数学代写|偏微分方程代写partial difference equations代考|MAT412

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|Laplace’s Equation

Perhaps the most important of all partial differential equations is
$$\Delta u:=u_{x_1 x_1}+u_{x_2 x_2}+\cdots+u_{x_n x_n}=0,$$
known as Laplace’s equation. You will find applications of it to problems in gravitation, elastic membranes, electrostatics, fluid flow, steady-state heat conduction and many other topics in both pure and applied mathematics.
As the remarks of the last section on ODEs indicated, the choice of boundary conditions is of paramount importance in determining the wellposedness of a given problem. The following two common types of boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^n$ yield well-posed problems and will be studied in a more general context in later chapters.
Dirichlet conditions. Given a function $f: \partial \Omega \rightarrow \mathbb{R}$, we require
$$u(\mathbf{x})=f(\mathbf{x}), \quad \mathbf{x} \in \partial \Omega .$$
In the context of elasticity, $u$ denotes a change of position, so Dirichlet boundary conditions are often referred to as displacement conditions.
Neumann conditions. Given a function $f: \partial \Omega \rightarrow \mathbb{R}$, we require
$$\frac{\partial u}{\partial n}(\mathbf{x})=f(\mathbf{x}), \quad \mathbf{x} \in \partial \Omega .$$
Here $\frac{\partial u}{\partial n}$ is the partial derivative of $u$ with respect to the unit outward normal of $\partial \Omega, \mathbf{n}$. In linear elasticity $\frac{\partial u}{\partial n}(\mathbf{x})=\nabla u(\mathbf{x}) \cdot \mathbf{n}(\mathbf{x})$ can be interpreted as a force, so Neumann boundary conditions are often referred to as traction boundary conditions.

We have been intentionally vague about the smoothness required of $\partial \Omega$ and $f$, and the function space in which we wish $u$ to lie. These are central areas of concern in later chapters.

数学代写|偏微分方程代写partial difference equations代考|Solution by separation of variables

The first method we present for solving Laplace’s equation is the most widely used technique for solving partial differential equations: separation of variables. The technique involves reducing a partial differential equation to a system of ordinary differential equations and expressing the solution of the PDE as a sum or infinite series.

Let us consider the following Dirichlet problem on a square in the plane. Let
$$\Omega=\left{(x, y) \in \mathbb{R}^2 \mid 0<x<1, \quad 0<y<1\right} .$$
We wish to find a function $u: \bar{\Omega} \rightarrow \mathbb{R}$ satisfying Laplace’s equation
$$u_{x x}+u_{y y}=0$$

at each point in $\Omega$ and satisfying the boundary conditions
\begin{aligned} & u(0, y)=0, \ & u(1, y)=0, \ & u(x, 0)=0, \ & u(x, 1)=f(x) . \end{aligned}
The key to separation of variables is to look for solutions of (1.36) of the form
$$u(x, y)=X(x) Y(y) .$$
When we put a function of this form into (1.36), the partial derivatives in the differential equation appear as ordinary derivatives on the functions $X$ and $Y$; i.e., (1.36) becomes
$$X^{\prime \prime}(x) Y(y)+X(x) Y^{\prime \prime}(y)=0 .$$
At any point $(x, y)$ at which $u$ is nonzero we can divide this equation by $u$ and rearrange to get
$$\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)} .$$
We now argue as follows: Since the right side of the equation does not depend on the variable $x$, neither can the left side; likewise, since the left side does not depend on $y$, neither does the right side. The only function on the plane that is independent of both $x$ and $y$ is a constant, so we must have
$$\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)}=\lambda .$$
This gives us
\begin{aligned} & X^{\prime \prime}=\lambda X, \ & Y^{\prime \prime}=-\lambda Y . \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|Laplace’s Equation

$$\Delta u:=u_{x_1 x_1}+u_{x_2 x_2}+\cdots+u_{x_n x_n}=0,$$

$$u(\mathbf{x})=f(\mathbf{x}), \quad \mathbf{x} \in \partial \Omega .$$

$$\frac{\partial u}{\partial n}(\mathbf{x})=f(\mathbf{x}), \quad \mathbf{x} \in \partial \Omega .$$

