4.3: Inhomogeneous Equations

Here we consider the initial value problem

$$\begin{eqnarray} \label{inh1} \Box u&=&w(x,t)\ \ \mbox{on}\ x\in\mathbb{R}^n,\ t\in\mathbb{R}^1\\ \label{inh2} u(x,0)&=&f(x)\\ \label{inh3} u_t(x,0)&=&g(x), \end{eqnarray}$$

where $\Box u:=u_{tt}-c^2\triangle u$. We assume $f\in C^3$, $g\in C^2$ and $w\in C^1$, which are given.

Set $u=u_1+u_2$, where $u_1$ is a solution of problem ($\ref{inh1}$)-($\ref{inh3}$) with $w:=0$ and $u_2$ is the solution where $f=0$ and $g=0$ in ($\ref{inh1}$)-($\ref{inh3}$). Since we have explicit solutions $u_1$ in the cases $n=1$, $n=2$ and $n=3$, it remains to solve

$$\begin{eqnarray} \label{duhu2gl} \Box u&=&w(x,t)\ \ \mbox{on}\ x\in\mathbb{R}^n,\ t\in\mathbb{R}^1\\ \label{duhu2in1} u(x,0)&=&0\\ \label{duhu2in2} u_t(x,0)&=&0. \end{eqnarray}$$

The following method is called Duhamel's principle which can be considered as a generalization of the method of variations of constants in the theory of ordinary differential equations.

To solve this problem, we make the ansatz

$$\begin{equation} \label{duh1} u(x,t)=\int_0^t\ v(x,t,s)\ ds, \end{equation}$$
where $v$ is a function satisfying
$$\begin{equation} \label{duh2} \Box v=0\ \ \mbox{for all}\ s \end{equation}$$

and

$$\begin{equation} \label{duh3} v(x,s,s)=0. \end{equation}$$

From ansatz ($\ref{duh1}$) and assumption ($\ref{duh3}$) we get

$$\begin{eqnarray} u_t&=&v(x,t,t)+\int_0^t\ v_t(x,t,s)\ ds,\nonumber\\ \label{duh4} &=&\int_0^t\ v_t(x,t,s). \end{eqnarray}$$

It follows $u_t(x,0)=0$. The initial condition $u(x,t)=0$ is satisfied because of the ansatz ($\ref{duh1}$). From ($\ref{duh4}$) and ansatz ($\ref{duh1}$) we see that

$$\begin{eqnarray*} u_{tt}&=&v_t(x,t,t)+\int_0^t\ v_{tt}(x,t,s)\ ds,\\ \triangle_x u&=&\int_0^t\ \triangle_x v(x,t,s)\ ds. \end{eqnarray*}$$

Therefore, since $u$ is an ansatz for ($\ref{duhu2gl}$)-($\ref{duhu2in2}$),

$$\begin{eqnarray*} u_{tt}-c^2\triangle_x u&=&v_t(x,t,t)+\int_0^t(\Box v)(x,t,s)\ ds\\ &=&w(x,t). \end{eqnarray*}$$

Thus necessarily $v_t(x,t,t)=w(x,t)$, see ($\ref{duh2}$). We have seen that the ansatz provides a solution of ($\ref{duhu2gl}$)-($\ref{duhu2in2}$) if for all $s$

$$\begin{equation} \label{duh5} \Box v=0,\ \ v(x,s,s)=0,\ \ v_t(x,s,s)=w(x,s). \end{equation}$$
Let $v^*(x,t,s)$ be a solution of
$$\begin{equation} \label{duh6} \Box v=0,\ \ v(x,0,s)=0,\ \ v_t(x,0,s)=w(x,s), \end{equation}$$
then

$$v(x,t,s):=v^*(x,t-s,s)\]

is a solution of ($\ref{duh5}$).
In the case $n=3$, where $v^*$ is given by, see Theorem 4.2,

$$v^*(x,t,s)=\frac{1}{4\pi c^2 t}\int_{\partial B_{ct}(x)}\ w(\xi,s)\ dS_\xi.\]

Then

$$\begin{eqnarray*} v(x,t,s)&=&v^*(x,t-s,s)\\ &=&\frac{1}{4\pi c^2 (t-s)}\int_{\partial B_{c(t-s)}(x)}\ w(\xi,s)\ dS_\xi. \end{eqnarray*}$$

from ansatz ($\ref{duh1}$) it follows

$$\begin{eqnarray*} u(x,t)&=&\int_0^t\ v(x,t,s)\ ds\\ &=&\frac{1}{4\pi c^2}\int_0^t\ \int_{\partial B_{c(t-s)}(x)}\ \frac{w(\xi,s)}{t-s}\ dS_\xi ds. \end{eqnarray*}$$

Changing variables by $\tau=c(t-s)$ yields

$$\begin{eqnarray*} u(x,t)&=&\frac{1}{4\pi c^2}\int_0^{ct}\ \int_{\partial B_{\tau}(x)}\ \frac{w(\xi,t-\tau/c)}{\tau}\ dS_\xi d\tau\\ &=&\frac{1}{4\pi c^2} \int_{ B_{ct}(x)}\ \frac{w(\xi,t-r/c)}{r}\ d\xi, \end{eqnarray*}$$

where $r=|x-\xi|$.

Formulas for the cases $n=1$ and $n=2$ follow from formulas for the associated homogeneous equation with inhomogeneous initial values for these cases.

Theorem 4.4. The solution of

$$\Box u=w(x,t),\ \ u(x,0)=0,\ \ u_t(x,0)=0,$$

where $w\in C^1$, is given by:

Case $n=3$:

$$u(x,t)=\frac{1}{4\pi c^2} \int_{ B_{ct}(x)}\ \frac{w(\xi,t-r/c)}{r}\ d\xi,$$

where $r=|x-\xi|$, $x=(x_1,x_2,x_3)$, $\xi=(\xi_1,\xi_2,\xi_3)$.

Case $n=2$:

$$u(x,t)=\frac{1}{4\pi c}\int_0^t\ \left( \int_{ B_{c(t-\tau)}(x)}\ \frac{w(\xi,\tau)}{\sqrt{c^2(t-\tau)^2-r^2}}\ d\xi\right)\ d\tau,$$

$x=(x_1,x_2)$, $\xi=(\xi_1,\xi_2)$.

Case $n=1$:

$$u(x,t)=\frac{1}{2c}\int_0^t\ \left(\int_{x-c(t-\tau)}^{x+c(t-\tau)}\ w(\xi,\tau)\ d\xi\right)\ d\tau.$$

Remark. The integrand on the right in formula for $n=3$ is called retarded potential. The integrand is taken not at $t$, it is taken at an earlier time $t-r/c$.

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.