### 数学代写|数值方法作业代写numerical methods代考| Lipschitz Continuous Functions

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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|数值方法作业代写numerical methods代考|Lipschitz Continuous Functions

We now examine functions that map one metric space into another one. In particular, we discuss the concepts of continuity and Lipschitz continuity.
It is convenient to discuss these concepts in the context of metric spaces.
Definition 1.7 Let $\left(X, d_{1}\right)$ and $\left(Y, d_{2}\right)$ be two metric spaces. A function $f$ from $X$ into $Y$ is said to be continuous at the point $\mathrm{a} \in X$ if for each $\varepsilon>0$ there exists a $\delta>0$ such that:
$$d_{2}(f(x), f(a))<\varepsilon \text { whenever } d_{1}(x, a)<\delta$$

This is a generalisation of the concept of continuity in Section $1.2$ (Definition 1.1). We should note that this definition refers to the continuity of a function at a single point. Thus, a function can be continuous at some points and discontinuous at other points.
Definition $1.8$ A function $f$ from a metric space $\left(X, d_{1}\right)$ into a metric space $\left(Y, d_{2}\right)$ is said to be a uniformly continuous on a set $E \subset X$ if for each $\varepsilon>0$ there exists a $\delta>0$ such that:
$$d_{2}(f(x), f(y))<\varepsilon \text { whenever } x, y \in E \text { and } d_{1}(x, y)<\delta$$
If the function $f$ is uniformly continuous, then it is continuous, but the converse is not necessarily true. Uniform continuity holds for all points in the set $E$, whereas normal continuity is only defined at a single point.

Definition 1.9 Let $f:[a, b] \rightarrow \mathbb{R}$ be a real-valued function and suppose that we can find two constants $M$ and $\alpha$ such that $|f(x)-f(y)| \leq M|x-y|^{\alpha}, \forall x, y \in[a, b]$. Then we say that $f$ satisfies a Lipschitz condition of order $\alpha$, and we write $f \in \operatorname{Lip}(\alpha)$.
We take an example. Let $f(x)=x^{2}$ on the interval $[a, b]$.
Then:
\begin{aligned} &|f(x)-f(y)|=\left|x^{2}-y^{2}\right|=|(x+y)(x-y)| \leq(|x|+|y|)|x-y| \ &\leq M|x-y| \text {, where } M=2 \max (|a|,|b|) . \end{aligned}
Hence $f \in \operatorname{Lip}(1)$.
A concept related to Lipschitz continuity is called a contraction.
Definition 1.10 Let $\left(X, d_{1}\right)$ and $\left(Y, d_{2}\right)$ be metric spaces. A transformation $T$ from $X$ into $Y$ is called a contraction if there exists a number $\lambda \in(0,1)$ such that:
$$d_{2}(T(x), T(y)) \leq \lambda d_{1}(x, y) \text { for all } x, y \in X$$
In general, a contraction maps a pair of points into another pair of points that are closer together. A contraction is always continuous.

The ability to discover and apply contraction mappings has considerable theoretical and numerical value. For example, it is possible to prove that stochastic differential equations (SDEs) have unique solutions by the application of fixed point theorems:

• Brouwer’s fixed point theorem
• Kakutani’s fixed point theorem
• Banach’s fixed point theorem
• Schauder’s fixed point theorem
Our interest here lies in the following fixed point theorem.

## 数学代写|数值方法作业代写numerical methods代考|INTRODUCTION AND OBJECTIVES

In this chapter we introduce a class of differential equations in which the highest order derivative is one. Furthermore, these equations have a single independent variable (which in nearly all applications plays the role of time). In short, these are termed ordinary differential equations (ODEs) precisely because of the dependence on a single variable.

ODEs crop up in many application areas, such as mechanics, biology, engineering, dynamical systems, economics and finance, to name just a few. It is for this reason that we devote two dedicated chapters to them.
The following topics are discussed in this chapter:

• Motivational examples of ODEs
• Qualitative properties of ODEs
• Common finite difference schemes for initial value problems for ODEs
• Some theoretical foundations.
In Chapter 3 we continue with our discussion of ODEs, including code examples in $\mathrm{C}++$ and Python.

## 数学代写|数值方法作业代写numerical methods代考|BACKGROUND AND PROBLEM STATEMENT

In this section we introduce the very first differential equation of this book. It is a scalar first-order linear ordinary differential equation (ODE), and we shall analyse it from several qualitative and quantitative viewpoints.

Consider a bounded interval $[0, T]$ where $T>0$. This interval could represent time or distance, for example. In most cases we shall view this interval as representing time values. In the interval we define the initial value problem (IVP) for an ODE:
\begin{aligned} &L u=u^{\prime}(t)+a(t) u(t)=f(t), t \in[0, T] \text { with } a(t) \geq \alpha>0, \forall t \in[0, T] \ &u(0)=A \end{aligned}
where $L$ is a first-order linear differential operator involving the derivative with respect to the time variable and $a=a(t)$ is a strictly positive function in $[0, T]$. The term $f(t)$ is called the inhomogeneous forcing term, and it is independent of $u$. Finally, the solution to the IVP must be specified at $t=0$; this is the so-called initial condition.
In general, the problem (2.1) has a unique solution given by:
\begin{aligned} &u(t)=I_{1}(t)+I_{2}(t) \ &I_{1}(t)=A \exp \left(-\int_{0}^{t} a(s) d s\right) \ &\left.I_{2}(t)=\exp \left(-\int_{0}^{t} a(s) d s\right) \int_{0}^{t} \exp \left(\int_{0}^{x} a(s) d s\right)\right) f(x) d x \end{aligned}
(See Hochstadt (1964), where the so-called integration factor is used to determine a solution.)

A special case of $(2.1)$ is when the right-hand term $f(t)$ is zero and $a(t)$ is constant; in this case the solution becomes a simple exponential term without any integrals, and this will be used later when we examine difference schemes to determine their feasibility. In particular, a scheme that behaves badly for the above special case will be unsuitable for more general or more complex problems unless some modifications are introduced.

## 数学代写|数值方法作业代写numerical methods代考|Lipschitz Continuous Functions

d2(F(X),F(一种))<e 每当 d1(X,一种)<d

d2(F(X),F(是))<e 每当 X,是∈和 和 d1(X,是)<d

|F(X)−F(是)|=|X2−是2|=|(X+是)(X−是)|≤(|X|+|是|)|X−是| ≤米|X−是|， 在哪里 米=2最大限度(|一种|,|b|).

d2(吨(X),吨(是))≤λd1(X,是) 对全部 X,是∈X

• Brouwer 不动点定理
• 角谷不动点定理
• 巴拿赫不动点定理
• Schauder 不动点定理
我们的兴趣在于以下不动点定理。

## 数学代写|数值方法作业代写numerical methods代考|INTRODUCTION AND OBJECTIVES

ODE 出现在许多应用领域，例如力学、生物学、工程、动力系统、经济学和金融学等。正是出于这个原因，我们用两个专门的章节来介绍它们。

• ODE 的励志示例
• ODE 的定性性质
• ODE 初值问题的常见有限差分格式
• 一些理论基础。
在第 3 章中，我们继续讨论 ODE，包括代码示例C++和 Python。

## 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。