EML 5060 Analysis in Mechanical Engineering 12/11/01
Closed book Van Dommelen 5:30-7:30 pm

Show all reasoning and intermediate results leading to your answer, or credit will be lost. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator and one handwritten standard size formula sheet.

1.

Classify the PDE

y² u_xx - 2 x y u_xy + x² u_yy + y u_x + x u_y = 0

Find new coordinates and that simplify this PDE. Draw the coordinate lines (lines of constant and of constant ). Indicate the form of the PDE in terms of the new coordinates.

Solution.

2.

Find the temperature u(x,t) in a bar initially at zero temperature if one end is kept at zero temperature but the temperature at the other end is linearly increased in time:

What is the form of the temperature distribution after long times?

Solution a. Solution b.

3.

Solve the convection equation

with initial and boundary conditions

(where f(x) is a known function) using the Laplace transform method. Clean up the answer.

Solution.