## 数学代写|复杂网络代写complex networks代考|COS496

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex networks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。我们的专家在代写复杂网络complex networks方面经验极为丰富，各种代写复杂网络complex networks相关的作业也就用不着说。

## 数学代写|复杂网络代写complex networks代考|Martingales

A sequence of integrable random variables $\left{M(t): t \in \mathbb{Z}{+}\right}$is called adapted to an increasing family of $\sigma$-fields $\left{\mathcal{F}_t: t \in \mathbb{Z}{+}\right}$if $M(t)$ is $\mathcal{F}t$-measurable for each $t$. The sequence is called a martingale if $\mathrm{E}\left[M(t+1) \mid \mathcal{F}_t\right]=M(t)$ for all $t \in \mathbb{Z}{+}$, and a supermartingale if $\mathrm{E}\left[M(t+1) \mid \mathcal{F}t\right] \leq M(t)$ for $t \in \mathbb{Z}{+}$.

A martingale difference sequence $\left{Z(t): t \in \mathbb{Z}{+}\right}$is an adapted sequence of random variables such that the sequence $M(t)=\sum{i=0}^t Z(i), t \geq 0$, is a martingale.
The following result is basic:
Theorem 1.3.4. (Martingale Convergence Theorem) Let $M$ be a supermartingale, and suppose that
$$\sup t \mathrm{E}[|M(t)|]<\infty .$$ Then ${M(t)}$ converges to a finite limit with probability one. If ${M(t)}$ is a positive, real valued supermartingale then by the smoothing property of conditional expectations (1.10), $$\mathrm{E}[|M(t)|]=\mathrm{E}[M(t)] \leq \mathrm{E}[M(0)]<\infty, \quad t \in \mathbb{Z}{+}$$
Hence we have as a direct corollary to the Martingale Convergence Theorem
Theorem 1.3.5. A positive supermartingale converges to a finite limit with probability one.

Since a positive supermartingale is convergent, it follows that its sample paths are bounded with probability one. The following result gives an upper bound on the magnitude of variation of the sample paths of both positive supermartingales, and general martingales.

## 数学代写|复杂网络代写complex networks代考|Markov models

The Markov chains that we consider evolve on a countable state space, denoted X. The chain itself is denoted $\boldsymbol{X}=\left{X(t): t \in \mathbb{Z}_{+}\right}$, with transition law defined by the transition matrix $P$ :
$$\mathrm{P}{X(t+1)=y \mid X(0), \ldots, X(t)}=P(x, y), \quad x, y \in \mathrm{X} .$$

Examples of Markov chains include both the reflected and unreflected random walks defined in Section 1.3.3. The independence of the $\mathcal{E}$ guarantees the Markovian property (1.16).

The transition matrix is viewed as a (possibly infinite-dimensional) matrix. Likewise, a function $c: \mathrm{X} \rightarrow \mathbb{R}$ can be viewed as a column vector, and we can express conditional expectations as a matrix-vector product,
$$\mathrm{E}[c(X(t+1)) \mid X(t)=x]=P c(x):=\sum_{y \in \mathbf{X}} P(x, y) c(y), \quad x \in \mathrm{X} .$$
More generally, the matrix product is defined inductively by $P^0(x, y)=\mathbf{1}{{x=y}}$ and for $n \geq 1$, $$P^n(x, y)=\sum P(x, z) P^{n-1}(z, y), \quad x, y \in \mathbf{X} .$$ Based on this we obtain the representation, $$\mathrm{E}[c(X(t+n)) \mid X(t)=x]=P^n c(x), \quad x \in \mathrm{X}, t \geq 0, n \geq 1 .$$ Central to the theory of Markov chains is the following generalization, known as the strong Markov property. Recall that a random time $\tau$ is called a stopping time if there exists a sequence of functions $f_n: \mathrm{X}^{n+1} \rightarrow{0,1}, n \geq 0$, such that the event ${\tau=n}$ can be expressed as a function of the first $n$ samples of $\boldsymbol{X}$, $${\tau=n}=f_n(X(0), \ldots, X(n)), \quad n \geq 0 .$$ We write this as ${\tau=n} \in \mathcal{F}_n$, where $\left{\mathcal{F}_k: k \geq 0\right}$ is the filtration generated by $\boldsymbol{X}$. We let $\mathcal{F}\tau$ denote the $\sigma$-field generated by the events “before $\tau$ “: that is,
$$\mathcal{F}_\tau:=\left{A: A \cap{\tau \leq n} \in \mathcal{F}_n, n \geq 0\right}$$

## 数学代写|复杂网络代写complex networks代考|Martingales

$$\sup t \mathrm{E}[|M(t)|]<\infty .$$然后${M(t)}$收敛到一个概率为1的有限极限。如果${M(t)}$是一个正的实值上鞅，则根据条件期望(1.10)的平滑性质，$$\mathrm{E}[|M(t)|]=\mathrm{E}[M(t)] \leq \mathrm{E}[M(0)]<\infty, \quad t \in \mathbb{Z}{+}$$

## 数学代写|复杂网络代写complex networks代考|Markov models

$$\mathrm{P}{X(t+1)=y \mid X(0), \ldots, X(t)}=P(x, y), \quad x, y \in \mathrm{X} .$$

$$\mathrm{E}[c(X(t+1)) \mid X(t)=x]=P c(x):=\sum_{y \in \mathbf{X}} P(x, y) c(y), \quad x \in \mathrm{X} .$$

## 广义线性模型代考

statistics-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|复杂网络代写complex networks代考|SSIE641

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex networks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。我们的专家在代写复杂网络complex networks方面经验极为丰富，各种代写复杂网络complex networks相关的作业也就用不着说。

## 数学代写|复杂网络代写complex networks代考|Random walks

Random walks are used in this book to model the cumulative arrival process to a network, as well as cumulative service at a buffer. The reflected random walk is a model for storage and queueing systems.

