### 数学代写|数值方法作业代写numerical methods代考|Ordinary Differential Equations

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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|Qualitative Properties of the Solution and Maximum Principle

Before we introduce difference schemes for (2.1), we discuss a number of results that allow us to describe how the solution $u$ behaves. First, we wish to conclude that if the initial value $A$ and inhomogeneous term $f(t)$ are positive, then the solution $u(t)$ should also be positive for any value $t$ in $[0, T]$. This so-called positivity or monotonicity result should be reflected in our difference schemes (not all schemes possess this property). Second, we wish to know how the solution $u(t)$ grows or decreases as a function of time. The following two results deal with these issues.

Lemma 2.1 (Positivity). Let the operator $L$ be defined in Equation (2.1), and let $w$ be a well-behaved function satisfying the inequalities:
\begin{aligned} &L w(t) \geq 0 \forall t \in[0, T] \ &w(0) \geq 0 \end{aligned}

Then the following result holds true:
$$w(t) \geq 0 \forall t \in[0, T] .$$
Roughly speaking, this lemma states that you cannot get a negative solution from positive input.
You can verify it by examining Equation (2.2) because all terms are positive. The following result gives bounds on the growth of $u(t)$.
Theorem 2.1 Let $u(t)$ be the solution of Equation (2.1). Then:
$$|u(t)| \leq \frac{N}{\alpha}+|A| \forall t \in[0, T]$$
where
$$|f(t)| \leq N \forall t \in[0, T] .$$
This result states that the value of the solution is bounded by the input data. In other words, it is a well-posed problem.

We wish to replicate these properties in our difference schemes for Equation (2.1). For completeness, we show the steps to be executed in order to produce the result in Equation (2.2).
$$\text { Let } I(t)=\exp \left(\int_{0}^{t} a(s) d s\right), \quad I^{-1}(t)=\exp \left(-\int_{0}^{t} a(s) d s\right) \text {. }$$
Then from Equation (2.1) we see:
$$I(t)\left(\frac{d u}{d t}+a u\right)=I(t) f(t)$$
or:
$$\frac{d}{d t}(I(t) u)=I(t) f(t) .$$
Integrating this equation between $t=0$ and $t=\xi$ gives:
\begin{aligned} &\left.\int_{0}^{\xi} \frac{d}{d t}(I(t) u) d t=\int_{0}^{\xi} I(t) f(t) d t \text { (and using the fact that } I(0)=1\right) \ &I(\xi) u(\xi)=u(0)+\int_{0}^{\xi} I(t) f(t) d t \ &u(\xi)=u(0) I^{-1}(\xi)+I^{-1}(\xi) \int_{0}^{\xi} I(t) f(t) d t \ &=\exp \left(-\int_{0}^{\xi} a(s) d s\right) u(0)+\exp \left(-\int_{0}^{\xi} a(s) d s\right) \int_{0}^{\xi} I(t) f(t) d t \end{aligned}

## 数学代写|数值方法作业代写numerical methods代考|Rationale and Generalisations

The IVP Equation (2.1) is a model for all the linear time-dependent differential equations that we encounter in this book. We no longer think in terms of scalar problems in which the functions in Equation (2.1) are scalar-valued, but we can view an ODE at different levels of abstraction. To this end, we focus on the generic homogeneous $O D E$ with solution $u(t)$ :
$$\frac{d u}{d t}=A u, t>0 .$$
This equation subsumes several special cases:

1. The variable $A$ is a square matrix, and then Equation (2.4) represents a system of ODEs. This is a very important area of research having many applications in science, engineering, and finance.
2. The variable $A$ is an ordinary or partial differential operator, and then Equation (2.4) represents an ODE in a Hilbert or Banach space.
3. The variable $A$ is a tridiagonal or block tridiagonal matrix that originates from a semi-discretisation in space of a time-dependent partial differential equation (PDE) using the Method of Lines (MOL) as discussed in Chapter $20 .$
4. The formal solution of $(2.4)$ is:
$$u(t)=u(0) e^{A t}, \quad t>0$$
In other words, we express the solution in terms of the exponential function of a matrix or of a differential operator. In the former case, there are many ways to compute the exponential of a matrix (see Moler and Van Loan (2003)).
5. The solution of Equation (2.4) can be simplified by matrix or operator splitting of the operator $A$ :
\begin{aligned} &A=A_{1}+A_{2} \ &\frac{d u}{d t}=A_{1} u \ &\frac{d u}{d t}=A_{2} u . \end{aligned}
For example, we can split a matrix $A$ into two simpler matrices, or we can split an operator $A$ into its convection and diffusion components. In other words, we solve Equation (2.4) as a sequence of simpler problems in (2.6). These topics will be discussed in Chapters 18,22 , and 23 .
6. The initial value problem (2.1) was originally used as a model test of finite difference methods in (Dahlquist (1956)). The resulting results and insights are helpful when dealing more complex IVPs.

