# 数学代写|偏微分方程代写partial difference equations代考|MATH1470

## 数学代写|偏微分方程代写partial difference equations代考|Three Prototypical Equations

It is largely believed that use of partial differential equations in solving problems concerning some physical phenomena started with Jean d’ Alembert’s work on vibrations of a string. Most concepts we shall discuss in rest of the book arose from the related vibrating string controversy ${ }^2$ that started in 1747 soon after Jean d’ Alembert (17171783) published his travelling wave solution of the one dimensional wave equation. The main issue was how to explain the connection between a physical problem and the proposed mathematical descriptions. The controversy spanned over the eighteenth century, and it got involved many eminent scientist with diverse backgrounds such as Daniel Bernoulli (1700-1782), Leonhard Euler(1707-1783), and Joseph-Louis Lagrange (1736-1813). As discussed later in Chap. 8, Lagrange resolved the main issue partially in 1759.

More generally, let $\Omega \subseteq \mathbb{R}^n$ be a nice domain. ${ }^3$ Suppose $u=u(\boldsymbol{x}, t): \Omega \times$ $\mathbb{R}^{+} \rightarrow \mathbb{R}$ is a sufficiently smooth function representing a physical quantity such as the density, velocity, pressure, viscosity, and temperature. A typical partial differential equation for a (unknown) function $u=u(\boldsymbol{x}, t)$ is an equation that gives a relation between the time rate of change $u_t=\partial u / \partial t$ of a physical quantity $u(\boldsymbol{x}, t)$, its flux across the boundary surface $S$, and a source or $\operatorname{sink}$ function $g(\boldsymbol{x}, t)$ representing the amount of quantity being created or destroyed within the region $\Omega$. Therefore, a mathematical formulation of various types of problems concerning continuum mechanics leads to partial differential equations. We will write the partial derivatives of a function $u=u(\boldsymbol{x}, t)$ as given below:
$$\begin{gathered} q=u_t=\partial_t u=\frac{\partial u}{\partial t} ; \quad p_i=u_{x_i}=\partial_{x_i} u=\frac{\partial u}{\partial x_i} \ r_{i j}=u_{x_i x_j}=\partial_{x_i x_j}^2 u=\frac{\partial^2 u}{\partial x_i \partial x_j}=\frac{\partial^2 u}{\partial x_j \partial x_i} ; \quad \text { etc. } \end{gathered}$$
Also, the del operator $\nabla$ in $n$ variables is written as
$$\nabla \equiv\left(\frac{\partial}{\partial x_1}, \cdots, \frac{\partial}{\partial x_n}\right)$$

## 数学代写|偏微分方程代写partial difference equations代考|Laplace and Poisson Equations

Let $\Omega \subset \mathbb{R}^3$ be a compact region, and $\boldsymbol{x}=(x, y, z)$. Recall that the differential operator
$$\nabla^2: \equiv \partial_{x x}+\partial_{y y}+\partial_{z z}$$
is a 3-dimensional Laplacian in variables $x, y, z$. In view of Eq. (4.2.26), when $u_t=0$, the equation modelling steady state of a process in a system without source or sink is given by
$$\nabla^2 u=u_{x x}+u_{y y}+u_{z z}=0 .$$
As said before, this is known as the Laplace equation, say for a function $u=u(\boldsymbol{x})$ representing the steady-state temperature distribution at the position $\boldsymbol{x}$. In addition to such type of steady-state cases of conservation equations modelling transport phenomena, we also come across Laplace equations in applications concerning gravitational potential, electrostatic, and electrodynamic potential.

Example 4.6 Suppose a point mass $m$ is at point $x=x_{(} t$ at time $t \geq 0$. It is assumed that every point mass around the (source) mass $m$ experience the force of attraction governed by Newton’s gravitational law. At any fixed time $t$, the one-parameter gravitational field $\mathbf{g}(\boldsymbol{x}, t)$ defined by the point mass $m$ is given by
$$\mathbf{g}(\boldsymbol{x}, t)=G m \frac{\boldsymbol{x}-\boldsymbol{x}(t)}{|\boldsymbol{x}-\boldsymbol{x}(t)|^3}, \quad \mathbf{x} \in \Omega \subset \mathbb{R}^3,$$
where $G$ is the (relative) gravitational constant. So, the force this field exerts on any other point mass $M$ located at a point $x \in \Omega$ is given by
$$\boldsymbol{f}(\boldsymbol{x}, t)=M \mathbf{g}(\boldsymbol{x}, t)=G m M \frac{\boldsymbol{x}-\boldsymbol{x}(t)}{|\boldsymbol{x}-\boldsymbol{x}(t)|^3} .$$
Writing $r(t)=|\boldsymbol{x}-\boldsymbol{x}(t)|$ for the distance between the point masses $m$ and $M$, the force function
$$u=u(\boldsymbol{x}, t)=\frac{c}{r}, \quad \text { with } c=G m M,$$
represents the gravitational potential at point $x$ and at time $t \geq 0$, because the (spatial) gradient of the function $u=u(\boldsymbol{x}, t)$ defines the field $\mathbf{g}(\boldsymbol{x}, t)$. More generally, if $m$ is unit mass, and $\rho=\rho(\boldsymbol{x})$ is the mass density of an infinitesimal volume $\mathrm{d} V$ around the point $\boldsymbol{x}=(X, Y, Z)$ in a continuous distribution of masses within the volume $V$ of the domain $\Omega$, then the function
$$u(\boldsymbol{x})=G \iiint_V \frac{\rho(\boldsymbol{x})}{r} \mathrm{~d} V$$

# 偏微分方程代考

## 数学代写|偏微分方程代写partial difference equations代考|Three Prototypical Equations

$$q=u_t=\partial_t u=\frac{\partial u}{\partial t} ; \quad p_i=u_{x_i}=\partial_{x_i} u=\frac{\partial u}{\partial x_i} r_{i j}=u_{x_i x_j}=\partial_{x_i x_j}^2 u=\frac{\partial^2 u}{\partial x_i \partial x_j}=\frac{\partial^2 u}{\partial x_j \partial x_i}$$

$$\nabla \equiv\left(\frac{\partial}{\partial x_1}, \cdots, \frac{\partial}{\partial x_n}\right)$$

## 数学代写|偏微分方程代写partial difference equations代考|Laplace and Poisson Equations

$$\nabla^2: \equiv \partial_{x x}+\partial_{y y}+\partial_{z z}$$

$$\nabla^2 u=u_{x x}+u_{y y}+u_{z z}=0 .$$

$$\mathbf{g}(\boldsymbol{x}, t)=G m \frac{\boldsymbol{x}-\boldsymbol{x}(t)}{|\boldsymbol{x}-\boldsymbol{x}(t)|^3}, \quad \mathbf{x} \in \Omega \subset \mathbb{R}^3,$$

$$\boldsymbol{f}(\boldsymbol{x}, t)=M \mathbf{g}(\boldsymbol{x}, t)=G m M \frac{\boldsymbol{x}-\boldsymbol{x}(t)}{|\boldsymbol{x}-\boldsymbol{x}(t)|^3}$$

$$u=u(\boldsymbol{x}, t)=\frac{c}{r}, \quad \text { with } c=G m M,$$

$$u(\boldsymbol{x})=G \iiint_V \frac{\rho(\boldsymbol{x})}{r} \mathrm{~d} V$$

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