Both are defined by taking successive sums of independent and identically distributed (i.i.d.) random variables.
Definition 1.3.1. Random Walks
Suppose that $\boldsymbol{X}=\left{X(k) ; k \in \mathbb{Z}{+}\right}$is a sequence of random variables defined by, $$X(k+1)=X(k)+\mathcal{E}(k+1), \quad k \in \mathbb{Z}{+}$$
where $X(0) \in \mathbb{R}$ is independent of $\mathcal{E}$, and the sequence $\mathcal{E}$ is i.i.d., taking values in $\mathbb{R}$. Then $\boldsymbol{X}$ is called a random walk on $\mathbb{R}$.
Suppose that the stochastic process $Q$ is defined by the recursion,
$$Q(k+1)=[Q(k)+\mathcal{E}(k+1)]{+}:=\max (0, Q(k)+\mathcal{E}(k+1)), \quad k \in \mathbb{Z}{+},$$
where again $Q(0) \in \mathbb{R}$, and $\mathcal{E}$ is an i.i.d. sequence of random variables taking values in $\mathbb{R}$. Then $Q$ is called the reflected random walk. .

Consider the following two models for comparison: For a fixed constant $a>0$, let $L^u$ denote the uniform distribution on the interval $[0, a]$, and $L^d$ the discrete distribution supported on the two points ${a / 3, a}$ with
$$L^d{a / 3}=1-L^d{a}=3 / 4 .$$

## 数学代写|复杂网络代写complex networks代考|Renewal processes

Renewal processes are used to model service-processes as well as arrivals to a network in standard books on queueing theory $[114,23]$. The general renewal process is defined as follows.
Definition 1.3.2. Renewal process
Let ${\mathcal{E}(1), \mathcal{E}(2), \ldots}$ be a sequence of independent and identical random variables with distribution function $\Gamma$ on $\mathbb{R}{+}$, and let $T$ denote the associated random walk defined by $T(n)=\mathcal{E}(1)+\cdots+\mathcal{E}(n), n \geq 1$, with $T(0)=0$. Then the (undelayed) renewal process is the continuous-time stochastic process, taking values in $\mathbb{Z}{+}$, defined by,
$$R(t)=\max {n: T(n) \leq t} .$$
The sample paths of a renewal process are piecewise constant, with jumps at the renewal times ${T(n): n \geq 1}$.

A renewal process $R$ takes on integer values and is non-decreasing, so that the quantity $R\left(t_1\right)-R\left(t_0\right), t_0, t_1 \in \mathbb{R}_{+}$, can be used to model the number of arrivals during the time-interval $\left(t_0, t_1\right]$, or the number of service completions for a server that is busy during this time-interval.

The most important example of a renewal process is the standard Poisson process, in which the process $\mathcal{E}$ has an exponential marginal distribution. The Poisson process is also another example of a stochastic process with independent increments, whose distribution is expressed as follows: For each $k \geq 0$ and $0 \leq t_0 \leq t_1<\infty$,
$$\mathrm{P}\left{R\left(t_1\right)-R\left(t_0\right)=k\right}=\frac{\left(\mu\left(t_1-t_0\right)\right)^k}{k !} e^{\mu\left(t_1-t_0\right)} .$$
Proposition 1.3.3 summarizes some basic results. More structure is described in Asmussen [23].

## 数学代写|复杂网络代写complex networks代考|Random walks

1.3.1.定义随机漫步

$$Q(k+1)=[Q(k)+\mathcal{E}(k+1)]{+}:=\max (0, Q(k)+\mathcal{E}(k+1)), \quad k \in \mathbb{Z}{+},$$

$$L^d{a / 3}=1-L^d{a}=3 / 4 .$$

## 数学代写|复杂网络代写complex networks代考|Renewal processes

1.3.2.定义更新流程

$$R(t)=\max {n: T(n) \leq t} .$$

$$\mathrm{P}\left{R\left(t_1\right)-R\left(t_0\right)=k\right}=\frac{\left(\mu\left(t_1-t_0\right)\right)^k}{k !} e^{\mu\left(t_1-t_0\right)} .$$

## 广义线性模型代考

statistics-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|复杂网络代写complex networks代考|CS-E5740

statistics-lab™ 为您的留学生涯保驾护航 在代写复杂网络complex networks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。我们的专家在代写复杂网络complex networks方面经验极为丰富，各种代写复杂网络complex networks相关的作业也就用不着说。

## 数学代写|复杂网络代写complex networks代考|Linear programs

In the theory of linear programming the standard primal problem is defined as the optimization problem,
$$\begin{array}{lrll} \max & c^{\mathrm{T}} x & & \ \text { s.t. } & \sum_j a_{i j} x_j & \leq b_i, & \text { for } i=1, \ldots, m ; \ & x_j & \geq 0, & \text { for } j=1, \ldots, n . \end{array}$$
Its dual is the linear program,
$$\begin{array}{lrl} \text { min } b^{\mathrm{T}} w & \ \text { s.t. } & \sum_j a_{j i} w_j \geq c_i, & \text { for } i=1, \ldots, n ; \ & w_j \geq 0, & \text { for } j=1, \ldots, m . \end{array}$$
The primal is usually written in matrix notation, $\max c^{\mathrm{T}} x$ subject to $A x \leq b, x \geq 0$; and the dual as $\min b^{\mathrm{T}} w$ subject to $A^{\mathrm{T}} w \geq c, w \geq 0$.

