I'm beginning my second year in my PhD program. I had to fight very hard to make it to the second year, as my program condenses the normal two year masters sequence that would make up the beginning of other programs into one year. They were also very eager to cut students using prelims, so the...
Homework Statement
Consider the following set of differential equations:
\begin{eqnarray*}
\dot{u} & = & b(v-u)(\alpha+u^2)-u \\
\dot{v} & = & c-u
\end{eqnarray*}
The parameters b \gg 1 and \alpha \ll 1 are fixed, with 8\alpha b < 1. Show that the system exhibits relaxation...
Homework Statement
Consider the radially symmetric wave equation in n dimensions
u_{tt} = u_{rr} + \frac{n-1}{r}u_r
Use induction to show that the solution is
u = \left(\frac{1}{r}\frac{\partial}{\partial r}\right)^{(n-3)/2} \frac{f(t-r)}{r}
for n odd and
u =...
Homework Statement
I'm working on a long problem and have come to the final step. The answer seems so simple, but I can't quite get to it. I need to evaluate this integral:
\int_0^{\infty}\ \left(e^{-k^2 t}/k\right)\sin(kx)\ dk
Homework Equations
Mathematica gives the result as...
Homework Statement
The equation for the probability distribution of the price of a call option is
\frac{\partial P}{\partial t} = \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 P}{\partial S^2} + rS\frac{\partial P}{\partial S} - rP
with the conditions P(0,t) = 0, P(S,0) = \max(S-K,0), and...
Backstory (feel free to skip): Last year, I applied to a bunch of PhD programs in applied math. I got accepted into a couple, but Northwestern offered me an interesting option. They rejected me from the PhD program, but they said that they would accept me into a masters program (which I'd have...
Homework Statement
Solve Laplace's equation
u_{xx} + u_{yy} = 0
on the semi-infinite domain -∞ < x < ∞, y > 0, subject to the boundary condition that u_y = (1/2)x u on y=0, with u(0,0) = 1. Note that separation of variables will not work, but a suitable transform can be applied...
Homework Statement
Find the solution of Laplace's equation for \phi(r,\theta) in the circular sector 0 < r < 1; 0 < \theta < \alpha with the boundary conditions \phi(r,0) = f(r), \phi(r,\alpha) = 0, \phi(1,\theta) = 0. (also, implicitly, the solution is bounded at r = 0). Use two different...
Homework Statement
The following is an integral form of the Bessel equation of order n:
J_n(x) = \frac{1}{\pi}\int_0^{\pi}\ \cos(x\sin(t)-nt)\ dt
Show by substitution that this satisfies the Bessel equation of order n.
Homework Equations
Bessel equation of order n: x^2y'' + xy' +...
Homework Statement
This is part of a larger problem, but in order to take what I believe is the first step, I need to take the Taylor series expansion of f(x,y) = \cos\sqrt{x+y} about (x,y) = (0,0)
On the other hand, the purpose of doing this expansion is to find an asymptotic expression for...
Homework Statement
Solve x^2\frac{du}{dx} = 0 in the sense of distributions.
Homework Equations
<u',f> = -<u,f'> for any test function f.
The Attempt at a Solution
My thinking is that since we want to see the action of the left hand side on a general test function f, we try...
Homework Statement
For α > 0, determine u(x) by the inverse Fourier transform
u(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ \frac{e^{ikx}}{ik+\alpha}\ dk
Homework Equations
The Attempt at a Solution
This seemed like a relatively simple residue problem. You just note that...
Suppose we have some ODE given by y' = G(x,y)/H(x,y). Let x and y depend on a third variable, t, so that x and y are parametrized in a way. Then applying the chain rule to y' gives
\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{G(x,y)}{H(x,y)}
Then comparing the numerators and...
Homework Statement
Find the mass of the plane region R in the first quadrant of the xy plane that is bounded by the hyperbolas xy=1, xy=2, x^2-y^2 = 3, x^2-y^2 = 5 where the density at the point x,y is \rho(x,y) = x^2 + y^2.
Homework Equations
The Attempt at a Solution
The...
Homework Statement
This isn't actually a homework problem, but rather a class of problems I'm running into as I study for prelims. I'm taking these from Greenspan's Calculus: An Introduction to Applied Mathematics. This type of problem has come up in the context of both volume and surface...
