### 数学代写|微分方程代写differential equation代考|MATH2065

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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|微分方程代写differential equation代考|Background Material

1.1.1. Partial Derivatives. In what is to come, we will be dealing with functions of both space and time. We may be interested in the electrolyte balance within muscle tissue, or the distribution of microorganisms occupying a lake; either way, we are studying how something occupying a region in space evolves over time. We often describe position in this space by using the Cartesian coordinates $x, y$, and $z$, and time by the variable $t$ (although other coordinate representations like polar or spherical coordinates are sometimes useful).

The quantity of interest may be the concentration (= number per unit volume) of calcium ions in a cell or the concentration of microorganisms in the lake, but it is typically denoted by some scalar function $u=u(x, y, z, t)$. If $u$ changes smoothly in time, then it has a time derivative $\frac{\partial u}{\partial t}$ defined by ${ }^{1}$
$$\frac{\partial u}{\partial t}=\lim {\Delta t \rightarrow 0} \frac{u(x, y, z, t+\Delta t)-u(x, y, z, t)}{\Delta t}$$ The fundamental theorem of calculus states that $$u(x, y, z, b)-u(x, y, z, a)=\int{a}^{b} \frac{\partial u}{\partial t} d t$$
In words, the cumulative change in $u$ over an interval of time can be measured by observing the difference between $u$ at the end and the beginning of the interval.

Similarly, if $u$ varies smoothly in space, spatial derivatives can be defined, such as
$$\frac{\partial u}{\partial x}=\lim _{\Delta x \rightarrow 0} \frac{u(x+\Delta x, y, z, t)-u(x, y, z, t)}{\Delta x},$$
and this is one component of the gradient of $u$, the vector-valued function
$$\nabla u=\left(\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}\right)$$
The gradient holds all of the information we need about how $u$ changes in space, but since there are an infinite number of directions in which we could move from a particular point, to find the derivative of $u$ in a particular direction, $\mathbf{v}$, where $\mathbf{v}$ is a unit vector, we define the directional derivative,
$$\frac{\partial u}{\partial \mathbf{v}}=\nabla u \cdot \mathbf{v} \text {. }$$
One important object that uses the gradient is
$$\mathbf{n}=\frac{\nabla u}{|\nabla u|},$$
which, provided $|\nabla u| \neq 0$, is a unit vector pointing in the direction of the greatest increase of the function $u$. The importance of this to a skier or snowboarder is obvious, pointing in the direction parallel to the “fall-line”. It is also noteworthy that $\mathbf{n}$ is perpendicular (orthogonal) to level surfaces of the function $u$. We can verify this by noting that if $(x(s), y(s), z(s))$ is a curve in space parametrized by $s$, the tangent direction of the curve is the vector $(\dot{x}(s), \dot{y}(s), \dot{z}(s))$. If the function $u(x, y, z)$ is a constant on this curve, $u(x(s), y(s), z(s))=C$, then differentiating this with respect to $s$, we find that
$$0=\frac{\partial u}{\partial x} \dot{x}(s)+\frac{\partial u}{\partial y} \dot{y}(s)+\frac{\partial u}{\partial z} \approx(s) \equiv \nabla u \cdot\left(\begin{array}{c} \dot{x}(s) \ \dot{y}(s) \ \dot{z}(s) \end{array}\right)$$
as claimed.

## 数学代写|微分方程代写differential equation代考|Ordinary Differential Equations

1.2.1. First Order Equations. An nrdinary differential equation specifies a relationship between the (time) derivative of some quantity $u$ and its values through, say,
$$\frac{d u}{d t}=f(u, t) .$$
This equation is autonomous if $f$ is independent of $t$, so that
$$\frac{d u}{d t}=f(u) .$$
Many of the problems discussed in this book are autonomous in time.
If $u$ is a scalar quantity, the solution of equation (1.25) can be readily understood using graphical means, i.e., by plotting $\frac{d u}{d t}$ vs. $u$. An example is shown in Figure $1.1$.
Figure 1.1. Plot of $\frac{d u}{d t}$ vs. $u$ for the bistable function $f(u)=a u(1-u)(u-\alpha)$ with $\alpha=0.25, a=10 .$
The first things to notice are the zeros of $f(u)$, i.e., the equilibria. For the example $f(u)=a u(u-1)(\alpha-u)$, shown in Figure 1.1, the equilibria are at $u_{0}=0, u_{0}=$ $\alpha$, and $u_{0}=1$. Next, one can determine the direction of movement if $u$ is not at an equilibrium. These are shown with arrows in Figure 1.1. For example, if $00$ so that $u$ is increasing there. This is our first indication that $u_{0}=0$ and $u_{0}=1$ are stable equilibria, while $u_{0}=\alpha$ is unstable.