Any linear programming problem can be placed in the standard form (1.8). For example, a minimization problem can be reformulated as a maximization problem by changing the sign of the objective function. An equality constraint $y=b$ can be represented as two inequality constraints, $y \leq b$ and $-y \leq-b$. In the resulting dual one finds that the two corresponding variables can be replaced by one variable that is unrestricted in sign.

## 数学代写|复杂网络代写complex networks代考|Some Probability Theory

Until Part III this book requires little knowledge of advanced topics in probability. It is useful to outline some of this advanced material here since, for example, the Law of Large Numbers and the Central Limit Theorem for martingales and renewal processes serves as motivation for the idealized network models developed in Parts I and II.
The starting point of probability theory is the probability space, defined as the triple $(\Omega, \mathcal{F}, \mathrm{P})$ with $\Omega$ an abstract set of points, $\mathcal{F}$ a $\sigma$-field of subsets of $\Omega$, and $\mathrm{P}$ a probability measure on $\mathcal{F}$. A mapping $X: \Omega \rightarrow \mathrm{X}$ is called a random variable if
$$X^{-1}{B}:={\omega: X(\omega) \in B} \in \mathcal{F}$$
for all sets $B \in \mathcal{B}(\mathrm{X})$ : that is, if $X$ is a measurable mapping from $\Omega$ to $\mathrm{X}$.
Given a random variable $X$ on the probability space $(\Omega, \mathcal{F}, \mathrm{P})$, we define the $\sigma$ field generated by $X$, denoted $\sigma{X} \subseteq \mathcal{F}$, to be the smallest $\sigma$-field on which $X$ is measurable.

If $X$ is a random variable from a probability space $(\Omega, \mathcal{F}, \mathrm{P})$ to a general measurable space $(\mathrm{X}, \mathcal{B}(\mathrm{X}))$, and $h$ is a real valued measurable mapping from $(\mathrm{X}, \mathcal{B}(\mathrm{X}))$ to the real line $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ then the composite function $h(X)$ is a real-valued random variable on $(\Omega, \mathcal{F}, \mathrm{P})$ : note that some authors reserve the term “random variable” for such real-valued mappings. For such functions, we define the expectation as
$$\mathrm{E}[h(X)]=\int_{\Omega} h(X(\omega)) \mathrm{P}(d w)$$
The set of real-valued random variables $Y$ for which the expectation is well-defined and finite is denoted $L^1(\Omega, \mathcal{F}, \mathrm{P})$. Similarly, we use $L^{\infty}(\Omega, \mathcal{F}, \mathrm{P})$ to denote the collection of essentially bounded real-valued random variables $Y$; That is, those for which there is a bound $M$ and a set $A_M \subset \mathcal{F}$ with $\mathrm{P}\left(A_M\right)=0$ such that ${\omega:|Y(\omega)|>M} \subseteq A_M$.
Suppose that $Y \in L^1(\Omega, \mathcal{F}, \mathrm{P})$ and $\mathcal{G} \subset \mathcal{F}$ is a sub- $\sigma$-field of $\mathcal{F}$. If $\hat{Y} \in$ $L^1(\Omega, \mathcal{G}, \mathrm{P})$ and satisfies
$$\mathrm{E}[Y Z]=\mathrm{E}[\hat{Y} Z] \quad \text { for all } Z \in L_{\infty}(\Omega, \mathcal{G}, \mathrm{P})$$
then $\hat{Y}$ is called the conditional expectation of $Y$ given $\mathcal{G}$, and denoted $\mathrm{E}[Y \mid \mathcal{G}]$. The conditional expectation defined in this way exists and is unique (modulo P-null sets) for any $Y \in L^1(\Omega, \mathcal{F}, \mathrm{P})$ and any sub $\sigma$-field $\mathcal{G}$.
Suppose now that we have another $\sigma$-field $\mathcal{H} \subset \mathcal{G} \subset \mathcal{F}$. Then
$$\mathrm{E}[Y \mid \mathcal{H}]=\mathrm{E}[\mathrm{E}[Y \mid \mathcal{G}] \mid \mathcal{H}] .$$

## 数学代写|复杂网络代写complex networks代考|Linear programs

$$\begin{array}{lrll} \max & c^{\mathrm{T}} x & & \ \text { s.t. } & \sum_j a_{i j} x_j & \leq b_i, & \text { for } i=1, \ldots, m ; \ & x_j & \geq 0, & \text { for } j=1, \ldots, n . \end{array}$$

$$\begin{array}{lrl} \text { min } b^{\mathrm{T}} w & \ \text { s.t. } & \sum_j a_{j i} w_j \geq c_i, & \text { for } i=1, \ldots, n ; \ & w_j \geq 0, & \text { for } j=1, \ldots, m . \end{array}$$

