### 数学代写|微分方程代写differential equation代考|МАTH2921

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。

• 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|Growth with a Carrying Capacity—Fisher’s Equation

Now we consider a diffusing and reproducing population for which there is population size-dependent death. We can represent this by the two reactions
$$U \stackrel{\alpha}{\longrightarrow} 2 U, \quad U+U \stackrel{\beta}{\longrightarrow} U .$$
For these reactions, the conservation equation is
$$\frac{\partial u}{\partial t}=D \frac{\partial^{2} u}{\partial x^{2}}+\alpha u-\beta u^{2} .$$
It is common to write this equation in the slightly different form
$$\frac{\partial u}{\partial t}=D \frac{\partial^{2} u}{\partial x^{2}}+\alpha u\left(1-\frac{u}{K}\right),$$
where $K=\frac{\alpha}{\beta}$ is called the carrying capacity. Rescaling the variables by setting $u=$ $\frac{\alpha}{\beta} v=K v, t=\frac{\tau}{\alpha}$, and $x=\sqrt{\frac{D}{\alpha}} \xi$, the equation simplifies to
$$\frac{\partial v}{\partial \tau}=\frac{\partial^{2} v}{\partial \xi^{2}}+v-v^{2}$$
with no free parameters.
A second derivation of this equation is related to SIR epidemic models. Here we suppose that there are two populations, denoted by $S$ and $I$, representing susceptible and infected populations, respectively. A susceptible individual can become infected by contact with another infected individual, but there is no possible recovery from the infection; an infected individual is permanently contagious. This process can be represented by the reaction
$$S+I \stackrel{\alpha}{\longrightarrow} 2 I$$ and the conservation equations for these two populations are
$$\frac{\partial s}{\partial t}=D_{s} \frac{\partial^{2} s}{\partial x^{2}}-\alpha s i, \quad \frac{\partial i}{\partial t}=D_{i} \frac{\partial^{2} i}{\partial x^{2}}+\alpha s i .$$
Under the assumption that the diffusion coefficient for both populations is the same, $D_{s}=D_{i}$, the quantity $s+i$ satisfies the diffusion equation and so has steady, constant solutions $s+i=S_{0}$. With this conserved quantity, the equation for $i$ becomes
$$\frac{\partial i}{\partial t}=D_{i} \frac{\partial^{2} i}{\partial x^{2}}+\alpha i\left(S_{0}-i\right)$$
Rescaling the variables by setting $i=S_{0} v, t=\frac{\tau}{\alpha S_{0}}, x=\sqrt{\frac{D_{i}}{\alpha S_{0}}} \xi$, the equation simplifies to $(6.30)$.

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

Now consider the situation in which organisms, say bacteria, consume a resource substrate, such as glucose, of which there is a finite supply. For example, suppose bacteria are grown on an agar gel on a Petrie dish. The reaction describing this is
$$U+S \stackrel{\alpha}{\longrightarrow} 2 U .$$
The units on $U$ and $S$ are such that one unit of $S$ converts into one unit of $U$. We assume that both the glucose and the bacteria move by diffusion. Consequently, the differential equations describing this evolution (in one spatial dimension) are
\begin{aligned} &\frac{\partial u}{\partial t}=D_{u} \frac{\partial^{2} u}{\partial x^{2}}+\alpha u s \ &\frac{\partial s}{\partial t}=D_{g} \frac{\partial^{2} s}{\partial x^{2}}-\alpha u s \end{aligned}
where $s$ represents the concentration of the resource substrate (i.e., the glucose). Numerical simulation of this system of equations is shown in Figure 6.13. The Matlab code for this simulation is titled CN_diffusion_gluc_micro_X.m. This simulation again suggests that there should be a traveling wave solution. The first step of the analysis is to simplify the equations by introducing scaled variables $t=\frac{\tau}{\alpha}, x=\sqrt{\frac{D_{u}}{\alpha}} \xi$, in terms of which the equations become
\begin{aligned} &\frac{\partial u}{\partial \tau}=\frac{\partial^{2} u}{\partial \xi^{2}}+u s \ &\frac{\partial s}{\partial \tau}=\delta \frac{\partial^{2} s}{\partial \xi^{2}}-u s \end{aligned}
where $\delta=\frac{D_{g}}{D_{u}}$.
Now, to examine the possibility of traveling wave solutions, we look for a solution of the form $u(\xi, \tau)=U(\xi-c \tau), s(\xi, \tau)=S(\xi-c \tau)$, and find the system of ordinary differential equations
\begin{aligned} 0 &=\frac{d^{2} U}{d \zeta^{2}}+c \frac{d U}{d \zeta}+U S \ 0 &=\delta \frac{d^{2} S}{d \zeta^{2}}+c \frac{d S}{d \zeta}-U S \end{aligned}

