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

如果你也在 怎样代写微分方程differential equation这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。


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

我们提供的微分方程differential equation及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|MATH2065

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) \
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
\frac{d u}{d t} &=f(u, v) \
\frac{d v}{d t} &=g(u, v)
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
&\frac{d u}{d t}=v \
&\frac{d v}{d t}=-f(u)
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|MATH2065


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)。

感兴趣的量可能是细胞中钙离子的浓度(=每单位体积的数量)或湖中微生物的浓度,但它通常由一些标量函数表示在=在(X,是,和,吨). 如果在随时间平滑变化,则有时间导数∂在∂吨被定义为1




这是梯度的一个组成部分在, 向量值函数

梯度包含我们需要的所有信息在空间的变化,但是由于我们可以从一个特定点移动的方向有无数个,因此要找到在在特定的方向,在, 在哪里在是单位向量,我们定义方向导数,


其中,提供|∇在|≠0, 是指向函数最大增量方向的单位向量在. 这对滑雪者或单板滑雪者的重要性是显而易见的,指向与“下降线”平行的方向。还值得注意的是n与函数的水平面垂直(正交)在. 我们可以通过注意到如果(X(s),是(s),和(s))是一条空间曲线,参数化为s,曲线的切线方向为向量(X˙(s),是˙(s),和˙(s)). 如果函数在(X,是,和)在这条曲线上是一个常数,在(X(s),是(s),和(s))=C,然后将其区分为s, 我们发现

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


这个方程是自治的,如果F独立于吨, 以便

如果在是一个标量,方程(1.25)的解可以很容易地使用图形方法来理解,即通过绘制d在d吨对比在. 一个例子如图所示1.1.
图 1.1。情节d在d吨对比在对于双稳态函数F(在)=一个在(1−在)(在−一个)和一个=0.25,一个=10.
首先要注意的是零F(在),即均衡。例如F(在)=一个在(在−1)(一个−在),如图 1.1 所示,平衡点位于在0=0,在0= 一个, 和在0=1. 接下来,可以确定运动的方向,如果在不处于平衡状态。这些在图 1.1 中用箭头表示。例如,如果00以便在那里正在增加。这是我们的第一个迹象表明在0=0和在0=1是稳定的平衡,而在0=一个不稳定。


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


d在d吨=F(在,在) d在d吨=G(在,在)



d在d吨=在 d在d吨=−F(在)
该系统的零斜线很容易确定,即在=0为了在零斜线,和F(在)=0为了在nullclines,即线条在=0,在=一个, 和在=1. 这些显示在图1.3作为虚线。

下一步是识别所有关键点,即d在d吨和d在d吨同时为零,因此是平衡点。当然,这些都是在和在空斜线。对于这个例子,它们是与在=0和在=0,一个和 1。

数学代写|微分方程代写differential equation代考 请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。







术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。



您的电子邮箱地址不会被公开。 必填项已用 * 标注