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## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Duhamel’s Principle

We state and prove the principle in the context of the wave equation. In Section 7.8.1, we will give a physical explanation for Duhamel’s Principle but in the context of the heat (diffusion) equation with a heat source.
First note that, by superposition, it suffices to solve
$$\left{\begin{array}{l} u_{t t}-c^{2} u_{x x}=f(x, t), \quad-\infty0, \ u(x, 0)=0, \quad-\inftys, \ w(x, s ; s)=0 \quad w_{t}(x, s ; s)=f(x, s), \quad-\infty<x<\infty \end{array}\right.$$

Effectively we are now solving the wave equation with time starting at $t=s$ (not $t=$ 0 ) and incorporating the source function $f$ into the “initial” data for the velocity at $t=s$. In invoking Duhamel’s Principle, we always place the source function $f$ in the initial data corresponding to one time derivative less than the total number of time derivatives. Note that the solution $w$ has physical dimensions of length per time (asopposed to $u$ which has dimensions of length). Duhamel’s Principle states that the solution to (3.12) is given by
$$u(x, t)=\int_{0}^{t} w(x, t ; s) d s$$
Let us check (effectively prove) Duhamel’s Principle for the wave equation by checking that $u(x, t)$ defined by (3.14) does indeed solve (3.12). In Exercise 3.17 you are asked to further show that (3.14) reduces to the exact same formula as in (3.11).

## 数学代写|偏微分方程作业代写Partial Differential Equations代考|Derivation via Green’s Theorem

As an alternate approach, we show how to prove Theorem $3.2$ by using the classical Green’s Theorem (Theorem A.2). This provides a nice example of a case where we effectively solve the PDE by integrating both sides! We follow the presentation in Strauss [6].

Proof. Since it is convenient to use $x$ and $t$ as dummy variables of integration, let’s denote the point at which we want to derive (3.11) by $\left(x_{0}, t_{0}\right)$. Thus, fix $\left(x_{0}, t_{0}\right)$ and let $D$ denote its associated domain of dependence in the $x$ – vs. $t$-plane (see Figure 3.6). We now assume a solution to (3.10) exists. We integrate both sides of the PDE over $D$; i.e., we have
$$\iint_{D} f(x, t) d x d t=\iint_{D}\left(u_{t t}-c^{2} u_{x x}\right) d x d t$$

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|DUHAMEL’S PRINCIPLE

$$\left{在吨吨−C2在XX=F(X,吨),−∞0, 在(X,0)=0,−\ infty, 在(X,s;s)=0在吨(X,s;s)=F(X,s),−∞<X<∞\对。$$

## 数学代写|偏微分方程作业代写PARTIAL DIFFERENTIAL EQUATIONS代考|DERIVATION VIA GREEN’S THEOREM

∬DF(X,吨)dXd吨=∬D(在吨吨−C2在XX)dXd吨