PHYSICS 312
Problem set 3 1999:
Solution

a:
The general solution to

$$\frac{d^2T_S}{dx^2}-\gamma^2(T_S-T_0)=0$$

is

$$T_S(x)=T_0+Ae^{\gamma x}+Be^{-\gamma x}$$

The boundary conditions T_S=0 for x=0 and x=a allow us to solve for A and B to obtain

$$T_S=T_0\left(1-\frac{e^{\gamma x}}{e^{\gamma a}+1}- \frac{e^{-\gamma x}}{e^{-\gamma a}+1}\right)$$

b:
Substituting

u=T-T_S

into

$$\frac{\partial^2T_S}{\partial x^2}-\gamma^2(T_S-T_0)=\frac{1}{k} \frac{\partial T}{\partial t}$$

gives

$$\frac{\partial^2 u}{\partial x^2}-\gamma^2u=\frac{\partial u}{\partial t}$$

c:
The initial temperature distribution for u is

$$u(x,0)=-T_0\left(1-\frac{e^{\gamma x}}{e^{\gamma a}+1}- \frac{e^{-\gamma x}}{e^{-\gamma a}+1}\right)$$

d:
The initial temperature distribution can be written on the form

$$u(x,0)=\sum_{n=1}^\infty c_n\sin(\frac{n\pi x}{a})$$

where

$$c_n=-\frac{2T_0}{a}\int_0^a\left(1-\frac{e^{\gamma x}}{e^{\ga... ... \frac{e^{-\gamma x}}{e^{-\gamma a}+1}\right)\sin{n\pi x}{a}dx$$

We can solve the time dependent problem formally by assuming that the Fourier coefficients are time dependent i.e.

$$u(x,t)=\sum_{n=1}^\infty \alpha_n(t)\sin(\frac{n\pi x}{a})$$

where $\alpha_n$ satisfies the initial condition

$$\alpha_n(0)=c_n$$

Substituting into the partial differential equation gives

$$(-\frac{n^2\pi^2}{a^2}-\gamma^2)\alpha_n=\frac{1}{k} \frac{d\alpha_n(t)}{dt}$$

with solution

$$\alpha_n(t)=c_n\exp[-(\frac{n^2\pi^2}{a^2}+\gamma^2)kt]$$

e:
In order to use the method of separation of variables the partial differential equations must be homogeneous. This is necessary in order to use the superposition principle.

It might be tempting to make the equation homogeneous by the substitution

u=T-T₀

We then get the homogeneous problem

$$\frac{\partial^2 u}{\partial x^2}-\gamma^2u=\frac{\partial u}{\partial t}$$

with initial condition

u(x,0)=-T₀

If we apply the method of separation of varables we will not get the correct solution!. The problem is that the method of separation of variables method give particular solutions to the partial differential equation. To these we should in principle add a time-independent solution to

$$\frac{\partial^2 u}{\partial x^2}-\gamma^2u=0$$

The appropriate solution to use is the steady state temperature. Only when we subtract this temperature, are we guaranteed that the particular solution is the complete solution.

For part f: see Maple worksheet.

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