## 数学代写|数值方法作业代写numerical methods代考|DISCRETISATION OF INITIAL VALUE PROBLEMS: FUNDAMENTALS

We now discuss finding an approximate solution to Equation (2.1) using the finite difference method. We introduce several popular schemes as well as defining standardised notation.

The interval or range where the solution of Equation $(2.1)$ is defined is $[0, T]$. When approximating the solution using finite difference equations, we use a discrete set of points in $[0, T]$ where the discrete solution will be calculated. To this end, we divide $[0, T]$ into $N$ equal intervals of length $k$, where $k$ is a positive number called the step size. (We also use the symbol $\Delta t$ to denote the step size in many cases.) We number these discrete points as shown in Figure 2.1. In general all coefficients and discrete functions will be defined at these mesh points only. We adopt the following notation:
\begin{aligned} &a^{n}=a\left(t_{n}\right), f^{n}=f\left(t_{n}\right) \ &a^{n, \theta}=a\left(\theta t_{n}+(1-\theta) t_{n+1}\right), 0 \leq \theta \leq 1,0 \leq n \leq N-1 \ &u^{n, \theta}=\theta u^{n}+(1-\theta) u^{n+1}, 0 \leq n \leq N-1 \text { (function to be calculated). } \end{aligned}
Not only do we have to approximate functions at mesh points, but we also have to come up with a scheme to approximate the derivative appearing in Equation (2.1). There are several possibilities, and they are based on divided differences. For example, the following divided differences approximate the first derivative of $u$ at the mesh point $t_{n}=n * k$;
$$\left.\begin{array}{l} D_{+} u^{n} \equiv \frac{u^{n+1}-u^{n}}{k} \ D_{-} u^{n} \equiv \frac{u^{n}-u^{n-1}}{k} \ D_{0} u^{n} \equiv \frac{u^{n+1}-u^{n-1}}{2 k} \end{array}\right}$$
The first two divided differences are called one-sided differences and give first-order accuracy to the derivative, while the last divided difference is called a centred approximation to the derivative. In fact, by using a Taylor’s expansion (assuming sufficient

smoothness of $u$ ), we can prove the following:
$$\left{\begin{array}{l} \left|D_{\pm} u\left(t_{n}\right)-u^{\prime}\left(t_{n}\right)\right| \leq M k, n=0,1, \ldots \ \left|D_{0} u\left(t_{n}\right)-u^{\prime}\left(t_{n}\right)\right| \leq M k^{2}, n=0,1, \ldots \end{array}\right.$$
Note that the first two approximations use two consecutive mesh points while the last formula uses three consecutive mesh points.

We now decide on how to approximate Equation (2.1) using finite differences. To this end, we need to introduce two new concepts:

• One-step and multistep methods
• Explicit and implicit schemes.
A one-step method is a finite difference scheme that calculates the solution at time-level $n+1$ in terms of the solution at time-level $n$. No information at levels $n-1$, $n-2$, or previous levels is needed in order to calculate the solution at level $n+1$. A multistep method, on the other hand, is a difference scheme where the solution at level $n+1$ is determined by values at levels $n, n-1$ and possibly previous time levels. Multistep methods are more complicated than one-step methods, and we concentrate solely on the latter methods in this book.

An explicit difference scheme is one where the solution at time $n+1$ can be calculated from the information at level $n$ directly. No extra arithmetic is needed: for example, using division or matrix inversion. An implicit finite difference scheme is one in which the terms involving the approximate solution at level $n+1$ are grouped together and only then can the solution at this level be found. Obviously, implicit methods are more difficult to program than explicit methods because we must solve a system of equations at each time step.