数学代写|偏微分方程代写partial difference equations代考|Solution by separation of variables

$$\Omega=\left{(x, y) \in \mathbb{R}^2 \mid 0<x<1, \quad 0<y<1\right} .$$

$$u_{x x}+u_{y y}=0$$

\begin{aligned} & u(0, y)=0, \ & u(1, y)=0, \ & u(x, 0)=0, \ & u(x, 1)=f(x) . \end{aligned}

$$u(x, y)=X(x) Y(y) .$$

$$X^{\prime \prime}(x) Y(y)+X(x) Y^{\prime \prime}(y)=0 .$$

$$\begin{array}{r} \int_0^{\infty} \int_{-\infty}^{\infty}\left[\mathbf{u}(x, t) \cdot \phi_t(x, t)+\mathbf{f}(\mathbf{u}(x, t)) \cdot \phi_x(x, t)\right] d x d t \ +\int_{-\infty}^{\infty} \mathbf{u}_0(x) \phi(x, 0) d x=0 \end{array}$$

$$C_0^1\left(\mathbb{R}^{2+}\right):=\left{\phi \in C^1\left(\mathbb{R}^{2+}\right) \mid \exists r>0 \text { s.t. } \operatorname{supp} \phi \subset B_r((0,0)) \cap \mathbb{R}^{2+}\right} .$$

数学代写|偏微分方程代写partial difference equations代考|The Rankine-Hugoniot Condition

$\mathbf{u}$ 是(3.5)在$N^l$和$N^r$的经典解，

$\mathbf{u}$ 在曲线$C$处经历跳跃不连续$[\mathbf{u}]$，且

$$s[\mathbf{u}]=[\mathbf{f}(\mathbf{u})]$$

$$\mathbf{u}:=\mathbf{u}^r(\mathbf{p})-\mathbf{u}^l(\mathbf{p}):=\lim {\left(x^r, t^r\right) \rightarrow \mathbf{p}} \mathbf{p}\left(x^r, t^r\right)-\lim {\left(x^l, t^l\right) \stackrel{l}{\rightarrow} \mathbf{p}} \mathbf{u}\left(x^l, t^l\right),$$

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|Math3357

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|A PDE without Solutions

Every now and then a paper appears with a title like “A method to solve all partial differential equations.” The content of such papers is always very far from satisfying the claims made in the title. It is rumored that a paper of this kind inspired Lewy to construct his famous example of a linear PDE which has no solutions at all. This example also highlights the importance of analyticity in the Cauchy-Kovalevskaya result.
Theorem 2.25. For a complex-valued function $u(x, y, z)$, let
$$L u=-u_x-i u_y+2 i(x+i y) u_z .$$
Then there is a real-valued function $f(x, y, z)$, of class $C^{\infty}\left(\mathbb{R}^3\right)$, such that the equation
$$L u=f(x, y, z)$$
has no solutions of class $C^1(\Omega)$ in any open subset $\Omega \subset \mathbb{R}^3$.
We note that when $f$ is analytic, the Cauchy-Kovalevskaya theorem applies and noncharacteristic initial-value problems for (2.104) have local solutions. In contrast, for nonanalytic $f$ there may be no solutions, even if no initial conditions are prescribed.

We shall not give a full proof of the theorem, but outline some of the main ideas. First, we shall prove the following lemma.

Lemma 2.26. Let $\psi \in C^{\infty}(\mathbb{R})$ be real-valued and such that $\psi$ is not real analytic at $z_0$. Then the equation
$$L u=\psi^{\prime}(z)$$
has no solution of class $C^1$ in any neighborhood of the point $\left(0,0, z_0\right)$.

Proof. Assume the contrary and let $u$ be a solution in a neighborhood of $\left(0,0, z_0\right)$, say for $x^2+y^2<\epsilon,\left|z-z_0\right|<\epsilon$. We set
$$v(r, \theta, z)=e^{i \theta} \sqrt{r} u(\sqrt{r} \cos \theta, \sqrt{r} \sin \theta, z) .$$
After some algebra, we find that $v$ satisfies the equation
$$-2 v_r-\frac{i}{r} v_\theta+2 i v_z=\psi^{\prime}(z) .$$

数学代写|偏微分方程代写partial difference equations代考|An Outline of the Main Idea

Consider a system of linear equations
$$a_{i j}^k(\mathbf{x}) \frac{\partial u_j}{\partial x_k}+b_{i j}(\mathbf{x}) u_j=0, \quad i=1, \ldots, N .$$
Let $\mathbf{u}=\left(u_1, \ldots, u_N\right)$ be a solution in a “lens-shaped” domain $\Omega \subset \mathbb{R}^n$ bounded by two surfaces $S$ and $Z$. Assume that $\mathbf{u}=0$ on $Z$ and that $S$ is noncharacteristic and analytic. We also assume that the coefficients in (2.121) are analytic.

