MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Kinetic Processes in Materials

3.21 Spring 2002

Samuel M. Allen and W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 3: Due Mon. March 4, 2002, Before 5PM in 4-049

Exercise 3.1

Please solve Excercise 5.2 in KPIM.

Exercise 3.2

Please solve Excercise 5.5 in KPIM.

Exercise 3.3

Please solve the one-dimensional diffusion equation on the infinite domain for the following initial and boundary conditions:

$$\begin{split} c(x,t=0) = & \left\{ \begin{array}{ll} c_o \... ...5in}} \frac{\partial c}{\partial x}(x=\infty,t) = 0 \end{split}$$

Figure 3-3-i: Trianglular initial conditions on the semi-infinite domain with zero-flux condtions at origin.

Part 1

Please find the time-dependent solution to this problem by using the superposition method.

Part 2

Please show how you would find the same solution using the Laplace transform. Find the solutions, $\hat{c}(x,p)$ which would need to be back-transformed to find the solution, but don't bother performing the back-transformation.