Math 46 Homework 2
Due April 9 at the beginning of class
(1) Page 53 # 6. You will see in part c why this is called a pitchfork bifurcation please
show on your plot.
(2) Page 54 # 10. (quick) This can be a sketch, but label clearly where stable and unstab
Worksheet #8: Initial layer
Consider the small mass damped spring equation
y + y + y = 0
y(0) = 0
t>0
y (0) = 1
1
(1) Write down and solve for the outer layer. [Hint: take = 0] Can you identify the constant?
(2) Rescale the ODE in terms of time taking =
t
Worksheet #9: WKB approximation
Consider 2 y xy = 0, where 1.
(1) For what domains is it oscillatory? Evanescent (growing/decaying)?
(2) Lets take x > 1. Write down k(x).
(3) Write down the general WKB solution.
(4) Find the coecients with the boundary co
Worksheet #4: Dimensional analysis II
In this worksheet we explore the fundamental solution for the heat equation (without calculus.)
A pulse of energy sized e is released at the origin at time t = 0. The medium has heat capacity
(energy per volume per de
Worksheet #3: ODE review
(1) Show that the transformation w = u1n makes the Bernoulli equation
u (t) + p(t)u(t) = q(t)un (t)
(which looks nonlinear) into a linear equation. In other words, equation is of the form
v (t) + p(t)v(t) = q (t). What are the fun
Worksheet #2: Scaling
Consider a chemical reactor tank with ow rate q, volume V , incoming concentration of reactant
ci . We stir the tank so concentration inside c(t) is uniform, so (chemical) mass inside is V c(t).
While inside the tank, the reactant de
Worksheet #5: Regular perturbation
Consider the rst order dierential equation
y = y + y 2
y(0) = 1.
(a) Plug y(t) = y0 (t) + y1 (t) + 2 y2 (t) + . . . into the ODE.
(b) Collect the 0 terms. What initial condition does y0 satisfy?
(c) Collect the 1 terms.
Worksheet #6: Asymptotic analysis
Part A
(1) Is tan = o() as 0?
(2) Is tan = O() as 0?
Part B
Let f (t, ) = tan t.
(1) Is f (t, ) uniformly convergent to zero on the interval (0, /4) as 0? (Hint: graph f
vs t.)
(2) Is f (t, ) uniformly convergent to zero
Worksheet #7: Dominant balancing
(1) Find the scaling of x with that makes two terms of equal order and others of lower
order, in:
x4 + x3 x2 + 2x 1 = 0
(2) Find the leading order term in each of the four roots.
(3) If you have time, nd the higher order c
Math 46: X hour of 5/12/11: Degenerate
Fredholm Equations
Alex Barnett
May 12, 2011
We used Section 4.3.3, particularly Thms 4.12 and 4.13, to determine if the
following had a solution, and then solve them. We made use of (4.31), the
starred equation in l
Worksheet #1: Dimensional Analysis
Say we suspect that drag force F depends only on a spheres radius a, its speed v, and the
surrounding uid density .
drag F
speed v
a
a) What are the dimensions of a, v, and F ?
Solution:
[a] = L
[v] = LT 1
[] = M L3
[F ]
Worksheet #1: Dimensional Analysis
Say we suspect that drag force F depends only on a spheres radius a, its speed v, and the
surrounding uid density .
drag F
speed v
a
a) What are the dimensions of a, v, and F ?
b) Create the dimensions matrix for this pr
Worksheet #12: Real or not. The story of Sturm-Liouville eigenvalues
Consider the Sturm-Liouville problem with p = 1 and q(x) real:
y + q(x)y = y,
a<x<b
with Dirichlet boundary conditions y(a) = y(b) = 0.
(a) Multiply the ODE by y .
(b) Multiply the conju
Worksheet #13: Volterra integral equations
(1) Convert the following integral equation into an IVP for u(t).
t
yu(y)dy u(t) = f (t)
on 0 t 1
0
(2) Prove the lemma:
x
x
s
f (y)(x y)dy
f (y)dyds =
a
[Hint: Let F (s) =
a
a
s
a f (y)dy
and use integration by
Worksheet #13: Volterra integral equations
(1) Convert the following integral equation into an IVP for u(t).
:3
® /0 yu(y)dy (11.5(15) 2 f(t)
log Emerylancrwl'au rm (:3 g} (r-Q (430+ 15 0 l%)d§> '2 u ME)
New mama clJlQQ/Qllokl 49 6 cigkML-H 8 1 *0 Pi ml
d
Math 46: Homework 9
Due May 29
(1) Page 372 # 5. This should be easy if you look up the radial part of the 3D Laplacian
operator.
(2) Page 372 # 6. Adapt the method from 1D. In fact is a positive operator. Note the
values would be eigenvalues of the Lapl