Let $v_i, i=1, \ldots, N$ be arbitrary functions in $C^1(\bar{\Omega})$. We multiply the $i$ th equation of (2.121) by $v_i$, sum over $i$, and integrate over $\Omega$. This yields
\begin{aligned} 0 & =\int_{\Omega} v_i(\mathbf{x}) a_{i j}^k(\mathbf{x}) \frac{\partial u_j}{\partial x_k}(\mathbf{x})+v_i(\mathbf{x}) b_{i j}(\mathbf{x}) u_j(\mathbf{x}) d \mathbf{x} \ & =\int_{\Omega}-\frac{\partial}{\partial x_k}\left[v_i(\mathbf{x}) a_{i j}^k(\mathbf{x})\right] u_j(\mathbf{x})+v_i(\mathbf{x}) b_{i j}(\mathbf{x}) u_j(\mathbf{x}) d \mathbf{x} \ & +\int_{\partial \Omega} a_{i j}^k(\mathbf{x}) v_i(\mathbf{x}) u_j(\mathbf{x}) n_k d S, \end{aligned}
where $\mathbf{n}$ is the outer normal to $\partial \Omega$.
Assume now that $\mathbf{v}$ satisfies the “adjoint” system of PDEs,
$$-\frac{\partial}{\partial x_k}\left(a_{i j}^k v_i\right)+b_{i j} v_i=0, \quad j=1, \ldots, N,$$
with initial conditions
$$v_i=f_i$$
on $S$. Then $(2.122)$ reduces to
$$0=\int_S a_{i j}^k(\mathbf{x}) f_i(\mathbf{x}) u_j(\mathbf{x}) n_k d S .$$

Assume now that $\mathbf{v}$ satisfies the “adjoint” system of PDEs,
$$-\frac{\partial}{\partial x_k}\left(a_{i j}^k v_i\right)+b_{i j} v_i=0, \quad j=1, \ldots, N,$$
with initial conditions
$$v_i=f_i$$
on $S$. Then (2.122) reduces to
$$0=\int_S a_{i j}^k(\mathbf{x}) f_i(\mathbf{x}) u_j(\mathbf{x}) n_k d S .$$

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|A PDE without Solutions

$$L u=-u_x-i u_y+2 i(x+i y) u_z .$$

$$L u=f(x, y, z)$$

$$L u=\psi^{\prime}(z)$$

$$v(r, \theta, z)=e^{i \theta} \sqrt{r} u(\sqrt{r} \cos \theta, \sqrt{r} \sin \theta, z) .$$

$$-2 v_r-\frac{i}{r} v_\theta+2 i v_z=\psi^{\prime}(z) .$$

数学代写|偏微分方程代写partial difference equations代考|An Outline of the Main Idea

$$a_{i j}^k(\mathbf{x}) \frac{\partial u_j}{\partial x_k}+b_{i j}(\mathbf{x}) u_j=0, \quad i=1, \ldots, N .$$

\begin{aligned} 0 & =\int_{\Omega} v_i(\mathbf{x}) a_{i j}^k(\mathbf{x}) \frac{\partial u_j}{\partial x_k}(\mathbf{x})+v_i(\mathbf{x}) b_{i j}(\mathbf{x}) u_j(\mathbf{x}) d \mathbf{x} \ & =\int_{\Omega}-\frac{\partial}{\partial x_k}\left[v_i(\mathbf{x}) a_{i j}^k(\mathbf{x})\right] u_j(\mathbf{x})+v_i(\mathbf{x}) b_{i j}(\mathbf{x}) u_j(\mathbf{x}) d \mathbf{x} \ & +\int_{\partial \Omega} a_{i j}^k(\mathbf{x}) v_i(\mathbf{x}) u_j(\mathbf{x}) n_k d S, \end{aligned}

$$-\frac{\partial}{\partial x_k}\left(a_{i j}^k v_i\right)+b_{i j} v_i=0, \quad j=1, \ldots, N,$$

$$v_i=f_i$$

$$0=\int_S a_{i j}^k(\mathbf{x}) f_i(\mathbf{x}) u_j(\mathbf{x}) n_k d S .$$

$$-\frac{\partial}{\partial x_k}\left(a_{i j}^k v_i\right)+b_{i j} v_i=0, \quad j=1, \ldots, N,$$

$$v_i=f_i$$

$$0=\int_S a_{i j}^k(\mathbf{x}) f_i(\mathbf{x}) u_j(\mathbf{x}) n_k d S .$$

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|MATH402

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|Scalar Equations of Second Order

