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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|Elements of calculus of variations

In this section we introduce the concept of space-time functionals. In the abstract sense, we mean a mapping or a correspondence that assigns a definite real number to each function belonging to some class or space. Space-time functionals are naturally functions of spatial coordinates as well as time. Such functionals play an important role in obtaining approximations of the theoretical solutions of IVPs encountered in mathematical physics, science, and engineering. In this section we study elements of the calculus of variations, a branch of mathematics that deals with extremums of spacetime functionals, i.e. maximums, saddle points, and minimums. We establish a correspondence between the solutions of IVPs and the extremums of the space-time functionals that are constructed using the IVPs. Therein lies our interest in studying the elements of the calculus of variations associated with space-time functionals. There are four basic lemmas that are important in this regard. We state these and provide their proofs.

Lemma 2.1 (Fundamental Lemma). If $\alpha(x, t)$ is continuous on $\bar{\Omega}{x t}=$ $\Omega{x t} \cup \Gamma$ and if
$$\int_{\Omega_{x t}} \alpha(x, t) h(x, t) d \Omega_{x t}=0 \quad \forall h(x, t) \in H^{(1)}: \quad h(\Gamma)=0$$
Then
$$\alpha(x, t)=0 \quad \forall(x, t) \in \bar{\Omega}{x t}$$ Proof. We construct the proof of this lemma by contradiction. Suppose $\alpha(x, t)$ is nonzero, say positive at some point in $\bar{\Omega}{x t}$. If we let
$$h(x, t)=\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right)$$
for some $(x, t) \in\left[x_{1}, x_{2}\right] \times\left[t_{1}, t_{2}\right]$ and $h(x, t)=0$ otherwise, then $h(x, t)$ satisfies the conditions of the lemma. However, we have
$$\int_{\bar{\Omega}{x t}} \alpha(x, t) h(x, t) d \Omega{x t}=\int_{t_{1} x_{1}}^{t_{2} x_{2}} \int_{x_{1}} \alpha(x)\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right) d x d t>0$$
Since the integrand is positive (except at $x_{1}, x_{2}, t_{1}, t_{2}$ ), this contradiction proves the lemma.

## 数学代写|有限元方法代写Finite Element Method代考|Concept of variation of a space-time functional

Variation means change or, in the sense of calculus, differential. Let $I(y)$ with $y=y(x, t)$ be a functional defined over some normed linear space and let
$$\Delta I(h)=I(y+h)-I(y)$$
be increment in $I$ corresponding to an increment $h=h(x, t)$ of the dependent variable $y=y(x, t)$. If $y$ is fixed, then $\Delta I(h)$ is a function of $h$, in general a non-linear functional. Suppose that
$$\Delta I(h)=\Phi(h)+\epsilon|h|$$
where $\Phi(h)$ is a linear functional and $\epsilon|h| \rightarrow 0$ as $|h| \rightarrow 0$. Then the functional $I$ is said to be differentiable and the principal linear part of the increment $\Delta I(h)$, i.e. the linear functional $\Phi(h)$ which differs from $\Delta I(h)$ by an infinitesimal of order higher than one relative to $|h|$, is called the variation of $I(y)$ denoted by $\delta I$ or $\delta I(h)$.
Theorem 2.7. The variation of a space-time functional is unique.
Theorem 2.8. A necessary condition for a space-time functional $I(y(x, t))$ to have an extremum for $y=y^{}$ is that its variation vanishes for $y=y^{}$, i.e. $\delta I(h)=0$ for $y=y *$ for all admissible $h=h(x, t)$.

# 有限元方法代考

## 数学代写|有限元方法代写Finite Element Method代考|Elements of calculus of variations

$$\int_{\Omega_{\boldsymbol{A} t}} \alpha(x, t) h(x, t) d \Omega_{x t}=0 \quad \forall h(x, t) \in H^{(1)}: \quad h(\Gamma)=0$$
$$\alpha(x, t)=0 \quad \forall(x, t) \in \bar{\Omega} x t$$

$$h(x, t)=\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right)$$

$$\int_{\bar{\Omega} x t} \alpha(x, t) h(x, t) d \Omega x t=\int_{t_{1} x_{1}}^{t_{2} x_{2}} \int_{x_{1}} \alpha(x)\left(x-x_{1}\right)\left(x_{2}-x\right)\left(t-t_{1}\right)\left(t_{2}-t\right) d x d t>0$$

## 数学代写|有限元方法代写Finite Element Method代考|Concept of variation of a space-time functional

$$\Delta I(h)=I(y+h)-I(y)$$

$$\Delta I(h)=\Phi(h)+\epsilon|h|$$

## 有限元方法代写

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## MATLAB代写

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

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