldsymbol{x}|$ is not a harmonic function. Rather than a linear function of $r$, assume that $u(\boldsymbol{x})=\psi(r)$ solves (2.1.1) for a sufficiently smooth function $\psi$ with non-zero derivatives. In this case,
$$
\frac{\partial u}{\partial x_j}=\psi^{\prime}(r) \frac{\partial r}{\partial x_j} \text { and } \frac{\partial^2 u}{\partial x_j^2}=\psi^{\prime}(r) \frac{\partial^2 r}{\partial x_j^2}+\psi^{\prime \prime}(r)\left(\frac{\partial r}{\partial x_j}\right)^2=\left(\frac{1}{r}-\frac{x_j^2}{r^3}\right) \psi^{\prime}(r)+\frac{x_j^2}{r^2} \psi^{\prime \prime}(r) \text {. }
$$

数学代写|偏微分方程代写partial difference equations代考|Green’s Theorem and the Modern Approach

Plainly, the radial variable $r=|\boldsymbol{x}|$ can be replaced by $r-r_o=\left|\boldsymbol{x}-\boldsymbol{x}o\right|$ in the family of harmonic functions (2.1.2) above. Hence, the function $\psi\left(r-r_o\right)$ is harmonic everywhere on $\Omega \subset \mathbb{R}^n$ except $\boldsymbol{x}=\boldsymbol{x}_o$. That is, $\psi\left(r-r_o\right)$ is harmonic for all $\boldsymbol{x} \in \Omega \backslash\left{\boldsymbol{x}_o\right}$. As noted in $\S 2.1$, the constants $b_n$ in (2.1.2) require Green’s Theorem to determine their value. Green’s Theorem: Let $u, v \in C^2(\bar{\Omega})$ on the domain $\Omega \subset \mathbb{R}^n$. Then (i) $\quad \int{\Omega} v \Delta u d \omega=\int_{\partial \Omega}\left(v \frac{\partial u}{\partial \boldsymbol{n}}-u \frac{\partial v}{\partial \boldsymbol{n}}\right) d \boldsymbol{\sigma}-\int_{\Omega} \nabla u \cdot \nabla v d \boldsymbol{\omega}$
(ii) $\int_{\Omega} v \Delta u d \omega=\int_{\partial \Omega}\left(v \frac{\partial u}{\partial \boldsymbol{n}}-u \frac{\partial v}{\partial \boldsymbol{n}}\right) d \sigma+\int_{\Omega} u \Delta v d \omega$ and
(iii) $\int_{\Omega} \nabla u \cdot \nabla u d \omega=\int_{\partial \Omega} u \frac{\partial u}{\partial \boldsymbol{n}} d \sigma-\int_{\Omega} u \Delta u d \omega$.
(Energy Identity)
The symbol $\int_{\Omega} f d \omega$ means the multi-dimensional integral of the function $f$ over the domain $\Omega$ with respect to the $n$-dimensional volume element $d \boldsymbol{\omega}$. Similarly, $\int_{\omega \Omega} f d \sigma$ is the integral over the domain boundary $\partial \Omega$ with respect to the $n$-dimensional surface element $d \boldsymbol{\sigma}$. Finally, $\boldsymbol{n}$ is the (outward) normal vector. To prove Green’s Theorem, recall the single field version of the Divergence Theorem (which can be found in any standard calculus text book). That is,

Divergence (Gauss-Green) Theorem: If $\psi \in C^1(\bar{\Omega})$ on the domain $\Omega \subset \mathbb{R}^n$, then $\int_{\Omega} \nabla \psi d \omega=$ $\int_\alpha \psi \cdot n d \sigma$

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|Laplace’s Equation and the Classical Approach