Let us consider a second-order PDE in two space dimensions,
\begin{aligned} L u= & a(x, y) u_{x x}+b(x, y) u_{x y}+c(x, y) u_{y y} \ & +d(x, y) u_x+e(x, y) u_y+f(x, y) u \ = & g(x, y) . \end{aligned}

The principal part of the symbol of $L$ is
$$L^p(x, y ; i \xi, i \eta)=-a(x, y) \xi^2-b(x, y) \xi \eta-c(x, y) \eta^2 .$$
Second-order PDEs are classified according to the behavior of $L^p$, viewed as a quadratic form in $\xi$ and $\eta$. The quadratic form given by (2.14) can be represented in matrix form as
$$L^p(x, y ; i \xi, i \eta)=(\xi, \eta)\left(\begin{array}{cc} -a(x, y) & -\frac{1}{2} b(x, y) \ -\frac{1}{2} b(x, y) & -c(x, y) \end{array}\right)\left(\begin{array}{l} \xi \ \eta \end{array}\right) .$$
Recall that a quadratic form is called definite if the associated symmetric matrix is (positive or negative) definite, it is called indefinite if the matrix has eigenvalues of both signs, and it is called degenerate if the matrix is singular.

Definition 2.3. The differential equation (2.13) is called elliptic if the quadratic form given by (2.14) is strictly definite, hyperbolic if it is indefinite and parabolic if it is degenerate.

The terms elliptic, parabolic and hyperbolic are motivated by the analogy with the classification of conic sections.

Example 2.4. Laplace’s equation is elliptic, the heat equation is parabolic and the wave equation is hyperbolic. For these three cases, the matrices associated with the principal part of the symbol are
$$\left(\begin{array}{cc} -1 & 0 \ 0 & -1 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 0 \ 0 & 1 \end{array}\right) \text { and }\left(\begin{array}{cc} -1 & 0 \ 0 & 1 \end{array}\right)$$
respectively.

数学代写|偏微分方程代写partial difference equations代考|Higher-Order Equations and Systems

The generalization of the definitions above to equations of higher order than second is straightforward.

Definition 2.8. Let $L$ be the $m$ th-order operator defined in (2.9). Characteristic surfaces are defined by the equation
$$L^p(\mathbf{x}, \nabla \phi)=0$$
An equation is called elliptic at $\mathbf{x}$ if there are no real characteristics at $\mathbf{x}$ or, equivalently, if
$$L^p(\mathbf{x}, i \boldsymbol{\xi}) \neq 0, \quad \forall \boldsymbol{\xi} \neq 0 .$$
An equation is called strictly hyperbolic ${ }^1$ in the direction $\mathbf{n}$ if

$L^p(\mathbf{x}, i \mathbf{n}) \neq 0$, and

all the roots $\omega$ of the equation
$$L^p(\mathbf{x}, i \boldsymbol{\xi}+i \omega \mathbf{n})=0$$
are real and distinct for every $\boldsymbol{\xi} \in \mathbb{R}^n$ which is not collinear with $\mathbf{n}$.
In applications, $\mathbf{n}$ is usually a coordinate direction associated with time. In this case, let us set $\mathbf{x}=\left(x_1, x_2, \ldots, x_{n-1}, t\right)$ and let $\boldsymbol{\xi}=\left(\xi_1, \ldots, \xi_{n-1}, 0\right)$ be a spatial vector.

For rapidly oscillating functions of small support, we may think of the coefficients of $L^p$ as approximately constant; let us assume they are constant. If $\omega$ is a root of $(2.30)$, then $u=\exp (i(\boldsymbol{\xi} \cdot \mathbf{x})+i \omega t)$ is a solution of $L^p u=0$. If $\omega$ has negative imaginary part, then this solution grows exponentially in time. Moreover, since $L^p$ is homogeneous of degree $m$, i.e., $L^p(\mathbf{x}, \lambda(i \boldsymbol{\xi}+i \omega \mathbf{n}))=\lambda^m L^p(\mathbf{x}, i \boldsymbol{\xi}+i \omega \mathbf{n})$ for any scalar $\lambda$, there are always roots with negative imaginary parts if there are any roots which are not real (if we change the sign of $\boldsymbol{\xi}$, we also change the sign of $\omega$ ). Moreover, if we multiply $\boldsymbol{\xi}$ by a scalar factor $\lambda$, then $\omega$ is multiplied by the same factor, and hence solutions would grow more and more rapidly the faster they oscillate in space. The condition that the roots in (2.30) are real is therefore a necessary condition for well-posedness of initial-value problems.
We now turn our attention to systems of $k$ partial differential equations involving $k$ unknowns $u_j, j=1,2, \ldots, k$ :
$$L_{i j}(\mathbf{x}, D) u_j=0, \quad i=1,2, \ldots, k$$

