5.6: Inhomogeneous Equations

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Consider a rod of length $2$m, laterally insulated (heat only flows inside the rod). Initially the temperature $u$ is

\[\frac{1}{k}\sin\left( \frac{\pi x}{2} \right) + 500 \text{ K}. \nonumber \]

The left and right ends are both attached to a thermostat, and the temperature at the left side is fixed at a temperature of $500\text{ K}$ and the right end at $100\text{ K}$. There is also a heater attached to the rod that adds a constant heat of $\sin\left( \frac{\pi x}{2}\right)$ to the rod. The differential equation describing this is inhomogeneous

\[\begin{aligned} \dfrac{\partial}{\partial t} u &= k \dfrac{\partial^2}{\partial x^2} u + \sin\left(\frac{\pi x}{2}\right),\nonumber\\[4pt] u(0,t) &= 500,\nonumber\\[4pt] u(2,t) &= 100,\nonumber\\[4pt] u(x,0) &= \frac{1}{k}\sin\left( \frac{\pi x}{2} \right) + 500.\end{aligned} \nonumber \]

ince the inhomogeneity is time-independent we write

\[u(x,t) = v(x,t) + h(x), \nonumber \]

where $h$ will be determined so as to make $v$ satisfy a homogeneous equation. Substituting this form, we find

\[\dfrac{\partial}{\partial t} v = k \dfrac{\partial^2}{\partial x^2} v + k h'' + \sin\left( \frac{\pi x}{2} \right). \nonumber \]

To make the equation for $v$ homogeneous we require \[h''(x) = - \frac{1}{k}\sin\left( \frac{\pi x}{2} \right), \nonumber \]

which has the solution

\[h(x) = C_1 x + C_2 + \frac{4}{k\pi^2}\sin\left( \frac{\pi x}{2} \right). \nonumber \]

At the same time we let $h$ carry the boundary conditions, $h(0)=500$, $h(2)=100$, and thus \[h(x) = -{200} x + 500 + \frac{4}{k\pi^2}\sin\left( \frac{\pi x}{2} \right). \nonumber \] The function $v$ satisfies

\[\begin{aligned} \dfrac{\partial}{\partial t} v &= k \dfrac{\partial^2}{\partial x^2} v, \nonumber\\[4pt] v(0,t) & = v(\pi,t) = 0, \nonumber\\[4pt] v(x,0) & = u(x,0) - h(x) = {200} x.\end{aligned} \nonumber \]

This is a problem of a type that we have seen before. By separation of variables we find

\[v(x,t) = \sum_{n=1}^\infty b_n \exp (- \frac{n^2\pi^2}{4}kt) \sin \frac{n\pi}{2}x. \nonumber \]

The initial condition gives

\[\sum_{n=1}^\infty b_n \sin nx = {200} x. \nonumber \]

from which we find

\[b_n = (-1)^{n+1} \frac{800}{n\pi}. \nonumber \]

And thus

\[u(x,t) = -\frac{200} x + 500 + \frac{4}{\pi^2 k} \sin \left(\frac{\pi x}{2}\right) + \frac{800}{\pi} \sum_{n=1}^\infty \frac{(-1)^n}{n+1} \sin \left(\frac{\pi nx}{2}\right) e^{-k (n \pi/2)^2 t}. \label{eq.10} \]

Note: as $t\rightarrow \infty$, $u(x,t) \rightarrow -\frac{400}{\pi} x + 500 + \frac{\sin \frac{\pi}{2}x}{k}$. As can be seen in Fig. $\PageIndex{1}$ this approach is quite rapid – we have chosen $k=1/500$ in that figure, and summed over the first 60 solutions.

$Inhomgen2.png$

Figure $\PageIndex{1}$: Time dependence of the solution to the inhomogeneous Equation \ref{eq.10}.