## 数学代写|数值方法作业代写numerical methods代考|Qualitative Properties of the Solution and Maximum Principle

|在(吨)|≤ñ一种+|一种|∀吨∈[0,吨]

|F(吨)|≤ñ∀吨∈[0,吨].

让 一世(吨)=经验⁡(∫0吨一种(s)ds),一世−1(吨)=经验⁡(−∫0吨一种(s)ds).

dd吨(一世(吨)在)=一世(吨)F(吨).

∫0Xdd吨(一世(吨)在)d吨=∫0X一世(吨)F(吨)d吨 （并使用以下事实 一世(0)=1) 一世(X)在(X)=在(0)+∫0X一世(吨)F(吨)d吨 在(X)=在(0)一世−1(X)+一世−1(X)∫0X一世(吨)F(吨)d吨 =经验⁡(−∫0X一种(s)ds)在(0)+经验⁡(−∫0X一种(s)ds)∫0X一世(吨)F(吨)d吨

## 数学代写|数值方法作业代写numerical methods代考|Rationale and Generalisations

IVP 方程（2.1）是我们在本书中遇到的所有线性时间相关微分方程的模型。我们不再考虑方程 (2.1) 中的函数是标量值的标量问题，但我们可以在不同抽象级别上查看 ODE。为此，我们专注于泛型同构这D和有溶液在(吨) :
d在d吨=一种在,吨>0.

1. 变量一种是一个方阵，则方程 (2.4) 表示一个 ODE 系统。这是一个非常重要的研究领域，在科学、工程和金融领域有许多应用。
2. 变量一种是普通或偏微分算子，则方程 (2.4) 表示希尔伯特或巴纳赫空间中的 ODE。
3. 变量一种是一个三对角矩阵或块三对角矩阵，它源自使用线法 (MOL) 对时间相关偏微分方程 (PDE) 进行空间半离散化，如第 1 章所述20.
4. 的正式解决方案(2.4)是：
在(吨)=在(0)和一种吨,吨>0
换句话说，我们用矩阵或微分算子的指数函数来表达解。在前一种情况下，有很多方法可以计算矩阵的指数（参见 Moler 和 Van Loan (2003)）。
5. 方程（2.4）的解可以通过矩阵或算子的算子拆分来简化一种 :
一种=一种1+一种2 d在d吨=一种1在 d在d吨=一种2在.
例如，我们可以拆分一个矩阵一种分成两个更简单的矩阵，或者我们可以拆分一个运算符一种分为对流和扩散成分。换句话说，我们将方程（2.4）求解为（2.6）中的一系列更简单的问题。这些主题将在第 18、22 和 23 章中讨论。
6. 初始值问题 (2.1) 最初在 (Dahlquist (1956)) 中用作有限差分方法的模型检验。在处理更复杂的 IVP 时，得到的结果和见解很有帮助。

## 数学代写|数值方法作业代写numerical methods代考|DISCRETISATION OF INITIAL VALUE PROBLEMS: FUNDAMENTALS

\left.\begin{array}{l} D_{+} u^{n} \equiv \frac{u^{n+1}-u^{n}}{k} \ D_{-} u^{ n} \equiv \frac{u^{n}-u^{n-1}}{k} \ D_{0} u^{n} \equiv \frac{u^{n+1}-u^{ n-1}}{2 k} \end{数组}\right}\left.\begin{array}{l} D_{+} u^{n} \equiv \frac{u^{n+1}-u^{n}}{k} \ D_{-} u^{ n} \equiv \frac{u^{n}-u^{n-1}}{k} \ D_{0} u^{n} \equiv \frac{u^{n+1}-u^{ n-1}}{2 k} \end{数组}\right}

$$\left{|D±在(吨n)−在′(吨n)|≤米ķ,n=0,1,… |D0在(吨n)−在′(吨n)|≤米ķ2,n=0,1,…\对。$$

• 一步法和多步法
• 显式和隐式方案。
一步法是一种有限差分方案，它在时间级计算解n+1就时间层面的解决方案而言n. 没有级别信息n−1, n−2，或者需要以前的级别才能计算级别的解决方案n+1. 另一方面，多步法是一种差分方案，其中水平的解决方案n+1由级别的值决定n,n−1可能还有以前的时间水平。多步法比一步法更复杂，本书只关注后一种方法。

## 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。

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## MATLAB代写

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