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# 数学代写|偏微分方程代考Partial Differential Equations代写|Poisson’s Equation (Modern Methods Revisited)

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Poisson’s Equation (Modern Methods Revisited)

The ideas presented thus far in this chapter as well as in Chapter 2 have provided the required mathematical tools to solve Laplace’s equation and its variants, namely the Laplace-Dirichlet, Laplace-Neumann, and Laplace-Robin equations. The next step is to solve the inhomogeneous version of Laplace’s equation, also called Poisson’s equation $\Delta u=\rho(\boldsymbol{x})$. Attempting the classical approach of separation of variables in $\mathbb{R}^3$ leads to $u(x)=u(x, y, z)=X(x) \cdot Y(y) \cdot Z(z)$ and $\rho(x)=$ $\rho(x, y, z)=\xi(x) \cdot \psi(y) \cdot \zeta(z)$, so that $\Delta(X(x) \cdot Y(y) \cdot Z(z))=\xi(x) \cdot \psi(y) \cdot \zeta(z)$ or $X^{\prime \prime}(x) \cdot Y(y) \cdot Z(z)+$ $X(x) \cdot Y^{\prime \prime}(y) \cdot Z(z)+X(x) \cdot Y(y) \cdot Z^{\prime}(z)=\xi(x) \cdot \psi(y) \cdot \zeta(z)$. Assuming that none of $X(x), Y(y)$, or $Z(z)$ is identically zero and then dividing through by $X(x) \cdot Y(y) \cdot Z(z)$ produces
$$\frac{X^{\prime \prime}(x)}{X(x)}+\frac{Y^{\prime \prime}(y)}{Y(y)}+\frac{Z^{\prime \prime}(z)}{Z(z)}=\frac{\xi(x) \cdot \psi(y) \cdot \zeta(z)}{X(x) \cdot Y(y) \cdot Z(z)} .$$
If the right-hand side of (3.4.1) is not constant ${ }^{13}$, then the separation of variables method will fail. Therefore, an alternative to the classical approach is required. This invites a return to the modern methods initiated in Chapter 2.
In particular, suppose that $G$ satisfies the Laplace distribution equation
$$\Delta G(x)=\delta\left(x-x_o\right), x \in \Omega \subset \mathbb{R}^n$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|The Heat Equation (A Blend of Modern and Classical)

The temperature distribution in a heat-conducting medium is described by the heat equation $\frac{\partial u}{\partial t}(\boldsymbol{x}, t)=\sigma^2 \Delta u(\boldsymbol{x}, t)$ for $\boldsymbol{x} \in \Omega \subset \mathbb{R}^n$ and $\kappa=\sigma^2$ is called the conductivity coefficient. Throughout this section, the conductivity coefficient will be treated as a constant. For $n=1$, the heat equation models the temperature distribution over an infinitely thin wire of length $L=|\Omega|$. When the wire is considered over an interval $\Omega=[a, b]$, then $L=|\Omega|=b-a$. As is demonstrated in the development of Laplace’s and Poisson’s equation, the initial and boundary conditions are crucial elements in the construction of solutions to the heat equation. The remainder of this section will focus on solutions of the heat equation when $\Omega$ is a compact set in $n=1,2$, and 3 spatial dimensions. This will be accomplished by the usual method of separation of variables. In the case in which the domain is unbounded, and especially $\Omega=\mathbb{R}^n$, then a new approach is required: The Fourier transform which will be deferred to Chapter 4.

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Poisson’s Equation (Modern Methods Revisited)

$$\frac{X^{\prime \prime}(x)}{X(x)}+\frac{Y^{\prime \prime}(y)}{Y(y)}+\frac{Z^{\prime \prime}(z)}{Z(z)}=\frac{\xi(x) \cdot \psi(y) \cdot \zeta(z)}{X(x) \cdot Y(y) \cdot Z(z)} .$$

$$\Delta G(x)=\delta\left(x-x_o\right), x \in \Omega \subset \mathbb{R}^n$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|The Heat Equation (A Blend of Modern and Classical)

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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。