Introduction

### Introduction

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).

#### Oscillatory integrals

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.

#### Asymptotic methods

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.

#### Perturbation methods

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 $\varepsilon \ll 1$?

#### An Art or a Science?

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

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