## 数学代写|复杂网络代写complex networks代考|Some Probability Theory

$$X^{-1}{B}:={\omega: X(\omega) \in B} \in \mathcal{F}$$

$$\mathrm{E}[h(X)]=\int_{\Omega} h(X(\omega)) \mathrm{P}(d w)$$

$$\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)} .$$

$$\frac{X^{\prime \prime}(x)}{X(x)}=-\frac{Y^{\prime \prime}(y)}{Y(y)}=\lambda .$$

\begin{aligned} & X^{\prime \prime}=\lambda X, \ & Y^{\prime \prime}=-\lambda Y . \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|MATH3403

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 $t \in[a, b]$.

## 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 $\mathbf{y} \in C^m(\mathbb{R})$.

Proof. Since $\mathbf{F}(s, \mathbf{y}(s))$ is continuous, we can use the Fundamental Theorem of Calculus to deduce that the right-hand side of (8.173) is continuously differentiable, so the left-hand side must be as well, and
$$\mathbf{y}^{\prime}(t)=\mathbf{F}(t, \mathbf{y}(t)) \text {. }$$
Thus, $\mathbf{y} \in C^1(\mathbb{R})$. If $\mathbf{F}$ is in $C^1$, we can repeat this process by noting that the right-hand side of (1.31) is differentiable (so the left-hand side is as well) and
$$\mathbf{y}^{\prime \prime}(t)=\mathbf{F}_{\mathbf{y}}(t, \mathbf{y}(t)) \cdot \mathbf{y}^{\prime}(t)+\mathbf{F}_t(t, \mathbf{y}(t)),$$
so $\mathbf{y} \in C^2(\mathbb{R})$. This can be repeated as long as we can take further continuous derivatives of $\mathbf{F}$. We conclude that, in general, $\mathbf{y}$ has one order of differentiablity more than $\mathbf{F}$.

# 偏微分方程代写

## 数学代写|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

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

$$\mathbf{y}^{\prime}(t)=\mathbf{F}(t, \mathbf{y}(t)) \text {. }$$

$$\mathbf{y}^{\prime \prime}(t)=\mathbf{F}_{\mathbf{y}}(t, \mathbf{y}(t)) \cdot \mathbf{y}^{\prime}(t)+\mathbf{F}_t(t, \mathbf{y}(t)),$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|偏微分方程代写partial difference equations代考|MATH476

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|Well-Posed Problems

We say that a problem is well-posed (in the sense of Hadamard) if

there exists a solution,

the solution is unique,

the solution depends continuously on the data.
If these conditions do not hold, a problem is said to be ill-posed. Of course, the meaning of the term continuity with respect to the data has to be made more precise by a choice of norms in the context of each problem considered.
In the course of this book we classify most of the problems we encounter as either well-posed or ill-posed, but the reader should avoid the assumption that well-posed problems are always “better” or more “physically realistic” than ill-posed problems. As we saw in the problem of buckling of a beam mentioned above, there are times when the conditions of a well-posed problem (uniqueness in this case) are physically unrealistic. The importance of ill-posedness in nature was stressed long ago by Maxwell [Max]:

For example, the rock loosed by frost and balanced on a singular point of the mountain-side, the little spark which kindles the great forest, the little word which sets the world afighting, the little scruple which prevents a man from doing his will, the little spore which blights all the potatoes, the little gemmule which makes us philosophers or idiots. Every existence above a certain rank has its singular points: the higher the rank, the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being may produce results of the greatest importance. All great results produced by human endeavour depend on taking advantage of these singular states when they occur.
We draw attention to the fact that this statement was made a full century before people “discovered” all the marvelous things that can be done with cubic surfaces in $\mathbb{R}^3$.

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

There is one way of proving existence of a solution to a problem that is more satisfactory than all others: writing the solution explicitly. In addition to the aesthetic advantages provided by a representation for a solution there are many practical advantages. One can compute, graph, observe, estimate, manipulate and modify the solution by using the formula. We examine below some representations for solutions that are often useful in the study of PDEs.
Variation of parameters
Variation of parameters is a formula giving the solution of a nonhomogeneous linear system of ODEs (1.13) in terms of solutions of the homogeneous problem (1.15). Although this representation has at least some utility in terms of actually computing solutions, its primary use is analytical.

The key to the variations of constants formula is the construction of a fundamental solution matrix $\Phi(t, \tau) \in \mathbb{R}^{n \times n}$ for the linear homogeneous system. This solution matrix satisfies
\begin{aligned} \frac{d}{d t} \Phi(t, \tau) & =\mathbf{A}(t) \Phi(t, \tau), \ \Phi(\tau, \tau) & =\mathbf{I}, \end{aligned}
where $\mathbf{I}$ is the $n \times n$ identity matrix. The proof of existence of the fundamental matrix is standard and is left as an exercise. Note that the unique solution of the initial-value problem (1.15), (1.14) for the homogeneous system is given by
$$\mathbf{y}(t):=\Phi\left(t, t_0\right) \mathbf{y}_0 .$$

The use of Leibniz’ formula reveals that the variation of parameters formula
$$\mathbf{y}(t):=\Phi\left(t, t_0\right) \mathbf{y}0+\int{t_0}^t \Phi(t, s) \mathbf{f}(s) d s$$
gives the solution of the initial-value problem (1.13), (1.14) for the nonhomogeneous system.
Cauchy’s integral formula
Cauchy’s integral formula is the most important result in the theory of complex variables. It provides a representation for analytic functions in terms of its values at distant points. Note that this representation is rarely used to actually compute the values of an analytic function; rather it is used to deduce a variety of theoretical results.