## 数学代写|微分方程代写differential equation代考|Spread of Rabies—SIR with Diffusion

It has been observed in England that rabid foxes tend to travel across much larger distances than rabies free animals. This observation has led to consideration of the spread of an infectious disease where the infected animals diffuse, but susceptible animals do not [51]. For this we consider the standard SIR disease dynamics
$$S+I \stackrel{\alpha}{\longrightarrow} 2 I, \quad I \stackrel{\beta}{\longrightarrow} R,$$
where $S$ represents the susceptible population, $I$ represents the infected population, and $R$ represents the recovered (or removed) population. The corresponding differential equations are
\begin{aligned} &\frac{\partial s}{\partial t}=-\alpha s i \ &\frac{\partial i}{\partial t}=\alpha s i-\beta i+D \frac{\partial^{2} i}{\partial x^{2}} . \end{aligned}
Introducing dimensionless variables $\sigma=\frac{s}{S_{0}}, u=\frac{i}{S_{0}}, t=\frac{\tau}{\alpha S_{0}}$, and $x=\sqrt{\frac{D}{\alpha S_{0}}} \xi$, we find the dimensionless equations
\begin{aligned} &\frac{\partial \sigma}{\partial \tau}=-\sigma u, \ &\frac{\partial u}{\partial \tau}=\sigma u-\eta u+\frac{\partial^{2} u}{\partial \xi^{2}}, \end{aligned}
depending on the single parameter $\eta=\frac{\beta}{\alpha S_{0}}=\frac{1}{R_{0}}$. A simulation of these equations is shown in Figure 6.15, and was computed using the Matlab code $\mathrm{CN}_{\text {_diffusion_SIR.m. }}$
As you can see from this figure, an initial amount of $u$ grows and spreads as a traveling wave, leading to a permanent decrease in the amount of $\sigma$, while the spreading bulge of $u$ is only temporary, as recovery eventually restores $u$ to zero. We would like to determine how fast this infection spreads and how much of the initial susceptible population is affected by it.

## 数学代写|微分方程代写differential equation代考|Growth with a Carrying Capacity—Fisher’s Equation

∂在∂吨=D∂2在∂X2+一个在−b在2.

∂在∂吨=D∂2在∂X2+一个在(1−在ķ),

∂在∂τ=∂2在∂X2+在−在2

∂s∂吨=Ds∂2s∂X2−一个s一世,∂一世∂吨=D一世∂2一世∂X2+一个s一世.

∂一世∂吨=D一世∂2一世∂X2+一个一世(小号0−一世)

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

∂在∂吨=D在∂2在∂X2+一个在s ∂s∂吨=DG∂2s∂X2−一个在s

∂在∂τ=∂2在∂X2+在s ∂s∂τ=d∂2s∂X2−在s

0=d2在dG2+Cd在dG+在小号 0=dd2小号dG2+Cd小号dG−在小号

## 数学代写|微分方程代写differential equation代考|Spread of Rabies—SIR with Diffusion

∂s∂吨=−一个s一世 ∂一世∂吨=一个s一世−b一世+D∂2一世∂X2.

∂σ∂τ=−σ在, ∂在∂τ=σ在−这在+∂2在∂X2,

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。