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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION E
Name:
This test has 8 pages. Work out everything as far as you can before making decimal approximations.
- Suppose that a string in a still fluid is tied down at x = 0 and x = 1. The force of resistance to motion of the string coming from the fluid is proportional to the velocity of the string. Which of the following is the equation of motion of this string?
(a) ∂u ∂t
= − 2 k
∂u ∂x
∂^2 u ∂x^2 (b) ∂^2 u ∂t^2
= − 2 k ∂u ∂x
- c^2 ∂^2 u ∂x^2 (c) ∂^2 u ∂t^2
= − 2 k
∂u ∂t
∂^2 u ∂x^2 (d) ∂u ∂t
= c^2 ∂^2 u ∂x^2 (e) ∂^2 u ∂t^2
= sinh u +
∂^2 u ∂x^2 Which two of the following are modes of the string?
(a) cos (πnct) sin (πnx) (b) (cos (πnct) − kt) sin (πnx) (c) cos (πnct) cos (πnx) (d) exp
− (πnc)^2 t
sin (πnx) (e) e−kt^ sin
t
π^2 c^2 n^2 − k^2
sin (πnx) (f) sin (πnct) sin (πnx)
Date: March 22, 2002. 1
(g) exp
− (πnc)^2 t
cos (πnx)
(h)
e−kt^ cos
t
π^2 c^2 n^2 − k^2
sin (πnx) (i) sinh(πnt) sin(πnx)
- Suppose that a wire of length L = π and diffusivity c = 1 with initial temperature 100o^ is placed in an insulating tube. One end is kept at 100o^ with a thermostat, while the other is kept at 0o.
(a) Find the temperature u(x, t). Your answer should be a sum of a steady state (in this case a linear function of x) and a solution of a similar problem with 0 o^ at both ends. (b) Draw a picture of what the temperature looks like at time t = 0 and at large time t when it has not quite reached the steady state.
- Consider the wave equation with gravitational force
∂^2 u ∂t^2
= c^2
∂^2 u ∂x^2
− g.
(a) Find the steady state (i.e. u = u(x) independent of time) which satisfies u = u 0 fixed (u 0 some constant) at x = 0 and u = u 1 (some other constant) at x = L. (b) Draw a picture of what it looks like for small positive g (for example, on the moon) and for large positive g (for example, on Jupiter).
- A wire of length L = 1 with insulated ends and diffusivity c = 1 has temperature f (x) = x at time t = 0. Find the temperature u(x, t) at time t and position x.
