Regular perturbation problems

Landau notation

Each of the following two definitions covers three cases. In other words, each definition is really three definitions in one; choose the relevant one when dealing with specific problems.

Definition of Big-O

Let \(f,g\) be real-valued functions. If there exist positive constants \(C\) and \(\delta\) such that \(|f(x)|\leq C|g(x)|\) for all

\[\begin{split} \begin{pmatrix} 0<x<\delta\\ x>\delta\\ |x-a|<\delta \end{pmatrix}, \end{split}\]

then we say \(f(x)=O(g(x))\) as

\[\begin{split} \begin{pmatrix} x\to 0\\ x\to \infty\\ x\to a \end{pmatrix}. \end{split}\]

Definition of Little-O

Let \(f,g\) be real-valued functions. If for every \(c>0\) there exists \(\delta=\delta(c)\) such that \(|f(x)|\leq c|g(x)|\) for all

\[\begin{split} \begin{pmatrix} 0<x<\delta\\ x>\delta\\ |x-a|<\delta \end{pmatrix}, \end{split}\]

then we say \(f(x)=o(g(x))\) as

\[\begin{split} \begin{pmatrix} x\to 0\\ x\to \infty\\ x\to a \end{pmatrix}. \end{split}\]

Definition of Asymptotic Equivalence

Let \(f,g\) be real-valued functions and \(g(\varepsilon)\neq 0\). If

\[ \lim_{\varepsilon\to 0+}\frac{f(\varepsilon)}{g(\varepsilon)}=1, \]

then we say \(f(\varepsilon)\sim g(\varepsilon)\) as \(\varepsilon\to 0\).

Note the difference between the first two definitions: for big-O, the inequality \(|f(x)|\leq C|g(x)|\) must hold for at least one value of \(C\), whereas for little-o, the inequality \(|f(x)|\leq c|g(x)|\) must hold for every \(c>0\).

Asymptotic sequences and expansions

  • A sequence \(\delta_n(\varepsilon)\), \(n=1,2,\ldots\), is called an asymptotic sequence if \(\delta_{n+1}(\varepsilon)=o(\delta_n(\varepsilon))\).

    An example of such an asymptotic sequence which we see frequently in this module is \(\delta_n(\varepsilon)=\varepsilon^n\).

  • A sum of the form

    \[ \sum_{n=1}^N a_n(x)\delta_n(\varepsilon) \]

    is called an asymptotic expansion of a function \(f(x,\varepsilon)\) as \(\varepsilon\to 0\) up to \(N\) asymptotic terms with respect to an asymptotic sequence \(\delta_n(\varepsilon)\), \(n=1,2,\ldots\), if

    \[ f(x,\varepsilon)-\sum_{n=1}^M a_n(x)\delta_n(\varepsilon) =o(\delta_M(\varepsilon)),\quad \varepsilon\to 0, \]

    for each \(M=1,2,\ldots,N\). If this holds for all \(N\), then we write

    \[ f(x,\varepsilon)\sim\sum_{n=1}^\infty a_n(x)\delta_n(\varepsilon), \quad \varepsilon\to 0. \]

  • For a given asymptotic sequence and a given function, the asymptotic coefficients \(a_n(x)\) are unique.

Regular perturbation problems

We’ve seen many examples in the lectures of regular perturbation problems involving a small parameter \(\varepsilon\). The general process for approximating algebraic equations and ODEs is very similar and can be summarised as:

  • Expand the unknown function as a power series in \(\varepsilon\), e.g.

    \[ u=u_0+\varepsilon u_1+\varepsilon^2 u_2+O(\varepsilon^3). \]

  • Substitute into the main equation.

  • Compare coefficients of different degrees of \(\varepsilon\). Comparing terms in \(\varepsilon^0\) yields \(u_0\), comparing terms in \(\varepsilon^1\) yields \(u_1\), and so on.

Note that for ODEs where boundary conditions are specified, we apply the boundary condition to the leading term in the expansion. For example, if the boundary condition is \(u(0)=\cos x\), then we prescribe

\[ u_0(0)=\cos x,\quad u_1(0)=0,\quad u_2(0)=0,\ldots \]