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# 数学代写|偏微分方程作业代写Partial Differential Equations代考|The Wave Equation on the Half-Line with a Fixed Boundary:Reflections

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## 数学代写|偏微分方程作业代写Partial Differential Equations代考|A Dirichlet (Fixed End) Boundary Condition and Odd Reflections

We model the situation of a fixed end for the left boundary of the semi-infinite string, i.e., the end labeled by $x=0$. We prescribe that
$$u(0, t)=0 \text { for all } t \geq 0 \text {. }$$
This boundary condition ensures that there is never any vertical displacement at the boundary point. Together, we obtain the following boundary/initial value problem:
$$\begin{cases}v_{t t}=c^{2} v_{x x}, & x \geq 0, \quad t \geq 0, \ v(x, 0)=\phi(x), & x \geq 0, \ v_{t}(x, 0)=\psi(x), & x \geq 0, \ v(0, t)=0, & t \geq 0 .\end{cases}$$
As we shall see shortly, there is a reason why we use $v$ for the displacement instead of $u$. What is different from our previous IVP (3.5) is precisely that we now only work on the half-line $x \geq 0$ and have an additional condition at the left end $(x=0)$.

Can we just blindly apply D’Alembert’s formula to (3.5)? There are two issues here:

• D’Alembert’s formula for $u(x, t)$ requires information about the initial data on the interval $(x-c t, x+c t)$, and even if $x \geq 0, x-c t$ might still be negative. We only have data for $x \geq 0$.
• How do we incorporate the fixed boundary condition with the D’Alembert approach?

## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Causality with Respect to the Fixed Boundary

We will now explore the preceding formula (3.21) from the perspective of causality and the fixed boundary. In (3.21), we are presented with a dichotomy depending on whether $x<c t$ or $x \geq c t$.

• If $x \geq c t$, then the solution and domains of dependence/influence are the same as before. In this instance, for the particular time $t$, the position $x$ is sufficiently far from the fixed boundary point $x=0$; consequently the fixed boundary has no influence on the displacement $u(x, t)$.
• If $x<c t$, the situation is different; the solution $u(x, t)$ will now depend on initial displacement and velocity for string position in the interval $[c t-x, x+c t]$. This is illustrated in Figure 3.7, where one notes that what happens initially to the string in the region $(0, c t-x)$ is irrelevant and has been shadowed by the fixed boundary. Alternatively, given any point $x_{0}$, the domain of influence of displacement/velocity at $x_{0}$ and $t=0$ is illustrated in Figure 3.8.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|The Plucked String and Hammer Blow Examples with a Fixed Left End

We now return to the examples of the plucked string and the hammer blow to observe the effect of including a boundary condition. In this scenario, one can think of a long string which is clamped down at $x=0$.

Example 3.8.1 (Plucked Semi-Infinite String with a Fixed Left End). We return to the example of a plucked string. Let $c=1$ and consider the initial data
$$\phi(x)= \begin{cases}1-|x-5|, & |x-5| \leq 1 \ 0, & |x-5|>1\end{cases}$$
and $\psi \equiv 0$. We plot the profiles in Figure 3.9. These can readily be achieved by inputting the formula into some mathematical software packages. Note that fixing the left boundary has the effect of an odd reflection on an incoming wave.

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|A DIRICHLET F一世X和d和nd边界条件和奇数反射

{在吨吨=C2在XX,X≥0,吨≥0, 在(X,0)=φ(X),X≥0, 在吨(X,0)=ψ(X),X≥0, 在(0,吨)=0,吨≥0.

• 达朗贝尔公式在(X,吨)需要有关区间上的初始数据的信息(X−C吨,X+C吨), 并且即使X≥0,X−C吨可能仍然是负面的。我们只有数据X≥0.
• 我们如何将固定边界条件与 D’Alembert 方法结合起来？

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|CAUSALITY WITH RESPECT TO THE FIXED BOUNDARY

• 如果X≥C吨，则依赖/影响的解决方案和域与以前相同。在这种情况下，对于特定时间吨, 位置X离固定边界点足够远X=0; 因此，固定边界对位移没有影响在(X,吨).
• 如果X<C吨，情况不同；解决方案在(X,吨)现在将取决于区间中弦位置的初始位移和速度[C吨−X,X+C吨]. 这在图 3.7 中进行了说明，其中人们注意到该区域中的字符串最初发生的情况(0,C吨−X)是不相关的，并且被固定的边界所遮蔽。或者，给定任何一点X0，位移/速度的影响域X0和吨=0如图 3.8 所示。

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|THE PLUCKED STRING AND HAMMER BLOW EXAMPLES WITH A FIXED LEFT END

φ(X)={1−|X−5|,|X−5|≤1 0,|X−5|>1