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# 数学代写|实分析代写Real Analysis代考|MATHS2100 The heat equation

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## 数学代写|实分析代写Real Analysis代考|Derivation of the heat equation

Consider an infinite metal plate which we model as the plane $\mathbb{R}^2$, and suppose we are given an initial heat distribution at time $t=0$. Let the temperature at the point $(x, y)$ at time $t$ be denoted by $u(x, y, t)$.

Consider a small square centered at $\left(x_0, y_0\right)$ with sides parallel to the axis and of side length $h$, as shown in Figure 9 . The amount of heat energy in $S$ at time $t$ is given by
$$H(t)=\sigma \iint_S u(x, y, t) d x d y,$$
where $\sigma>0$ is a constant called the specific heat of the material. Therefore, the heat flow into $S$ is
$$\frac{\partial H}{\partial t}=\sigma \iint_S \frac{\partial u}{\partial t} d x d y,$$
which is approximately equal to
$$\sigma h^2 \frac{\partial u}{\partial t}\left(x_0, y_0, t\right),$$
since the area of $S$ is $h^2$. Now we apply Newton’s law of cooling, which states that heat flows from the higher to lower temperature at a rate proportional to the difference, that is, the gradient.

## 数学代写|实分析代写Real Analysis代考|Steady-state heat equation in the disc

After a long period of time, there is no more heat exchange, so that the system reaches thermal equilibrium and $\partial u / \partial t=0$. In this case, the time-dependent heat equation reduces to the steady-state heat equation
$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$
The operator $\partial^2 / \partial x^2+\partial^2 / \partial y^2$ is of such importance in mathematics and physics that it is often abbreviated as $\Delta$ and given a name: the Laplace operator or Laplacian. So the steady-state heat equation is written as
$$\Delta u=0,$$
and solutions to this equation are called harmonic functions.
Consider the unit disc in the plane
$$D=\left{(x, y) \in \mathbb{R}^2: x^2+y^2<1\right},$$
whose boundary is the unit circle $C$. In polar coordinates $(r, \theta)$, with $0 \leq r$ and $0 \leq \theta<2 \pi$, we have
$$D={(r, \theta): 0 \leq r<1} \quad \text { and } \quad C={(r, \theta): r=1}$$
The problem, often called the Dirichlet problem (for the Laplacian on the unit disc), is to solve the steady-state heat equation in the unit disc subject to the boundary condition $u=f$ on $C$. This corresponds to fixing a predetermined temperature distribution on the circle, waiting a long time, and then looking at the temperature distribution inside the disc.

## 数学代写|实分析代写Real Analysis代考|Derivation of the heat equation

$$H(t)=\sigma \iint_S u(x, y, t) d x d y,$$

$$\frac{\partial H}{\partial t}=\sigma \iint_S \frac{\partial u}{\partial t} d x d y,$$

$$\sigma h^2 \frac{\partial u}{\partial t}\left(x_0, y_0, t\right),$$

## 数学代写|实分析代奇eal Analysis代考|Steady-state heat equation in the disc

$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$

$$\Delta u=0,$$

\left 缺少或无法识别的分隔符

$$D=(r, \theta): 0 \leq r<1 \quad \text { and } \quad C=(r, \theta): r=1$$

## 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。