Problem set #6

Consider the diffusion equation in 1-dimension with fixed boundary conditions.

$$\begin{eqnarray*}u_t & = & \frac{1}{1000} u_{xx} \\ u(0,t) & = & 0 \\ u(2,t) & = & 0 \end{eqnarray*}$$

1.

Use separation of variables to find a solution to this problem in terms of Fourier series.

2.

Solve with problem with initial conditions

$$u(x,0) = \begin{cases} 2 & \frac{1}{3} \leq x \leq \frac{2}{3... ...{3} \leq x \leq \frac{5}{3} \\ 0 & {\rm otherwise} \end{cases}$$

Examine plots of the solution at t=0 and t=1 using the first 5 terms, the first 10 terms, the first 20 terms and the first 50 terms of the series solution.

3.

Solve with problem with initial conditions

$$u(x,0) = \begin{cases} 24\left(x-\frac{1}{3}\right) & \frac{1... ...{2} \leq x \leq \frac{5}{3} \\ 0 & {\rm otherwise} \end{cases}$$

Examine plots of the solution at t=0 and t=1 using the first 5 terms, the first 10 terms, the first 20 terms and the first 50 terms.

4.

Both of the initial conditions describe initial concentrations near x=1/2 and x=3/2. For the two different initial conditions above, which solution converges faster. Explain your conclusion in terms of the Fourier coefficients. Hint: Look at the rate of convergences of the coefficients.