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EGN 5456 Homework 1 Fall 1996

1. A bar of length 2 has a coefficient of heat conduction 3. Initially the bar has a temperature of 100C; however, one end of the bar is welded to a large steel container of ice water. The other end of the bar is insulated. List the PDE, draw the region in which it applies, list the boundary conditions and initial conditions in the figure. State what the type of the equation is. Show the qualitative appearance of the solution, in the T,x-plane for t=0, ( but small), and .
2. Air is flowing through a duct of length 5 at a speed 2. The initial air in the duct has entropy s=1, but between times t=1 and t=2 hotter air enters the duct with entropy s=2. List the PDE, draw the region in which it applies, list any boundary conditions and initial conditions in the figure. State what the type of the equation is. Show the qualitative appearance of the solution, in the s,x-plane for , t=1.5, , and but small). At which time has all air in the duct returned to entropy s=1?
3. A guitar string of length is pushed up at the center using a pick and released from this position. The tension force in the string is 2 and the mass per unit length is 1. List the PDE, draw the region in which it applies, list any boundary conditions and initial conditions in the figure. State what the type of the equation is. Show the qualitative appearance of the solution, in the T,x-plane at time and at time .
4. A plate with length 2 and height 1 is welded at three sides to containers of ice water, but at the fourth side to a container of boiling water. List the PDE, draw the region in which it applies, list any boundary conditions and initial conditions in the figure. State what the type of the equation is. Draw isotherms, (contour lines of constant temperature), qualitatively. Describe precisely where the temperature is discontinuous and where it is continuous.
5. One half of a copper ring is initially at a temperature 100C while the other half is at temperature 0C. List the PDE, draw the region in which it applies, list any boundary conditions and initial conditions in the figure. State what the type of the equation is. Draw the solution qualitatively in the T,x-plane for times t=0, , and .
6. A disk is at half of it circumference in contact with ice water, at the other half with boiling water. List the PDE, draw the region in which it applies, list the boundary conditions and initial conditions in the figure. State what the type of the equation is. Draw lines of constant temperature in the disk: don't get fooled.
7. A disk is initially at temperature 100C, but is at its perimeter welded to a container of ice water. List the PDE, draw the region in which it applies, list the boundary conditions and initial conditions in the figure. State what the type of the equation is. Draw contour lines and T,r-planes at times and .

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