# 偏微分方程代写

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

\begin{aligned} \frac{d}{d t} \Phi(t, \tau) & =\mathbf{A}(t) \Phi(t, \tau), \ \Phi(\tau, \tau) & =\mathbf{I}, \end{aligned}

$$\mathbf{y}(t):=\Phi\left(t, t_0\right) \mathbf{y}_0 .$$

$$\mathbf{y}(t):=\Phi\left(t, t_0\right) \mathbf{y}0+\int{t_0}^t \Phi(t, s) \mathbf{f}(s) d s$$

## 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|常微分方程代写ordinary differential equation代考|MAT-328

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation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。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

## 数学代写|常微分方程代写ordinary differential equation代考|Linear Second Order ODEs

The general form of such equations is, according to the introduction (see e.g.(15))
$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=b(x),$$
where $a_0, a_1, a_2, b$ are real functions defined on a real interval $I \subseteq \Re$. We may consider these functions continuous on $I$.
If $a_0(x) \neq 0, \forall x \in I$, we can divide both members of (1.2.1) by it, thus getting an equation whose leading coefficient is 1

$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=f(x)$$
where we used the notations $p(x)=\frac{a_1(x)}{a_0(x)}, q(x)=\frac{a_2(x)}{a_0(x)}, f(x)=\frac{b(x)}{a_0(x)}$. Obviously, if the coefficients of (1.2.1) are of class $\mathrm{C}^0(I)$, so are $p, q$ and $f$.
We see that, if $a_0(x)=0, \forall x \in I$, the equation is no more of second order, and, at the points at which $a_0(x)=0$, it has singularities. For the moment, we shall not deal with such situations, such that we consider that the given equation may be brought to the form (1.2.2).
Let us denote by
$$\mathrm{L} y \equiv y^{\prime \prime}+p(x) y^{\prime}+q(x) y .$$
The operator $\mathrm{L}$ is defined on $\mathrm{C}^2(I)$, with range in $\mathrm{C}^0(I)$, and we can easily prove that it is linear.
The kernel of this operator is a subset of $\mathrm{C}^2(I)$, containing functions cancelled by $\mathrm{L}$
$$\operatorname{ker} \mathrm{L}=\left{y \in \mathrm{C}^2(I) \mid \mathrm{L} y=0\right} .$$

## 数学代写|常微分方程代写ordinary differential equation代考|HOMOGENEOUS EQUATIONS

Let us take the associated to (1.2.1) homogeneous equation
$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=0 .$$
If we know a particular solution of this equation, say $Y(x)$, we can completely solve (1.2.7). Indeed, let us perform the change of function
$$y(x)=z(x) Y(x),$$
$z(x)$ being the new unknown function. Replacing this in (1.2.7), we get

$$a_0(x) Y z^{\prime \prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] z^{\prime}+\left[a_0(x) Y^{\prime \prime}+a_1(x) Y^{\prime}+a_2(x) Y\right] z=0 .$$
As $Y$ is a solution of (1.2.7), it follows that $u=z^{\prime}$ must satisfy
$$a_0(x) Y u^{\prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] u=0 ;$$
this is a linear first order ODE.
We conclude that if we know a particular solution, we can reduce the order of the given equation by one unit.
Suppose now that $Y_1(x)$ is a known particular solution of the homogeneous equation, associated to (1.2.2)
$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$
and suppose moreover that $Y_1$ does not vanish on $I$. Using the change of function $y=Y_1 z$, we find that $u=z^{\prime}$ must satisfy
$$u^{\prime}+\left(2 \frac{Y_1^{\prime}(x)}{Y_1(x)}+p(x)\right) u=0,$$
i.e., a linear first order homogeneous ordinary differential equation. According to Sec.1.2, it allows the general integral
$$u(x)=C_1 \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)},$$
where $\int p(x) \mathrm{d} x$ is a primitive of $p(x)$ and $C_1$ is an arbitrary constant. Getting back to $y$, we deduce
$$y(x)=C_1 Y_1(x) \int \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)} \mathrm{d} x .$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Linear Second Order ODEs

$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=b(x),$$

$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=f(x)$$

$$\mathrm{L} y \equiv y^{\prime \prime}+p(x) y^{\prime}+q(x) y .$$

$$\operatorname{ker} \mathrm{L}=\left{y \in \mathrm{C}^2(I) \mid \mathrm{L} y=0\right} .$$

