### 数学代写|matlab代写|CS1132

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

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• 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|matlab代写|Steady-state flow of heat

When the inner and outer walls of a body, for example the inner and outer walls of a house, are maintained at different constant temperatures, heat will flow from the warmer wall to the colder one. When each surface parallel to a wall has attained a constant temperature, the flow of heat has reached a steady state. In a steady-state flow of heat, each surface parallel to a wall, because its temperature is now constant, is referred to as an isothermal surface. Isothermal surfaces at different distances from an interior wall will have different temperatures. In many cases the temperature of an isothermal surface is only a function of its distance $x$ from the interior wall, and the rate of flow of heat $Q$ in a unit time across such a surface is proportional both to the area $A$ of the surface and to $d T / d x$, where $T$ is the temperature of the isothermal surface. Hence,
$$Q=-\kappa A \frac{d T}{d x},$$
where $\kappa$ is called the thermal conductivity of the material between the walls.
In place of a flat wall, let us consider a hollow cylinder whose inner and outer surfaces are located at $r=r_{1}$ and $r=r_{2}$, respectively. At steady state, Equation $1.2 .27$ becomes
$$Q_{r}=-\kappa A \frac{d T}{d r}=-\kappa(2 \pi r L) \frac{d T}{d r},$$
assuming no heat generation within the cylindrical wall.
We can find the temperature distribution inside the cylinder by solving Equation 1.2.28 along with the appropriate conditions on $T(r)$ at $r=r_{1}$ and $r=r_{2}$ (the boundary conditions). To illustrate the wide choice of possible boundary conditions, let us require that the inner surface is maintained at the temperature $T_{1}$. We assume that along the outer surface,heat is lost by convection to the environment, which has the temperature $T_{\infty}$. This heat loss is usually modeled by the equation
$$\left.\kappa \frac{d T}{d r}\right|{r=\mathrm{r}{2}}=-h\left(T-T_{\infty}\right),$$
where $h>0$ is the convective heat transfer coefficient. Upon integrating Equation $1.2 .28$,
$$T(r)=-\frac{Q_{r}}{2 \pi \kappa L} \ln (r)+C,$$
where $Q_{r}$ is also an unknown. Substituting Equation 1.2.30 into the boundary conditions, we obtain
$$T(r)=T_{1}+\frac{Q_{r}}{2 \pi \kappa L} \ln \left(r_{1} / r\right),$$
with
$$Q_{r}=\frac{2 \pi \kappa L\left(T_{1}-T_{\infty}\right)}{\kappa / r_{2}+h \ln \left(r_{2} / r_{1}\right)} .$$

## 数学代写|matlab代写|Logistic equation

The study of population dynamics yields an important class of first-order, nonlinear, ordinary differential equations: the logistic equation. This equation arose in Pierre François Verhulst’s (1804-1849) study of animal populations. ${ }^{3}$ If $x(t)$ denotes the number of species in the population and $k$ is the (constant) environment capacity (the number of species that can simultaneously live in the geographical region), then the logistic or Verhulst’s equation is
$$x^{\prime}=a x(k-x) / k,$$
where $a$ is the population growth rate for a small number of species.
To solve Equation 1.2.41, we rewrite it as
$$\frac{d x}{(1-x / k) x}=\frac{d x}{x}+\frac{x / k}{1-x / k} d x=r d t .$$
Integration yields
$$\ln |x|-\ln |1-x / k|=r t+\ln (C),$$
or
$$\frac{x}{1-x / k}=C e^{r t}$$
If $x(0)=x_{0}$,
$$x(t)=\frac{k x_{0}}{x_{0}+\left(k-x_{0}\right) e^{-r t}} .$$
As $t \rightarrow \infty, x(t) \rightarrow k$, the asymptotically stable solution.

## 数学代写|matlab代写|Steady-state flow of heat

$$Q=-\kappa A \frac{d T}{d x},$$

$$Q_{r}=-\kappa A \frac{d T}{d r}=-\kappa(2 \pi r L) \frac{d T}{d r},$$

$$\kappa \frac{d T}{d r} \mid r=\mathrm{r} 2=-h\left(T-T_{\infty}\right),$$

$$T(r)=-\frac{Q_{r}}{2 \pi \kappa L} \ln (r)+C,$$

$$T(r)=T_{1}+\frac{Q_{r}}{2 \pi \kappa L} \ln \left(r_{1} / r\right),$$

$$Q_{r}=\frac{2 \pi \kappa L\left(T_{1}-T_{\infty}\right)}{\kappa / r_{2}+h \ln \left(r_{2} / r_{1}\right)} .$$

## 数学代写|matlab代写|Logistic equation

$$x^{\prime}=a x(k-x) / k,$$

$$\frac{d x}{(1-x / k) x}=\frac{d x}{x}+\frac{x / k}{1-x / k} d x=r d t$$

$$\ln |x|-\ln |1-x / k|=r t+\ln (C),$$

$$\frac{x}{1-x / k}=C e^{r t}$$

$$x(t)=\frac{k x_{0}}{x_{0}+\left(k-x_{0}\right) e^{-r t}} .$$

## 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。