The next thing to do is to linearize the equations about the equilibria. Linearization is a very important procedure by which one reduces a nonlinear equation to a linear equation. ${ }^{3}$ It is a good idea to understand it thoroughly, because it is used often in this text.

## 数学代写|微分方程代写differential equation代考|Systems of first order equations

1.2.2. Systems of first order equations. We now turn our attention to systems of first order equations, which can still be written in the form of (1.24) provided we recognize that $u$ is a vector, rather than a scalar, quantity. The most important example for this text is when there are two unknown scalar functions $u(t)$ and $v(t)$ and the equations describing their evolution are in the form
\begin{aligned} \frac{d u}{d t} &=f(u, v) \ \frac{d v}{d t} &=g(u, v) \end{aligned}
As with first order equations, a useful way to proceed is with a graphical, or phase plane, analysis. The first step of this analysis is to plot the nullclines, the curves in the $u-v$ plane along which either $u$ or $v$ do not change, i.e., $\frac{d u}{d t}=0$ or $\frac{d v}{d t}=0$.

There are many examples of this procedure in this book, however, for purposes of illustration, let’s look at solutions of the second order differential equation
$$\frac{d^{2} u}{d t^{2}}+f(u)=0$$
where $f(u)=a u(1-u)(u-\alpha)$, the same function as used above. To write this equation as a first order system, we set $v=\frac{d u}{d t}$, and then the equations are
\begin{aligned} &\frac{d u}{d t}=v \ &\frac{d v}{d t}=-f(u) \end{aligned}
The nullclines for this system are easily determined, being the line $v=0$ for the $u$ nullcline, and $f(u)=0$ for the $v$ nullclines, i.e., the lines $u=0, u=\alpha$, and $u=1$. These are shown plotted in Figure $1.3$ as dashed lines.

The next step is to identify all the critical points, i.e., the points at which $\frac{d u}{d t}$ and $\frac{d v}{d t}$ are simultaneously zero, hence, points of equilibrium. These are, of course, all the intersections of the $u$ and $v$ nullclines. For this example, they are the points with $v=0$ and $u=0, \alpha$ and 1 .

## 数学代写|微分方程代写differential equation代考|Background Material

1.1.1。偏导数。在接下来的内容中，我们将处理空间和时间的功能。我们可能对肌肉组织内的电解质平衡或占据湖泊的微生物分布感兴趣；无论哪种方式，我们都在研究占据空间区域的事物如何随着时间的推移而演变。我们经常使用笛卡尔坐标来描述这个空间中的位置X,是， 和和, 和时间由变量吨（尽管有时极坐标或球坐标等其他坐标表示很有用）。

∂在∂吨=林Δ吨→0在(X,是,和,吨+Δ吨)−在(X,是,和,吨)Δ吨微积分基本定理指出

∂在∂X=林ΔX→0在(X+ΔX,是,和,吨)−在(X,是,和,吨)ΔX,

∇在=(∂在∂X,∂在∂是,∂在∂和)

∂在∂在=∇在⋅在.

n=∇在|∇在|,

0=∂在∂XX˙(s)+∂在∂是是˙(s)+∂在∂和≈(s)≡∇在⋅(X˙(s) 是˙(s) 和˙(s))

## 数学代写|微分方程代写differential equation代考|Ordinary Differential Equations

1.2.1。一阶方程。一个普通微分方程指定了某个量的（时间）导数之间的关系在以及它的价值，比如说，

d在d吨=F(在,吨).

d在d吨=F(在).

## 数学代写|微分方程代写differential equation代考|Systems of first order equations

1.2.2。一阶方程组。我们现在将注意力转向一阶方程组，只要我们认识到在是一个向量，而不是一个标量，数量。本文最重要的例子是当有两个未知的标量函数时在(吨)和在(吨)描述它们演化的方程是

d在d吨=F(在,在) d在d吨=G(在,在)

d2在d吨2+F(在)=0

d在d吨=在 d在d吨=−F(在)

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