偏微分方程代写

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

\begin{aligned} L u= & a(x, y) u_{x x}+b(x, y) u_{x y}+c(x, y) u_{y y} \ & +d(x, y) u_x+e(x, y) u_y+f(x, y) u \ = & g(x, y) . \end{aligned}

$L$符号的主体部分是
$$L^p(x, y ; i \xi, i \eta)=-a(x, y) \xi^2-b(x, y) \xi \eta-c(x, y) \eta^2 .$$

$$L^p(x, y ; i \xi, i \eta)=(\xi, \eta)\left(\begin{array}{cc} -a(x, y) & -\frac{1}{2} b(x, y) \ -\frac{1}{2} b(x, y) & -c(x, y) \end{array}\right)\left(\begin{array}{l} \xi \ \eta \end{array}\right) .$$

2.3.定义如果由式(2.14)给出的二次形式是严格确定的，则称微分方程(2.13)为椭圆型，如果是不定的，则称为双曲型，如果是退化的，则称为抛物线型。

$$\left(\begin{array}{cc} -1 & 0 \ 0 & -1 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 0 \ 0 & 1 \end{array}\right) \text { and }\left(\begin{array}{cc} -1 & 0 \ 0 & 1 \end{array}\right)$$

数学代写|偏微分方程代写partial difference equations代考|Higher-Order Equations and Systems

2.8.定义设$L$为(2.9)中定义的$m$第th阶运算符。特征曲面由方程定义
$$L^p(\mathbf{x}, \nabla \phi)=0$$

$$L^p(\mathbf{x}, i \boldsymbol{\xi}) \neq 0, \quad \forall \boldsymbol{\xi} \neq 0 .$$

$L^p(\mathbf{x}, i \mathbf{n}) \neq 0$，和

$$L^p(\mathbf{x}, i \boldsymbol{\xi}+i \omega \mathbf{n})=0$$

$$L_{i j}(\mathbf{x}, D) u_j=0, \quad i=1,2, \ldots, k$$以上翻译结果来自有道神经网络翻译（YNMT）· 通用场景

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|M-541

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|Estimation

When we speak of an estimate for a solution we refer to a relation that gives an indication of the solution’s size or character. Most often these are inequalities involving norms of the solution. We distinguish between the following two types of estimate. An a posteriori estimate depends on knowledge of the existence of a solution. This knowledge is usually obtained through some sort of construction or explicit representation. An a priori estimate is one that is conditional on the existence of the solution; i.e., a result of the form, “If a solution of the problem exists, then it satisfies …” We present here an example of each type of estimate.
Gronwall’s inequality and energy estimates
In this section we derive an a priori estimate for solutions of ODEs that is related to the energy estimates for PDEs that we examine in later chapters. The uniqueness theorem 1.4 is an immediate consequence of this result. To derive our estimate we need a fundamental inequality called Gronwall’s inequality.
Lemma 1.10 (Gronwall’s inequality). Let
\begin{aligned} & u:[a, b] \rightarrow[0, \infty), \ & v:[a, b] \rightarrow \mathbb{R}, \end{aligned}
be continuous functions and let $C$ be a constant. Then if
$$v(t) \leq C+\int_a^t v(s) u(s) d s$$
for $t \in[a, b]$, it follows that
$$v(t) \leq C \exp \left(\int_a^t u(s) d s\right)$$
for $t \in[a, b]$.
The proof of this is left as an exercise.

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

One of the most important modern techniques for proving the existence of a solution to a partial differential equation is the following process.

1. Convert the original PDE into a “weak” form that might conceivably have very rough solutions.
2. Show that the weak problem has a solution.
3. Show that the solution of the weak equation actually has more smoothness than one would have at first expected.
4. Show that a “smooth” solution of the weak problem is a solution of the original problem.

We give a preview of parts one, two, and four of this process in Section 1.2.1 below, but in this section let us consider precursors of the methods for part three: showing smoothness.
Smoothness of solutions of ODEs
The following is an example of a “bootstrap” proof of regularity in which we use the fact that $\mathbf{y} \in C^0$ to show that $\mathbf{y} \in C^1$, etc. Note that this result can be used to prove the regularity portion of Theorem 1.1 (which asserted the existence of a $C^1$ solution).

