What are "Asymptotic and Perturbation Methods"? Why we need to study them? And why we study them together?

First of all, we study some ODEs and PDEs (as you know many phenomenae are described by ODEs and PDEs).

We start from some integrals of the type \begin{equation*} I(k):=\int f(x)e^{k \phi(x)}\,dx \end{equation*} and \begin{equation*} I(k)=\int f(x)e^{ik \phi(x)}\,dx \end{equation*} with real-valued function $\phi(x)$ (both $\phi$ and $f$ are infinitely smooth) and their complex and multidimensional versions. We are interested how such integrals behave as $k \to +\infty$. We are interested in this because in many cases we get approximate solutions in this form.

How solutions of ODEs behave near singular point? F.e. how behave solutions of the following equations \begin{gather*} \sqrt{t}y'=f(t,y),\\ t y'=f(t,y),\\ t^2y'=f(t,y) \end{gather*} as $t\to+ 0$?

How solutions behave near infinity (as $t\to +\infty$?)

Assume that equation includes a small (say, $\varepsilon \ll 1$)or a large parameter (say, $\lambda \gg 1$). How solutions behave as $\varepsilon \to +0$ or $\lambda \to +\infty$?

What is a proper description?

And we could consider similar problems for PDEs.

Let us consider ODE or PDE containing a small or a large parameter--may be even not in equation but in the initial conditions. Assume that we know how to solve this equation as $\varepsilon=0$. How to solve it as $\varepsilon \ll 1$?

The simplest answer would be
\begin{equation}
X=X_0+X_1\varepsilon + X_2\varepsilon^2+\ldots
\label{eq-1.1.1}
\end{equation}
but in many cases it would be not true. If (\ref{eq-1.1.1}) holds perturbation is *regular*. But even here not everything is clear-cut: Let $X(t)$ be a solution of ODE and we are interested in the long term asymptotics (that meas, for $t\gg1$). But pretty often $X_n(t)=O(t^n)$ and therefore (\ref{eq-1.1.1}) provides a good approximation only for $\varepsilon t\ll 1$. Can we get a good approximation under less restrictive assumption: say $\varepsilon^2 t\ll 1$? Or better without any restriction at all (that means for all $t)?

But the case of the *singular perturbation* is even more interesting. F.e. consider the following two-point problem for ODE:
\begin{align}
-&\varepsilon^2 u'' + u=0\qquad 0 < x < l,
\label{eq-1.1.2}\\
&u(0)=b_1,\ u (l)=b_2.
\label{eq-1.1.3}
\end{align}
One can prove easily that the solution exists for all $\varepsilon>0$ and is uniformly bounded. But does it mean that $u=u_\varepsilon (x)\to u$ as $\varepsilon \to +0$ which solves the same problem as $\varepsilon=0$? The answer is "yes" but the convergence is not uniform. Indeed, as $\varepsilon =0$ equation (\ref{eq-1.1.2}) becomes $u=f$ and for this equation conditions (\ref{eq-1.1.3}) cannot be imposed. Thus, unless $f(0)=b_1$ and $f(l)=b_2$ convergence $u_\varepsilon \to f$ cannot be uniform.

The better apprximation (with $O(\varepsilon)$ error) is given by
\begin{equation}
u_\varepsilon = f + \underbracket{(b_1-f(0))e^{-x/\varepsilon}} + \underbracket{(b_2-f(l))e^{-(l-x)/\varepsilon}}
\label{eq-1.1.4}
\end{equation}
where selected are *boundary layer types* terms (they are negligible as $x\gg \varepsilon$ and $l-x\gg \varepsilon$ respectively).

But we want a better, multi-term approximation similar to (\ref{eq-1.1.1}) but with the *boundary layer types* terms. We could beinterested in the different BVP.

And also in multidimensional problems: \begin{align} -&\varepsilon^2 \Delta u'' + u=0\qquad x\in \Omega\label{eq-1.1.5}\\ &u|_{\partial \Omega}=g\label{eq-1.1.6} \end{align} where $\Omega$ is a domain and $\partial\Omega$ its boundary.

Or we can consider a Neumann (or Robin) boundary problem on the whole boundary or on its part.

**Remark.**
As Dirichlet boundary problem is given on $\Gamma_1\subset \partial\Omega$ and Neumann boundary problem is given on $\Gamma_2= \partial\Omega\setminus \Gamma_1$ and $\Gamma_1$ and $\Gamma_2$ are not disjoint, this is a singular problem even as $\varepsilon=1$ and asymptotics near $\Gamma_1\cap\Gamma_2$ could be studied.

Let us change sign at $\varepsilon^2\Delta$ in (\ref{eq-1.1.5}). Situation changes drastically, it becomes even more complicated. We get *Helmholtz equation*
\begin{equation}
\Delta u + k^2 u=0
\label{eq-1.1.7}
\end{equation}
with $k=1/\varepsilon$.

This equation could be obtained from *wave equation*
\begin{equation}
\Delta u -c^{-2}\partial_t^2 u=0
\label{eq-1.1.8}
\end{equation}
after substitution $u=e^{i\omega t}v(x)$ with $\omega =c k$.

Solutions of wave equation satisfyin initial conditions
\begin{equation}
u|_{t=o} = a(x) e^{i\phi_0 (x)k},\qquad u|_{t=o} = kb(x) e^{i\phi_0 (x)k}
\label{eq-1.1.9}
\end{equation}
is constructed as
\begin{equation}
u(x,t)= A^+(x,t,k) e^{i\phi^+ (x,t)k} + A^-(x,t,k) e^{i\phi^- (x,t)k}
\label{eq-1.1.10}
\end{equation}
where *eikonals* $\phi_\pm$ satisfy *eikonal equation*
\begin{equation}
\phi^\pm_t= \pm c|\nabla_x\phi^{\pm}|
\label{eq-1.1.11}
\end{equation}
with initial data
\begin{equation}
\phi^\pm|_{t=0}=\phi_0
\label{eq-1.1.12}
\end{equation}
and amplitudes are
\begin{equation}
A^\pm (x,t,k)\sim \sum_{k=0} ^\infty a^\pm_{j}(x,t)k^{-j}
\label{eq-1.1.13}
\end{equation}
and $a^\pm_j$ satisfy *transport equations*. The series here is *asymptotic* (the notion we'll learn from the very beginning). This is a *short wave approximation*.

The construction seems to be straightgforward, but there is a pitfall: eikonal is constructed by geometrical *ray construction* which itself works for all $t$, eikonal may become non-smooth due to *caustics* or *focussing* of the rays and short wave approximation fails there. We will answer the following questions:

- what to do near caustics and
- what to do after it.

Oscillotary integrals will be handy!

Similarly we can consider short-wave approximations for Maxwell's equations. We also consider *semiclassical approximation* (i.e. as $\hbar \ll 1$) fo *Schrödinger equation*
\begin{equation}
i\hbar \psi_t = -\frac{\hbar^2}{2m}\Delta \psi + V\psi .
\label{eq-1.1.14}
\end{equation}

Related topic: calculate approximately (as $\hbar\ll 1$) eigenvalues of $1$-dimensional *Schrödinger operator*
\begin{equation}
\hat{H}= -\frac{\hbar^2}{2m}\partial_x^2 + V .
\label{eq-1.1.15}
\end{equation}

Guessing the form in which we are looking for approximation is often than not an art (well, here we are talking about an original research). I have been privileged to know probably the greatest artist in this Arlen Il'in

https://ru.wikipedia.org/wiki/Ильин,_Арлен_Михайлович

(sorry, Russian only but you can use Google translate)