Homework Statement
I have two separate problems, but I get the same answer for each. I feel like this must be wrong.
Question 1: Find the leading term of a uniformly valid (for t > 0) asymptotic expansion of the solution of the IVP \ddot{x} + \epsilon\dot{x}(x^2-1)+x = 0 \mbox{ } x(0) =...
Homework Statement
A thin rod of length ∏ is heated at one end to temperature T_0. It is insulated along its length and cooled at the other end by convection in a fluid of temperature T_f . Find the transient and steady-state temperature distribution in the rod, assuming unit thermal diff...
Homework Statement
Consider a bead of mass m that is confined to move on a circular hoop of radius r. The axis of symmetry of the hoop is horizontal, and the hoop is rotating about a vertical axis at a uniform rate \omega. Neglect friction and assume a constant gravitational acceleration of...
Homework Statement
Consider the second order wave equation
u_{tt} = 4u_{xx}
There are initial and boundary conditions attached, but I'm less concerned with those for the moment. I think I can figure those out if I can figure out where to get started.
Rewrite this as a system of first order...
Homework Statement
Consider the system of equations
\begin{eqnarray*}
e_1 &=& Ak^p + Bh^q \\
e_2 &=& A(k/2)^p + Bh^q \\
e_3 &=& Ak^p + B(h/2)^q \\
e_4 &=& A(k/2)^p + B(h/2)^q
\end{eqnarray*}
Suppose that the e_i are known, as well as k and h. Find A, B, p, and q.
Homework...
Homework Statement
This problem arises from the following ODE:
\epsilon y'' + y' + y = 0, y(0) = \alpha, y(1) = \beta
where 0 < x < 1, 0 < \epsilon \ll 1
Find the exact solution and expand it in a Taylor series for small \epsilon
Homework Equations
I guess knowing the Taylor...
Homework Statement
Consider
\frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2} subject to
u(0,t) = A(t),\ u(L,t) = 0,\ u(x,0) = g(x). Assume that u(x,t) has a Fourier sine series. Determine a differential equation for the Fourier coefficients (assume appropriate continuity)...
Last semester really killed my GPA (went from a 3.7 to a 3.35). I'm deathly afraid that this screwed me over for graduate school, so I'm trying to find ideas for something I can fall back on if this doesn't work out. I'm a math major with a very heavy concentration in physics (in fact, I would...
Homework Statement
You want to boil a pot of water at 20C by heating it to 100C. I suggest a way of heating the pot in a reversible manner: simply inserting a Carnot engine between the reservoir (at 100C) and the pot. The Carnot engine operates between two temperatures, absorbing heat dQ_1...
Homework Statement
A possible ideal gas cycle operates as follows:
(i) From an initial state (p_1,V_1), the gas is cooled at constant pressure to (p_1,V_2);
(ii) the gas is heated at constant volume to (p_2,V_2);
(iii) the gas expands adiabatically back to (p_1,V_1).
Assuming...
I am (hopefully!) going to be entering grad school in the fall. I will be going for a PhD in applied mathematics. Many grad schools recommend that you take a course in probability before entering (it is strongly hinted that doing so will improve your odds of admission). Here's where the...
Homework Statement
Let E,E' be metric spaces, f:E\rightarrow E' a function, and suppose that S_1,S_2 are closed subsets of E such that E = S_1 \cup S_2. Show that if the restrictions of f to S_1,S_2 are continuous, then f is continuous. Also, if the restriction that S_1,S_2 are closed is...
Homework Statement
For a particle of mass m moving in the potential V(x) = \frac{1}{2}m\omega^2x^2 (i.e. a harmonic oscillator), it is often convenient to express the position and momentum operators in terms of the ladder operators a_{\pm}:
x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_-)
p =...
Homework Statement
Consider the sequence a_1,a_2,..., such that \lim_{n\rightarrow\infty} a_n = a (with a_i \in R). Show that \lim_{n\rightarrow\infty}\left(\frac{\sum_{i=1}^n a_i}{n}\right) = a
In other words, it's given that for some \epsilon > 0,d(a_n,a) < \epsilon\ \forall n > N...
I'm entering my senior year of college, and I want to go to grad school. I'm starting to panic because I don't know what I want to do. First off, money is a huge deal to me - I know I shouldn't go in thinking that, but I can't help it. I'm getting my BS in math/BA in physics at the University...