Theorem 1.13. If $\mathbf{F}: \mathbb{R} \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ is in $C^{m-1}\left(\mathbb{R} \times \mathbb{R}^n\right)$ for some integer $m \geq 1$, and $\mathbf{y} \in C^0(\mathbb{R})$ satisfies the integral equation
$$\mathbf{y}(t)=\mathbf{y}\left(t_0\right)+\int_{t_0}^t \mathbf{F}(s, \mathbf{y}(s)) d s,$$
then in fact $\mathbf{y} \in C^m(\mathbb{R})$.

偏微分方程代写

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

\begin{aligned} & u:[a, b] \rightarrow[0, \infty), \ & v:[a, b] \rightarrow \mathbb{R}, \end{aligned}

$$v(t) \leq C+\int_a^t v(s) u(s) d s$$

$$v(t) \leq C \exp \left(\int_a^t u(s) d s\right)$$

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

ode解的光滑性

$$\mathbf{y}(t)=\mathbf{y}\left(t_0\right)+\int_{t_0}^t \mathbf{F}(s, \mathbf{y}(s)) d s,$$

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。

数学代写|偏微分方程代写partial difference equations代考|Green’s Function for the Wave Equation

In this section we shall show how the solution of the space form of the wave equation under certain boundary conditions can be made to depend on the determination of the appropriate Green’s function.
Suppose that $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ satisfies the equation
$$\left(\frac{\partial^2}{\partial x^{\prime 2}}+\frac{\partial^2}{\partial y^{\prime 2}}+\frac{\partial}{\partial z^{\prime 2}}\right) G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)+k^2 G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=0$$
and that it is finite and continuous with respect to either the variables $x, y, z$ or to the variables $x^{\prime}, y^{\prime}, z^{\prime}$ for points $\mathbf{r}, \mathbf{r}^{\prime}$ belonging to a region $V$ which is bounded by a closed surface $S$ except in the neighborhood of the point $r$, where it has a singularity of the same type as
$$\frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$$
as $\mathbf{r}^{\prime} \rightarrow \mathbf{r}$. Then proceeding as in the derivation of equation (4) of the last section, we can prove that, if $\mathbf{r}$ is the position vector of a point within $V$, then
$$\Psi(\mathbf{r})=\frac{1}{4 \pi} \int_S\left{G\left(\mathbf{r}, \mathbf{r}^{\prime}\right) \frac{\partial \Psi\left(\mathbf{r}^{\prime}\right)}{\partial n}-\Psi\left(\mathbf{r}^{\prime}\right) \frac{\partial G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)}{\partial n}\right} d S^{\prime}$$
where $\mathbf{n}$ is the outward-drawn normal to the surface $S$.
It follows immediately from equation (3) that if $G_1\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ is such a function and if it satisfies the boundary condition
$$G_1\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=0$$
if the point with position vector $\mathbf{r}^{\prime}$ lies on $S$, then
$$\Psi(\mathbf{r})=-\frac{1}{4 \pi} \int_S \Psi\left(\mathbf{r}^{\prime}\right) \frac{\partial G_1\left(\mathbf{r}, \mathbf{r}^{\prime}\right)}{\partial n} d S^{\prime}$$
by means of which the value of $\Psi$ at any point $\mathbf{r}$ within $S$ can be calculated in terms of the values of $\Psi$ on the boundary.

数学代写|偏微分方程代写partial difference equations代考|The Nonhomogeneous Wave Equation

The second-order hyperbolic equation
$$\llcorner\psi=f(\mathbf{r}, t)$$
where
$$\mathrm{L}=\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2$$
which arises in electromagnetic theory and other branches of mathematical physics is called the nonhomogeneous wave equation. It is readily seen that if $\psi_1$ is any solution of the nonhomogeneous equation (1) and $\psi_2$ is any solution of the wave equation, then
$$\psi=\psi_1+\psi_2$$
is also a solution of equation (1).
Suppose that a function $\psi$ satisfies equation (1) in the finite region bounded by a closed surface $S$ and that we wish to find the value of the function at a point $P$, with position vector $\mathbf{r}$, which lies within $S$. If we denote by $\Omega$ the region bounded by $S$ and the sphere $C$ of center $P$ and small radius $\varepsilon$, we may write Green’s theorem in the form
$$\int_{\Omega}\left(\psi \nabla^2 \phi-\phi \nabla^2 \psi\right) d \tau^{\prime}=\left(\int_C+\int_S\right)\left(\psi \frac{\partial \phi}{\partial n}-\phi \frac{\partial \psi}{\partial n}\right) d S^{\prime}$$
where the normals $\mathbf{n}$ are in the directions shown in Fig. 23. In equation (4) we take $\psi\left(\mathbf{r}^{\prime}\right)$ to be a solution of equation (1), so that
$$\nabla^2 \psi\left(\mathbf{r}^{\prime}\right)=\frac{1}{c^2} \frac{\partial^2}{\partial t^2} \psi\left(\mathbf{r}^{\prime}\right)-f\left(\mathbf{r}^{\prime}, t\right)$$
and assume that
$$\phi=\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} F\left(t-t^{\prime}+\frac{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}{c}\right)$$
where $t^{\prime}$ is a constant and the function $F$ is arbitrary. It follows that $\mathrm{L} \phi=0$, so that
$$\nabla^2 \phi=\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2}$$
Substituting from equation (5) and (7) into equation (4), we find that
\begin{aligned} \frac{1}{c^2} \frac{\partial}{\partial t} \int_{\Omega}\left(\psi \frac{\partial \phi}{\partial t}-\phi \frac{\partial \psi}{\partial t}\right) d \tau^{\prime} & +\int_{\Omega} f\left(\mathbf{r}^{\prime}, t\right) \phi d \tau^{\prime} \ & =\left(\int_C+\int_S\right)\left(\psi \frac{\partial \phi}{\partial n}-\phi \frac{\partial \psi}{\partial n}\right) d S^{\prime} \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|Green’s Function for the Wave Equation