## 数学代写|常微分方程代写ordinary differential equation代考|HOMOGENEOUS EQUATIONS

$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=0 .$$

$$y(x)=z(x) Y(x),$$
$z(x)$是新的未知函数。在(1.2.7)中替换它，我们得到

$$a_0(x) Y z^{\prime \prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] z^{\prime}+\left[a_0(x) Y^{\prime \prime}+a_1(x) Y^{\prime}+a_2(x) Y\right] z=0 .$$

$$a_0(x) Y u^{\prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] u=0 ;$$

$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$

$$u^{\prime}+\left(2 \frac{Y_1^{\prime}(x)}{Y_1(x)}+p(x)\right) u=0,$$

$$u(x)=C_1 \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)},$$

$$y(x)=C_1 Y_1(x) \int \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)} \mathrm{d} x .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|M-544

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation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。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

## 数学代写|常微分方程代写ordinary differential equation代考|THE METHOD OF VARIATION OF PARAMETERS

Except for $Y(x)$, formula (1.1.14) refers only to the coefficients of (1.1.1). Lagrange remarked that $Y(x)$ can be obtained in terms of these coefficients if we search it under the form
$$Y(x)=C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x},$$
that is, shaping it according to the general solution of the associated to (1.1.1) homogeneous equation. Introducing this in (1.1.1) yields
$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}-p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}+p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$
from which we deduce that $C(x)$ must satisfy
$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$
$$C^{\prime}(x)=f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} .$$
This is an equation considered at Sec.1.1. It follows that the general integral of (1.1.18) is written in the form
$$C(x)=K+\int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x$$
In this expression, $\mathrm{K}$ is an arbitrary constant and the integral in the right member is a primitive of the function $f(x) \mathrm{e}^{\int p(x) \mathrm{d} x}$. Actually, we don’t need the general solution of (1.1.18) for our purpose; all we need is a particular solution, which can be found giving to $K$ an arbitrarily chosen value, e.g. $K=0$. With this, we get
$$Y(x)=\mathrm{e}^{-\int p(x) \mathrm{d} x} \int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x .$$

## 数学代写|常微分方程代写ordinary differential equation代考|DIFFERENTIAL POLYNOMIALS

Let us denote by $\mathrm{D}$ the operator indicating the derivative of first order of a function

$$\mathrm{D} \equiv \frac{\mathrm{d}}{\mathrm{d} x}$$
and by E the identity
$$\mathrm{E} y=y$$
Then L may be also expressed as
$$\mathrm{L} y=\mathrm{P}1(x, \mathrm{D}) y, \quad \mathrm{P}_1(x, \mathrm{D}) \equiv \mathrm{D}+p(x) \mathrm{E} .$$ The operator defined in (1.1.29) is a formal polynomial of first order in D and it is called a differential polynomial. Let now $\mathbf{y}=\left\lfloor y_j\right\rfloor{j=1, n}, \mathbf{f}=\left\lfloor f_j\right\rfloor_{j=1, n}$ be vector functions and assume that we must solve the vector equation
$$\mathrm{Ly} \equiv \dot{\mathbf{y}}+p(x) \mathbf{y}=\mathbf{f}, \quad p \in \mathrm{C}^0(I), \mathbf{f} \in\left(\mathrm{C}^0(I)\right)^n .$$
Writing (1.1.30) componentwisely, this means, in fact, that one has to solve $n$ uncoupled ODEs
$$\mathrm{L} y_j \equiv \dot{y}_j+p(x) y_j=f_j, \quad j=\overline{1, n} .$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|THE METHOD OF VARIATION OF PARAMETERS

$$Y(x)=C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x},$$

$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}-p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}+p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$

$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$

$$C^{\prime}(x)=f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} .$$

$$C(x)=K+\int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x$$

$$Y(x)=\mathrm{e}^{-\int p(x) \mathrm{d} x} \int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x .$$

## 数学代写|常微分方程代写ordinary differential equation代考|DIFFERENTIAL POLYNOMIALS

$$\mathrm{D} \equiv \frac{\mathrm{d}}{\mathrm{d} x}$$
E表示恒等式
$$\mathrm{E} y=y$$

$$\mathrm{L} y=\mathrm{P}1(x, \mathrm{D}) y, \quad \mathrm{P}1(x, \mathrm{D}) \equiv \mathrm{D}+p(x) \mathrm{E} .$$(1.1.29)中定义的算子是D中的一阶形式多项式，称为微分多项式。现在让$\mathbf{y}=\left\lfloor y_j\right\rfloor{j=1, n}, \mathbf{f}=\left\lfloor f_j\right\rfloor{j=1, n}$是向量函数假设我们必须解向量方程
$$\mathrm{Ly} \equiv \dot{\mathbf{y}}+p(x) \mathbf{y}=\mathbf{f}, \quad p \in \mathrm{C}^0(I), \mathbf{f} \in\left(\mathrm{C}^0(I)\right)^n .$$

$$\mathrm{L} y_j \equiv \dot{y}_j+p(x) y_j=f_j, \quad j=\overline{1, n} .$$

## 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|常微分方程代写ordinary differential equation代考|CRN18324

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation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。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