设$\Omega \subset \mathbb{R}^n$为边界为$\partial \Omega$的光滑域(即开放的连通集)。域$\Omega$的闭包$\bar{\Omega}$是闭集，闭集是该集合与其边界$\bar{\Omega}=\Omega \cup \partial \Omega$的并集。如果$\Omega$有界，则边界$\partial \Omega$是一条曲线(在$n=2$空间维度上)或曲面${ }^8(n \geq 3)$。如上所述，满足拉普拉斯方程(2.1.1)的函数$u \in C^2(\Omega)$称为谐波
$$
\Delta u=\nabla^2 u=\sum_{j=1}^n \frac{\partial^2 u}{\partial x_j^2}=0 .
$$
如何构造调和函数?也就是说，如何解决(2.1.1)?对于一维问题(即$n=1$)，显然$u(x)=x+1$满足$\frac{d^2 u}{d x^2}=0$。对于$u(x)$的任何标量倍数也是如此。当$n>1$ ?John[43]和Evans[23]对调和函数的发展导致了对一般情况的考虑。由于一维调和函数沿$y=x+1$线，因此似乎可以合理地猜测$n$维调和函数沿径向长度$r=|\boldsymbol{x}|=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$。观察到$\partial r / \partial x_j=x_j / r=x_j \cdot r^{-1}$和$\partial^2 r / \partial x_j^2=1 / r-x_j^2 / r^3$, $r$的拉普拉斯式采用简单的形式$\Delta r=(n-1) / r$。显然，径向长度的拉普拉斯函数不等于0，因此$r=$$|\boldsymbol{x}|$不是调和函数。假设$u(\boldsymbol{x})=\psi(r)$不是$r$的线性函数，而是求解具有非零导数的足够光滑的函数$\psi$(2.1.1)。在这种情况下，
$$
\frac{\partial u}{\partial x_j}=\psi^{\prime}(r) \frac{\partial r}{\partial x_j} \text { and } \frac{\partial^2 u}{\partial x_j^2}=\psi^{\prime}(r) \frac{\partial^2 r}{\partial x_j^2}+\psi^{\prime \prime}(r)\left(\frac{\partial r}{\partial x_j}\right)^2=\left(\frac{1}{r}-\frac{x_j^2}{r^3}\right) \psi^{\prime}(r)+\frac{x_j^2}{r^2} \psi^{\prime \prime}(r) \text {. }
$$

数学代写|偏微分方程代写partial difference equations代考|Green’s Theorem and the Modern Approach

显然，在上述调和函数族(2.1.2)中，径向变量$r=|\boldsymbol{x}|$可以用$r-r_o=\left|\boldsymbol{x}-\boldsymbol{x}o\right|$代替。因此，除了$\boldsymbol{x}=\boldsymbol{x}o$, $\psi\left(r-r_o\right)$函数在$\Omega \subset \mathbb{R}^n$上处处是谐波的。也就是说，$\psi\left(r-r_o\right)$是所有$\boldsymbol{x} \in \Omega \backslash\left{\boldsymbol{x}_o\right}$的谐波。如$\S 2.1$所述，(2.1.2)中的常数$b_n$需要格林定理来确定它们的值。格林定理，设$u, v \in C^2(\bar{\Omega})$在域$\Omega \subset \mathbb{R}^n$上。然后(i) $\quad \int{\Omega} v \Delta u d \omega=\int{\partial \Omega}\left(v \frac{\partial u}{\partial \boldsymbol{n}}-u \frac{\partial v}{\partial \boldsymbol{n}}\right) d \boldsymbol{\sigma}-\int_{\Omega} \nabla u \cdot \nabla v d \boldsymbol{\omega}$
(ii) $\int_{\Omega} v \Delta u d \omega=\int_{\partial \Omega}\left(v \frac{\partial u}{\partial \boldsymbol{n}}-u \frac{\partial v}{\partial \boldsymbol{n}}\right) d \sigma+\int_{\Omega} u \Delta v d \omega$和
(iii) $\int_{\Omega} \nabla u \cdot \nabla u d \omega=\int_{\partial \Omega} u \frac{\partial u}{\partial \boldsymbol{n}} d \sigma-\int_{\Omega} u \Delta u d \omega$。
(能源认同)
符号$\int_{\Omega} f d \omega$表示函数$f$在域$\Omega$上对$n$维体积元$d \boldsymbol{\omega}$的多维积分。类似地，$\int_{\omega \Omega} f d \sigma$是域边界$\partial \Omega$上对$n$维曲面元$d \boldsymbol{\sigma}$的积分。最后，$\boldsymbol{n}$是(向外)法向量。为了证明格林定理，回想一下散度定理的单场版本(它可以在任何标准微积分教科书中找到)。也就是说，

散度(高斯-格林)定理:如果$\psi \in C^1(\bar{\Omega})$在域$\Omega \subset \mathbb{R}^n$上，则 $\int_{\Omega} \nabla \psi d \omega=$ $\int_\alpha \psi \cdot n d \sigma$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|偏微分方程代写partial difference equations代考|LAPLACE’S EQUATION

Posted on 2023年5月11日2023年5月11日 by statistics-lab

数学代写|偏微分方程代写partial difference equations代考|LAPLACE’S EQUATION

One of the most important partial differential equations in physics and engineering applications is Laplace’s equation, given by $\nabla^2 u=u_{x x}+u_{y y}+u_{z z}=0$.

Here, $x, y, z$ are Cartesian coordinates in space. The expression $\nabla^2 u$ is called the Laplacian of $u$. The theory of the solutions of the Laplace equation is called a potential theory. Solutions that have continuous second partial derivatives are known as harmonic functions.

Laplace’s equation occurs mainly in gravitation, electrostatics, steady-state heat flow, and fluid flow.

Practical problems involving Laplace’s equation are boundary value problems in a region $T$ in space with boundary surface $S$. Such problems can be grouped into three types:
I. First boundary value problem or Dirichlet problem if $u$ is prescribed on $S$. II. Second boundary value problem or Neumann problem if the normal a derivative is prescribed on $S$.
III. Third or mixed boundary value problem or Robin problem if $u$ is prescribed on a portion of $S$ and the remaining portion of $S$.