$$\left(\frac{\partial^2}{\partial x^{\prime 2}}+\frac{\partial^2}{\partial y^{\prime 2}}+\frac{\partial}{\partial z^{\prime 2}}\right) G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)+k^2 G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=0$$

$$\frac{e^{i k\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$$

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

$$G_1\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=0$$

$$\Psi(\mathbf{r})=-\frac{1}{4 \pi} \int_S \Psi\left(\mathbf{r}^{\prime}\right) \frac{\partial G_1\left(\mathbf{r}, \mathbf{r}^{\prime}\right)}{\partial n} d S^{\prime}$$

数学代写|偏微分方程代写partial difference equations代考|The Nonhomogeneous Wave Equation

$$\llcorner\psi=f(\mathbf{r}, t)$$

$$\mathrm{L}=\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2$$

$$\psi=\psi_1+\psi_2$$

$$\int_{\Omega}\left(\psi \nabla^2 \phi-\phi \nabla^2 \psi\right) d \tau^{\prime}=\left(\int_C+\int_S\right)\left(\psi \frac{\partial \phi}{\partial n}-\phi \frac{\partial \psi}{\partial n}\right) d S^{\prime}$$

$$\nabla^2 \psi\left(\mathbf{r}^{\prime}\right)=\frac{1}{c^2} \frac{\partial^2}{\partial t^2} \psi\left(\mathbf{r}^{\prime}\right)-f\left(\mathbf{r}^{\prime}, t\right)$$

$$\phi=\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} F\left(t-t^{\prime}+\frac{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}{c}\right)$$

$$\nabla^2 \phi=\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2}$$

\begin{aligned} \frac{1}{c^2} \frac{\partial}{\partial t} \int_{\Omega}\left(\psi \frac{\partial \phi}{\partial t}-\phi \frac{\partial \psi}{\partial t}\right) d \tau^{\prime} & +\int_{\Omega} f\left(\mathbf{r}^{\prime}, t\right) \phi d \tau^{\prime} \ & =\left(\int_C+\int_S\right)\left(\psi \frac{\partial \phi}{\partial n}-\phi \frac{\partial \psi}{\partial n}\right) d S^{\prime} \end{aligned}

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|The index of a mapping

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|The index of a mapping

In this section we transfer the index-sum formula from the case $n=2$ to the situation of arbitrary dimensions. In this context we derive that the degree of mapping gives us an integer. We begin with the easy

