Problem set 4

Asymptotic approximations

1. Asymptotics for polynomials

For the quintic equation, beware the root around $x = 0$. The correct re-scaling of $x$ will balance the quadratic and $\epsilon$ terms. For the cubic equation, beware the roots around infinity. Re-scale $x$ to balance the cubic and linear terms.

Check that the roots of the tan problem are given by \[ x_n = n\pi + \frac{\epsilon}{n\pi} - \frac{\epsilon^3}{(n\pi)^3} + \ldots \]

and \[ x = \pm \epsilon^{1/2} (1 - \frac{\epsilon}{6} + \ldots) \]

2. Exponential smallness

For the outer expansion, remember that terms that are exponentially small should be ignored. For the inner expansion, verify that \[ f = \frac{1}{\epsilon}\left(\frac{e^{-X} - 1}{X}\right) + 1 - \frac{\epsilon X}{3} + \ldots, \]

where $x = \epsilon X$.

3. Matched asymptotics

The outer approximation is $y \sim 2/x + \ldots$. The inner approximation is $Y \sim 2(1 - e^{-X})$ where you should verify that $x = 1 + \epsilon X$.

4. Matched asymptotics

You will not be able to solve the outer problem neatly in closed form. The solution is given by solving \[ u_0 + u_0^3 = 2 e^{x-1}. \]

For the inner problem, re-scale according to $x = \epsilon X$. Make sure that as $X \to \infty$ from the inner layer, that it matches with the inner limit of the outer solution, $u_0(0)$ , as $x \to 0$.

4. Semi-infinite problem

Verify that \[ y = \frac{1}{x} + \frac{2\epsilon}{x}(\sqrt{x} - 1) + \ldots \]