## 数学代写|常微分方程代写ordinary differential equation代考|EQUATIONS OF THE FORM $y^{\prime}=f(x)$

This is the simplest form of (1.1.1). The solutions of this equation may be obviously regarded as primitives of $f$. Consequently, its general solution (integral) is
$$y(x)=\int f(x) d x+C,$$
where $\int f(x) d x$ is one of the primitives of $f$ and $C$ is an arbitrary constant. The representation (1.1.2) is obviously obtained by integrating both members of $y^{\prime}=f(x)$. If we wish to get the solution passing through the point $\left(x_0, y_0\right)$, where $x_0 \in I$, then it is convenient to choose $\int_{x_0}^x f(\xi) \mathrm{d} \xi$ among the primitives of $f$. Indeed, with this choice, the solution passes through $\left(x_0, y_0\right)$ if
$$C+\int_{x_0}^{x_0} f(\xi) \mathrm{d} \xi=y_0$$
therefore if $C=y_0$. This yields
$$y(x)=\int_{x_0}^x f(\xi) \mathrm{d} \xi+y_0 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|THE LINEAR HOMOGENEOUS EQUATION

This equation is also a particular case of (1.1.1), where the free term is identically null, that is
$$y^{\prime}+p(x) y=0$$
Dividing by $y$ both terms of this equation, we immediately get
$$\frac{\mathrm{d}}{\mathrm{d} x}(\ln |y|)=-p(x) .$$
This means that $\ln |y|$ satisfies an equation of the previously considered type. Thus, the general solution of (1.1.6) is, by using directly (1.1.2),
$$\ln |y|=\widetilde{C}-\int p(x) \mathrm{d} x,$$
where $\widetilde{C}$ is an arbitrary constant and $\int p(x) \mathrm{d} x$ – one of the primitives of $p$. From (1.1.7) we see that $y$ is the general solution of (1.1.5) and is expressed by
$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x},$$
with $C$ arbitrary constant.

Let us get back to the equation (1.1.1), in which the functions $f$ and $p$, defined on $I \subseteq \Re$, are not identically null. Suppose that we know a particular solution of (1.1.1), $Y(x)$ say, and let us perform the change of function
$$y(x)=z(x)+Y(x) .$$
Introducing this in (1.1.1) immediately involves
$$z^{\prime}+p(x) z+Y^{\prime}+p(x) Y=f(x) ;$$
thus, $z$ satisfies the homogeneous equation
$$z^{\prime}+p(x) z=0$$
which was studied at Sec.1.2 and whose general solution is
$$z(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x} .$$
Getting back to (1.1.10), we see that the general solution of (1.1.1) may be expressed in the form
$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x}+Y(x),$$
where $Y(x)$ is a particular solution of the non-homogeneous equation (1.1.1). This form is very important, as it is characteristic for linear ODEs in general; we shall discuss it further.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|EQUATIONS OF THE FORM $y^{\prime}=f(x)$

$$y(x)=\int f(x) d x+C,$$

$$C+\int_{x_0}^{x_0} f(\xi) \mathrm{d} \xi=y_0$$

$$y(x)=\int_{x_0}^x f(\xi) \mathrm{d} \xi+y_0 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|THE LINEAR HOMOGENEOUS EQUATION

$$y^{\prime}+p(x) y=0$$

$$\frac{\mathrm{d}}{\mathrm{d} x}(\ln |y|)=-p(x) .$$

$$\ln |y|=\widetilde{C}-\int p(x) \mathrm{d} x,$$

$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x},$$

$$y(x)=z(x)+Y(x) .$$

$$z^{\prime}+p(x) z+Y^{\prime}+p(x) Y=f(x) ;$$

$$z^{\prime}+p(x) z=0$$

$$z(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x} .$$

$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x}+Y(x),$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|运筹学作业代写Operational Research代考|TIE2110

statistics-lab™ 为您的留学生涯保驾护航 在代写运筹学operational research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。我们的专家在代写运筹学operational research代写方面经验极为丰富，各种代写运筹学operational research相关的作业也就用不着说。

## 数学代写|运筹学作业代写operational research代考|Encoding and evaluating the network reliability by $B D D$

The K-terminal network reliability function can be represented by a boolean function $f$ defined as follows:
$$\left{\begin{array}{l} f\left(x_1, x_2, \ldots, x_m\right)=1 \text { if nodes in } K \text { are linked by edges } e_i \text { with } x_i=1 \ f\left(x_1, x_2, \ldots, x_m\right)=0 \text { otherwise } \end{array}\right.$$
where boolean variable $x_i$ stands for the state of the link $e_i(1 \leq i \leq m)$. For instance, the boolean formula encoded by the BDD structure in figure 3 is:
$$x_1\left(\bar{x}_2\left(\bar{x}_3 x_4 x_5 x_6+x_3\left(\bar{x}_4 x_5 x_6+x_4\right)\right)+x_2\left(\bar{x}_4 x_5 x_6+x_4\right)\right)+\bar{x}_1 x_2\left(x_3\left(\bar{x}_4 x_5 x_6+x_4\right)+\bar{x}_3 x_5 x_6\right)$$
Our aim is to encode this reliability function by BDD. The algorithm is developed in Section 3.3. In figure 3(b), we explain the definition of BDD through an example of BDD representing the K-terminal reliability of network $\mathrm{G}$ (see figure

