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MATH348 - April 15, 2011 NAME: Exam II - 50 Points
In order to receive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all reasoning and work is provided. When applicable, please enclose your final answers in boxes.
- (10 Points) Conceptual Questions. For the following questions, assume that we are considering the physical problem on a bounded domain, x ∈ (0, 1).
(a) Write down the heat and wave equations and any initial conditions needed for unique solutions.
(b) Suppose we are given the boundary conditions u(0, t) = 0 and ux(1, t) = 0 for each problem. Explain the physical meaning of each boundary condition for both the heat equation and wave equation.
(c) How do solutions of these heat and wave equations behave/evolve in time?
(d) If u(x, t) is an equilibrium solution,
∂u ∂t = 0, for all t, to the heat equation then is it a solution to the wave equation? Explain.
- (10 Points) Quickies
(a) Given,
F ′′(x) + λF (x) = 0, λ ∈ [0, ∞).
The following table contains different boundary conditions for the ODE. Fill in each table element with either a yes or a no. Boundary value prob- lem has a cosine solu- tion
Boundary value prob- lem has a sine solution
Boundary value prob- lem has a nontrivial constant solution F ′(0) = 0, F ′(L) = 0 F (0) = 0, F ′(L) = 0 F (0) = 0, F (L) = 0 F ′(0) = 0, F (L) = 0 (b) Suppose that we know that,
Gn(t) = Bne−knc (^2) t , Bn ∈ R, Fn(x) = cos (knx) , kn = nπ, n = 0, 1 , 2 , · · · ,
are the temporal and spatial solutions to some heat equation. Assuming that u(x, 0) = f (x): i. Write down the general solution to the PDE.
ii. Solve for any unknown constants in terms of f (x).
iii. What is long term behavior of the temperature of this one-dimensional object?
- (10 Points) Solve the following partial differential equation,
∂^2 u ∂t^2
∂u ∂t
∂^2 u ∂x^2 , x ∈
, t ∈ (0, ∞),
u(0, t) = 0, u
, t
= 0, u(x, 0) = 0, ut(x, 0) = g(x).