Proposition 1. Let $\Omega_j \subset \mathbb{R}^n$ for $j=1,2$ denote two bounded open disjoint sets and $\Omega:=\Omega_1 \cup \Omega_2$ their union. Furthermore, let $f(x) \in A^0(\Omega)$ represent a continuous mapping with the property
$$f(x) \neq 0 \quad \text { for all points } \quad x \in \partial \Omega_1 \cup \partial \Omega_2 .$$
Then we have the identity
$$d(f, \Omega)=d\left(f, \Omega_1\right)+d\left(f, \Omega_2\right)$$
Proof: When we choose the quantity $\varepsilon>0$ sufficiently small, we obtain $|f(x)|>\varepsilon$ for all points $x \in \partial \Omega_1 \cup \partial \Omega_2$. Furthermore, we have a sequence of functions $\left{f_k\right}_{k=1,2, \ldots} \subset A^1(\Omega)$ satisfying $f_k \rightarrow f$ uniformly on $\bar{\Omega}$ as well as $\left|f_k(x)\right|>\varepsilon$ for all points $x \in \partial \Omega_1 \cup \partial \Omega_2$ and all indices $k \geq k_0$. Now we utilize the admissible test function $\omega \in C_0^0((0, \varepsilon), \mathbb{R})$ with the property $\int_{\mathbb{R}^n} \omega(|y|) d y=1$, and we easily see the following equation for all indices $k \geq k_0$ :
\begin{aligned} d\left(f_k, \Omega\right) & =\int_{\Omega} \omega\left(\left|f_k(x)\right|\right) J_{f_k}(x) d x \ & =\int_{\Omega_1} \omega\left(\left|f_k(x)\right|\right) J_{f_k}(x) d x+\int_{\Omega_2} \omega\left(\left|f_k(x)\right|\right) J_{f_k}(x) d x \ & =d\left(f_k, \Omega_1\right)+d\left(f_k, \Omega_2\right) . \end{aligned}
This implies the desired identity $d(f, \Omega)=d\left(f, \Omega_1\right)+d\left(f, \Omega_2\right) \quad$ q.e.d.

数学代写|偏微分方程代写partial difference equations代考|The product theorem

Let the function $f \in A^1(\Omega)$ with $0<\varepsilon<\inf {x \in \partial \Omega}|f(x)|$ be given. Furthermore, we take an admissible test function $\omega \in C_0^0((0, \varepsilon), \mathbb{R})$ satisfying $$\int{\mathbb{R}^n} \omega(|y|) d y=1$$
Then we have the identity
$$\int_{\Omega} \omega(|f(x)|) J_f(x) d x=d(f, \Omega) \int_{\mathbb{R}^n} \omega(|y|) d y .$$
Now we shall generalize this identity to the class of arbitrary test functions $\varphi \in C_0^0\left(\mathbb{R}^n \backslash f(\partial \Omega), \mathbb{R}\right)$. Then we utilize this result to determine the degree of mapping $d(g \circ f, \Omega, z)$ for a composed function $g \circ f$ with the generators $f, g \in C^0\left(\mathbb{R}^n\right)$, and we obtain the so-called product theorem.

Definition 1. Let $\mathcal{O} \subset \mathbb{R}^n$ denote an open set and assume $x \in \mathcal{O}$. Then we call the following set
$$G_x:=\left{y \in \mathcal{O}: \begin{array}{l} \text { There exists a path } \varphi(t):[0,1] \rightarrow \mathcal{O} \in C^0([0,1]) \ \text { satisfying } \varphi(0)=x, \varphi(1)=y \end{array}\right}$$
the connected component of $x$ in $\mathcal{O}$.

Remarks:

The connected component $G_x$ represents the largest open connected subset of $\mathcal{O}$ which contains the point $x$.

When we consider two connected components with $G_x$ and $G_y$, only the alternative $G_x \cap G_y=\emptyset$ or $G_x=G_y$ is possible.

偏微分方程代写

数学代写|偏微分方程代写partial difference equations代考|The index of a mapping

$$f(x) \neq 0 \quad \text { for all points } \quad x \in \partial \Omega_1 \cup \partial \Omega_2 .$$

$$d(f, \Omega)=d\left(f, \Omega_1\right)+d\left(f, \Omega_2\right)$$

\begin{aligned} d\left(f_k, \Omega\right) & =\int_{\Omega} \omega\left(\left|f_k(x)\right|\right) J_{f_k}(x) d x \ & =\int_{\Omega_1} \omega\left(\left|f_k(x)\right|\right) J_{f_k}(x) d x+\int_{\Omega_2} \omega\left(\left|f_k(x)\right|\right) J_{f_k}(x) d x \ & =d\left(f_k, \Omega_1\right)+d\left(f_k, \Omega_2\right) . \end{aligned}

数学代写|偏微分方程代写partial difference equations代考|The product theorem

$$\int_{\Omega} \omega(|f(x)|) J_f(x) d x=d(f, \Omega) \int_{\mathbb{R}^n} \omega(|y|) d y .$$

$$G_x:=\left{y \in \mathcal{O}: \begin{array}{l} \text { There exists a path } \varphi(t):[0,1] \rightarrow \mathcal{O} \in C^0([0,1]) \ \text { satisfying } \varphi(0)=x, \varphi(1)=y \end{array}\right}$$
$\mathcal{O}$中$x$的连接组件。

有限元方法代写

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MATLAB代写

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