2). The BDD can represent the SDP implicitly avoiding huge storage for large number of SDP. A useful property of BDD is that all the paths from the root to the leaves are disjoint. If $f$ represents the system reliability expression, based on this property, the K-terminal network reliability $R_K$ of $G$ can be recursively evaluated by:
\begin{aligned} & \forall i \in{1, \ldots, m}: \ & R_K(p ; G)=\operatorname{Pr}(f=1) \ & R_K(p ; G)=\operatorname{Pr}\left(x_i \cdot f_{x_i=1}=1\right)+\operatorname{Pr}\left(\bar{x}i, f{x_i=0}=1\right) \ & R_K(p ; G)=p_i \cdot \operatorname{Pr}\left(f_{x_i=1}=1\right)+q_i \cdot \operatorname{Pr}\left(f_{x_i=0}=1\right) \ & \end{aligned}
with $p=\left(p_1, \ldots, p_m\right)$.
For instance, in figure $3(\mathrm{~b})$, the K-terminal network reliability is then defined as follows:
$$R_K(p ; G)=p_1\left(q_2\left(q_3 p_4 p_5 p_6+p_3\left(q_4 p_5 p_6+p_4\right)\right)+p_2\left(q_4 p_5 p_6+p_4\right)\right)+q_1 p_2\left(p_3\left(q_4 p_5 p_6+p_4\right)+q_3 p_5 p_6\right)$$
The next section presents our BDD-based algorithm for the K-terminal network reliability problem.

## 数学代写|运筹学作业代写operational research代考|Construction of the $B D D$ representing the $K$-terminal reliability function

We remind that the order of the variables is very important for BDD generation (see Section 2). Time and space complexity of BDD closely depend on variable ordering. This paper is not concerned with this kind of problem and we use a breadth-first-search (BFS) ordering.
In short, our algorithm follows three steps:

• 1 The edges are ordered by using a heuristic.
• 2 The BDD is generated to encode the network reliability. The following shows the construction of the BDD encoding the K-terminal network reliability.
• 3 From this BDD structure, we obtain the K-terminal network reliabilities (whatever $p_i, i \in[1 \ldots m]$ ) as shown in the previous section.

The top-down construction process can be represented as a binary tree such that the root corresponds to the original graph $G$ and children correspond to graphs obtained by deletion /contraction of edges. Nodes in the binary tree correspond to subgraphs of $G$. At the root, we consider the edge $e_1$, construct the subgraph $G_{-1}$, that is $G$ with $e_1$ deleted and the subgraph $G_{* 1}$ that is $G$ with $e_1$ contracted. Then at the second step, from $G_{-1}$, we construct $G_{-1-2}$ where $e_2$ is deleted and $G_{-1 * 2}$ where $e_2$ is contracted and so on from each created subgraphs until the vertices of $K$ are fully connected or at least one vertex of $K$ is disconnected. There are $2^n$ possible states and isomorphic graphs appear in the computation process. For the graph $G$ pictured in Fig. 2, its subgraphs $G_{* 1 * 2}$ and $G_{-1 * 2 * 3}$ are isomorphic. Our aim is to provide an efficient method in order to avoid redundant computation due to the appearance of isomorphic subproblems during the process. We use the method introduced by Carlier and Lucet ${ }^{15}$ for representing graph by partition which is an efficient way for solving this kind of problem. By identifying the isomorphic subgraphs an expansion tree is modified as a rooted acyclic graph which is a BDD (see figure $3(\mathrm{~b})$ ).

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|Encoding and evaluating the network reliability by $B D D$

k端网络可靠性函数可以用布尔函数$f$表示，定义如下:
$$\left{\begin{array}{l} f\left(x_1, x_2, \ldots, x_m\right)=1 \text { if nodes in } K \text { are linked by edges } e_i \text { with } x_i=1 \ f\left(x_1, x_2, \ldots, x_m\right)=0 \text { otherwise } \end{array}\right.$$

$$x_1\left(\bar{x}_2\left(\bar{x}_3 x_4 x_5 x_6+x_3\left(\bar{x}_4 x_5 x_6+x_4\right)\right)+x_2\left(\bar{x}_4 x_5 x_6+x_4\right)\right)+\bar{x}_1 x_2\left(x_3\left(\bar{x}_4 x_5 x_6+x_4\right)+\bar{x}_3 x_5 x_6\right)$$

2). BDD可以隐式地表示SDP，避免大量SDP占用巨大的存储空间。BDD的一个有用的性质是从根到叶的所有路径都是不相交的。若$f$表示系统可靠性表达式，则根据该性质，$G$的k端网络可靠性$R_K$可递归求出:
\begin{aligned} & \forall i \in{1, \ldots, m}: \ & R_K(p ; G)=\operatorname{Pr}(f=1) \ & R_K(p ; G)=\operatorname{Pr}\left(x_i \cdot f_{x_i=1}=1\right)+\operatorname{Pr}\left(\bar{x}i, f{x_i=0}=1\right) \ & R_K(p ; G)=p_i \cdot \operatorname{Pr}\left(f_{x_i=1}=1\right)+q_i \cdot \operatorname{Pr}\left(f_{x_i=0}=1\right) \ & \end{aligned}

$$R_K(p ; G)=p_1\left(q_2\left(q_3 p_4 p_5 p_6+p_3\left(q_4 p_5 p_6+p_4\right)\right)+p_2\left(q_4 p_5 p_6+p_4\right)\right)+q_1 p_2\left(p_3\left(q_4 p_5 p_6+p_4\right)+q_3 p_5 p_6\right)$$

## 数学代写|运筹学作业代写operational research代考|Construction of the $B D D$ representing the $K$-terminal reliability function

2生成BDD对网络可靠性进行编码。BDD编码k端网络可靠性的构造如下图所示。

## 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。