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MATH348 - November 21, 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, π).
(a) Write down the heat and wave equations and any initial conditions needed for unique solutions. For each, what does the unknown function u measure?
(b) Suppose we are given the boundary conditions ux(0, t) = 0 and ux(π, t) = 0 for each problem. Explain the physical meaning of each boundary condition for both the heat equation and wave equation.
(c) The following graph gives the only nonzero initial configuration for each problem. Describe and/or draw the associated heat and wave dynamics. If there is an equilibrium state then be sure to state it.
6
-
@ @ @ @ @ x = π
y = π f^ (x) =^ u(x,^ 0)
y
x
- (10 Points)
(a) The following table contains different boundary conditions for the ODE, F ′′^ + λF = 0, λ ∈ [0, ∞). 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) Calculate the Fourier transform of f (x) = δ− 2 (x) − δ 2 (x).
(c) Noting that F {f ′} = −iωF {f }, use Fourier transforms to find the general solution to ut = uxx in the Fourier domain.
(d) Find the value of λ such that F ′′^ + λF = 0 subject to F (0) = 0, F ′(π) = 0 has a non-zero solution.
(e) Assuming u(x, t) = F (x, y)G(t), find the time ODE and space PDE consistent with utt + ut = uxx + uyy.
- (10 Points) Find the unique solution to,
∂^2 u ∂t^2
∂^2 u ∂x^2
x ∈ (0, π) t ∈ (0, ∞)
u(0, t) = 0, u(π, t) = 0, (2) u(x, 0) = 0, (3) ut(x, 0) = g(x). (4)
Math348 - November 21, 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, π).
(a) Write down the heat and wave equations and any initial conditions needed for unique solutions. For each, what does the unknown function u measure?
(b) Suppose we are given the boundary conditions ux(0, t) = 0 and u(π, t) = 0 for each problem. Explain the physical meaning of each boundary condition for both the heat equation and wave equation.
(c) The following graph gives the only nonzero initial configuration for each problem. Describe and/or draw the associated heat and wave dynamics. If there is an equilibrium state then be sure to state it.
6
-
@ @ @ @ @ x = π
y = π f^ (x) =^ u(x,^ 0)
y
x
- (10 Points)
(a) Show that u(x, t) = ln(x^2 + y^2 ) is a solution to uxx + uyy = 0.
(b) Show that u(r) = r−^1 is a solution to urr + 2r−^1 ur = 0.
- (10 Points) Using separation of variables, find the three ODEs consistent with the PDE utt + ut = uxx + uyy.
- (10 Points) Find the unique solution to,
∂u ∂t
∂^2 u ∂x^2
x ∈ (0, π) t ∈ (0, ∞)
ux(0, t) = 0, ux(π, t) = 0, (6) u(x, 0) = f (x). (7)
- (10 Points)
(a) The following table contains different boundary conditions for the ODE, F ′′^ + λF = 0, λ ∈ [0, ∞). 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) Calculate the Fourier transform of f (x) = δ 2 (x) − δ− 2 (x).
(c) Noting that F {f ′} = −iωF {f }, use Fourier transforms to find the general solution to ut = uxx in the Fourier domain.
(d) Assuming u(x, t) = F (x, y)G(t), find the time ODE and space PDE consistent with utt + ut = uxx + uyy.
(e) Given that uh(x, t) =
∑^ ∞
n=
sin(nx)e−n (^2) t is the homogeneous solution to ut = uxx +F (x, t) where F (x, t) =
∑^ ∞
n=
fn(t) sin(nx), find the ODEs associated with the time dynamics of the general solution to the PDE.
- (10 Points) Show that u(x, y, z) = (x^2 + y^2 + z^2 )−^1 /^2 is a solution to uxx + uyy + uzz = 0.
- (10 Points) Given,
F ′′^ + λF = 0, λ ∈ [0, ∞)
Find the non-zero functions, F , and values for λ that satisfy the following boundary conditions.
(a) F ′(0) = 0 and F ′(π) = 0
(b) F ′(0) = 0 and F (π) = 0
(c) F (0) = 0 and F ′(π) = 0
