Iterative method

Faster if expansion sequence is unknown (i.e. we don't know it it's a power series or a log series for instance); slower, if the expansion sequence is known.

$x_{n+1} = g(x_n; \epsilon)$

Espansion method

Pose (guess) expansion. For instance a power series in small parameter, $\epsilon$:

$x=1+\epsilon x_1 + \epsilon^2 x_2 + ...$ as $\epsilon \rightarrow 0$

and substitute in algebraic equation, and equate terms of equal order because asymptotic expansions (using a fixed set of functions of $\epsilon$) are unique.

Easier than the iterative method, specially when working to higher orders, but must assume form of expansion.

Singular perturbations

When limit problem ($\epsilon =0$) differs in an important way from the limit $\epsilon \rightarrow 0$). Main method:

Regularization method: Scale variables so that the problem becomes regular.

Non-integral powers

When power expansion fails (one of the coefficients seems to need to be $\infty$..), an expansion in non-integral powers may be necessary.

This happens for example when the roots at limit problem ($\epsilon =0$) are a double root. As he says from example given in the notes, we could have guessed that an order $\epsilon^{1/2}$ change in $x$ would be required to produce and order $\epsilon$ change in a function at its minimum. Yeah if we are perturbing the parabola by an order $\epsilon$, then the new root would be the same as perturbing $x$ in such a way as to get the order $\epsilon$ change in the original parabola. At the minimum of the parabola, from Taylor expanding, we see we need a larger $\epsilon^{1/2}$ change in $x$ to get the $\epsilon$ change in the function.

Finding the right expansion sequence

We first pose the general expansion:

$x=1+\delta_1 x_1$,   $\delta_1(\epsilon) \ll 1$

Substitute into the algebraic equation, and look for dominant balances in the result. This will involve looking for the largest terms with and without $\delta_1(\epsilon)$

Once we have the first, term, we add a term to the expansion:

$x=1+\delta_1 x_1+\delta_2 x_2$,   $\delta_2(\epsilon) \ll \delta_1$

And we repeat this process

Again, the iterative method is very useful when the expansion sequence is not known, and can be faster than the above method involving unknown expansion functions, $\delta$.

Logarithms

Normally appears in transcendental equations.

Use iterative method as expansion method is hard to guess.

In his example, "over this range the term $x$ is slowly varying while $e^{-x}$ is rapidly varying. This suggests rewriting the equation as $e^{-x} = \frac{\epsilon}{x}$. I think this is so that we control/determine the faster changing term.