ECH5842 CHEMICAL ENGINEERING ANALYSIS R.Chella

HOMEWORK Fall'97

1. Consider the inhomogeneous, two-dimensional diffusion equation

$$u_t - u_{xx} - u_{yy} = e^{-\vert x\vert}y\sin \omega t, \qquad \ -\infty < x < \infty, \qquad 0 \leq y \leq \pi$$

The boundary conditions are:

and the initial condition is:

$$u(x,y,0) = \left\{\begin{array} {ll} a = {\rm constant} &\m... ... x\vert \leq 1$}\\ 0 & \mbox{otherwise} \end{array} \right.$$

Use Fourier transforms with respect to x and eigenfunction expansions with respect to y to derive an integral representation of the form:

$$u(x,y,t) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}\ \overline{u}(\omega,y,t) e^{-i \omega x}d\omega$$

for the solution. You need not evaluate the inversion integral explicitly. However, give the explicit series form for $\overline{u}(\omega,y,t)$.

2. Solve using Fourier transforms:

$$u_t - u_{xx} = 0, \qquad 0 \leq x < \infty, t \gt 0$$

with initial condition:

u(x,0) = 0

and boundary conditions:

Compare with the solution obtained in class using the Laplace Transform technique.

3. Show using Fourier transforms that the solution to:

$$u_t - u_{xx} + tu = 0, \qquad 0 \leq x < \infty, t \gt 0$$

with initial condition:

u(x,0) = e^-x

and boundary conditions:

is

$$u(x,t)= \frac{2}{\pi}e^{-t^2/2} \int_0^\infty \ \frac{e^{-\omega^2 t} \cos \omega x}{1+\omega^2} \ d\omega$$

4.Consider the diffusion equation with variable coefficient:

Solve using integral transforms. (a) What is the most general choice for the constants C₁, C₂, and C₃ for which the solution can be obtained in similarity form? (b) For the choice of constants in part (a), calculate the solution and evaluate all integration constants explicitly.