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# 数学代写|偏微分方程作业代写Partial Differential Equations代考|Some Final Remarks

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## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Nonsmooth Solutions

3.12.1. Nonsmooth Solutions. When solving the IVP (3.5) to arrive at D’Alembert’s formula, we implicitly made some assumptions on the initial data $\phi$ and $\psi$. Alternatively, in order to show that D’Alembert’s formula does indeed solve the wave equation, we would need $\phi$ to be $C^{2}$ and $\psi$ to be $C^{1}$. On the other hand, we do not need these assumptions to just plug $\phi$ and $\psi$ into D’Alembert’s formula: Indeed, in our examples of the plucked string and the hammer blow, we simply applied D’Alembert’s formula to, respectively, a $\phi$ which was not $C^{1}$ and a $\psi$ which was not even continuous. While we discussed the resulting solutions and plotted profiles, in what sense do we have a solution? Do the discontinuities count or can they be ignored? The answer to these questions will be revealed once we introduce the notion of a solution in the sense of distributions (cf. Chapter 9). This more integral-focused notion of a solution will indeed allow us to conclude that our placement of these nonsmooth functions in D’Alembert’s formula does indeed yield a solution, albeit a distributional one.

## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Heterogeneous Media and Scattering

Wave propagation through a heterogeneous string leads to a wave equation in which the speed parameter $c$ is spatially dependent:
$$u_{t t}=c^{2}(x) u_{x x}$$
This spatial dependence of $c$ is determined by the physical nature of the string. With certain general assumptions on the coefficient function $c(x)$, the resulting IVPs are well-posed. However, closed form solution formulas are not usually available. An interesting feature here is the scattering of waves, whereby aspects of the inhomogeneous medium can cause incoming waves to have a change in their outgoing behavior. For example, consider the $1 \mathrm{D}$ wave equation with
$$c(x)= \begin{cases}c_{1}, & x<0 \\ c_{2}, & x \geq 0\end{cases}$$ where $c_{1}>0, c_{2}>0$ with $c_{1} \neq c_{2}$. You can think of this as a model for a vibrating string made of two materials of differing mass densities, one for $x<0$ and the other for $x>0$. A signal concentrated at say $x_{0}<0$ will propagate in the positive $x$ direction until it interacts with the change in medium at $x=0$. This interaction will cause the wave to be scattered and give rise to both a change in its amplitude and speed. In Exercise 3.22, We give more details and outline the steps in order to find an exact solution.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|Finite Propagation Speed, Other “Wave” Equations, and Dispersion

The solution to the wave equation exhibits finite propagation speed. This was illustrated in the domain of influence where we saw that information can propagate through space $(x)$ with different speeds, but no speed greater than $c$. The name “the wave equation” is mainly for historical reasons and one often loosely calls any timedependent PDE in which information (data) propagates at finite speeds “a wave equation”. For example, the simple first-order transport equations of Chapter 2 can be thought of as wave equations. On the other hand, there are also far more complicated linear and nonlinear PDEs for wave propagation, for example,

• the inviscid Burgers equation which we saw in Section 2.3,
• the Schrödinger, Klein-Gordon, Telegraph, and KdV equations.
The latter two PDEs exhibit a fundamental process associated with the propagation of waves known as dispersion. Dispersion for our vanilla wave equation proves rather trivial. It is true that light propagation (as an electromagnetic wave) satisfies the 3D wave equation with $c$ being the speed of light; however, it does so in a vacuum. This is rather uninteresting as there is literally nothing that “can be seen”. On the other hand, when lights hits a glass prism, the waves of different wavenumbers have different velocities and the resulting dispersion produces a spectrum of colors. In general, dispersion describes the way in which an initial disturbance of the wave medium distorts over time, explicitly specifying how composite plane wave components of an initial signal evolve temporally for different wavenumbers.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|NONSMOOTH SOLUTIONS

3.12.1。不光滑的解决方案。解决IVP时3.5为了得出 D’Alembert 公式，我们隐含地对初始数据做了一些假设φ和ψ. 或者，为了证明 D’Alembert 公式确实解决了波动方程，我们需要φ成为C2和ψ成为C1. 另一方面，我们不需要这些假设来插入φ和ψ进入 D’Alembert 公式：确实，在我们的拨弦和锤击示例中，我们简单地将 D’Alembert 公式分别应用于φ这不是C1和一个ψ这甚至不是连续的。虽然我们讨论了最终的解决方案和绘制的配置文件，但在什么意义上我们有解决方案？不连续性是否重要，还是可以忽略不计？一旦我们在分布的意义上引入解决方案的概念，这些问题的答案就会揭晓CF.CH一种p吨和r9. 这种更注重积分的解决方案概念确实使我们能够得出结论，我们将这些非光滑函数置于 D’Alembert 公式中确实会产生一个解决方案，尽管是一个分布解决方案。

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|HETEROGENEOUS MEDIA AND SCATTERING

C(X)={C1,X<0C2,X≥0在哪里C1>0,C2>0和C1≠C2. 您可以将其视为由两种不同质量密度的材料制成的振动弦的模型，一种用于X<0另一个为X>0. 一个信号集中在sayX0<0将向正面传播X方向直到它与介质的变化相互作用X=0. 这种相互作用将导致波散射并引起其幅度和速度的变化。在练习 3.22 中，我们给出了更多细节并概述了步骤，以便找到一个精确的解决方案。

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|FINITE PROPAGATION SPEED, OTHER “WAVE” EQUATIONS, AND DISPERSION

• 我们在第 2.3 节中看到的无粘性 Burgers 方程，
• 薛定谔、克莱因-戈登、电报和 KdV 方程。
后两个偏微分方程表现出与称为色散的波传播相关的基本过程。我们的普通波动方程的色散证明是微不足道的。光的传播是真的一种s一种n和l和C吨r这米一种Gn和吨一世C在一种在和满足 3D 波动方程C是光速；但是，它是在真空中进行的。这是相当无趣的，因为实际上没有“可以看到”的东西。另一方面，当光线照射到玻璃棱镜上时，不同波数的波具有不同的速度，由此产生的色散会产生光谱。一般来说，色散描述了波介质的初始扰动随时间扭曲的方式，明确指定了初始信号的复合平面波分量如何针对不同的波数随时间演变。