In general, when we want to solve a boundary value problem, we have to first select the appropriate coordinates in which the boundary surface $S$ has a simple representation.

数学代写|偏微分方程代写partial difference equations代考|Laplacian in CyuIndrical Coordinates

The first step in solving a boundary value problem is generally the introduction of coordinates in which the boundary surface $S$ has a simple representation. Cylindrical symmetry (a cylinder as a region $T$ ) calls for cylindrical coordinates $r, \theta, z$ related to $x, y, z$ by $x=r \cos \theta, y=r \sin \theta, z=z$.
So, from the Laplace equation, we can write,
$$
\nabla^2 u=u_{r r}+\frac{1}{r} u_r+\frac{1}{r^2} u_{\theta \theta}+u_{z z}
$$

Spherical symmetry (a ball as region $\mathrm{T}$ bounded by a sphere $\mathrm{S}$ ) requires spherical coordinates $r, \theta, \phi$ related to $x, y, z$ by $x=r \cos \theta \sin \phi, y=r \sin \theta \sin \phi, z=r \cos \phi$.
Using the concept of chain rule, we obtain spherical coordinates
$$
\nabla^2 u=u_{r r}+\frac{2}{r} u_r+\frac{1}{r^2} u_{\phi \phi}+\frac{\cot \phi}{r^2} u_\phi+\frac{1}{r^2 \sin ^2 \phi} u_{\theta \theta}
$$
Sometimes, it can be written as
$$
\therefore \nabla^2 u=\frac{1}{r^2}\left[\frac{\partial}{\partial r} r^2\left(\frac{\partial u}{\partial r}\right)+\frac{1}{\sin \phi} \frac{\partial}{\partial \phi}\left(\sin \phi \frac{\partial u}{\partial \phi}\right)+\frac{1}{\sin ^2 \phi} \frac{\partial^2 u}{\partial \phi^2}\right] .
$$

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|LAPLACE’S EQUATION

在物理和工程应用中最重要的偏微分方程之一是拉普拉斯方程，由$\nabla^2 u=u_{x x}+u_{y y}+u_{z z}=0$给出。

这里$x, y, z$是空间中的笛卡尔坐标。表达式$\nabla^2 u$称为$u$的拉普拉斯式。拉普拉斯方程解的理论叫做势理论。具有连续二阶偏导数的解称为调和函数。

拉普拉斯方程主要出现在万有引力、静电学、稳态热流和流体流动中。

涉及拉普拉斯方程的实际问题是边界面为$S$的空间区域$T$中的边值问题。这类问题可分为三类:
1 .在$S$上规定$u$时，第一边值问题或Dirichlet问题。2第二边值问题或诺伊曼问题，如果法线导数规定在$S$。
3第三边值问题或混合边值问题或Robin问题，如果$S$的一部分和$S$的其余部分规定了$u$。

一般来说，当我们要解决边界值问题时，我们必须首先选择合适的坐标，其中边界表面$S$具有简单的表示。

数学代写|偏微分方程代写partial difference equations代考|Laplacian in CyuIndrical Coordinates

解决边界值问题的第一步通常是引入坐标，其中边界表面$S$具有简单的表示。柱对称(一个圆柱体作为一个区域$T$)需要通过$x=r \cos \theta, y=r \sin \theta, z=z$与$x, y, z$相关的柱坐标$r, \theta, z$。
根据拉普拉斯方程，我们可以写，
$$
\nabla^2 u=u_{r r}+\frac{1}{r} u_r+\frac{1}{r^2} u_{\theta \theta}+u_{z z}
$$

球对称(一个球作为一个区域$\mathrm{T}$，由一个球体$\mathrm{S}$包围)需要球坐标$r, \theta, \phi$与$x, y, z$通过$x=r \cos \theta \sin \phi, y=r \sin \theta \sin \phi, z=r \cos \phi$相关联。
利用链式法则的概念，我们得到了球坐标
$$
\nabla^2 u=u_{r r}+\frac{2}{r} u_r+\frac{1}{r^2} u_{\phi \phi}+\frac{\cot \phi}{r^2} u_\phi+\frac{1}{r^2 \sin ^2 \phi} u_{\theta \theta}
$$
有时，它可以写成
$$
\therefore \nabla^2 u=\frac{1}{r^2}\left[\frac{\partial}{\partial r} r^2\left(\frac{\partial u}{\partial r}\right)+\frac{1}{\sin \phi} \frac{\partial}{\partial \phi}\left(\sin \phi \frac{\partial u}{\partial \phi}\right)+\frac{1}{\sin ^2 \phi} \frac{\partial^2 u}{\partial \phi^2}\right] .
$$

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