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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Singular Perturbation

In considered previously problems, the addition of the small perturbation did not significantly alter the equation or the form of the solution. We name this a regular perturbation, and usually the solution can be expressed as a power series in ε having a non-vanishing radius of convergence: the exact solution smppthly approaches the unperturbed solution as ε → 0.

Regular perturbation expansions with respect to a small parameter ε do not always produce a complete approximate solution. In fact, there are several situations where this method fails, in particular:

  1. When the small parameter multiplies the highest derivative in an ordinary differential equation problem.
  2. When algebraic equations change degree when ε = 0.
  3. When ε = 0 changes the character of the problem, as in the case of a partial differential equation changing type (from elliptic to parabolic, for example). In other words, the solution for ε = 0 is fundamentally different in character from the solution for ε close to zero.
  4. When the problems occur on infinite domains, giving rise to so-called secular terms.

Such problems fall in the general category of singular perturbation, where a common feature is that the equations have multiple time or spatial scales each covering only a part of the solution domain.

For differential equations, in particular in fluid mechanics, problems involving boundary layers are common. If there is a boundary layer, a leading-order regular perturbation term found by setting ε = 0 in the equations often provides a valid approximation in a large outer region, but fails severely in the boundary layer. A valid approximation in the boundary layer is found by a rescaling of the equations, as discussed below. Often the inner and outer approximations with respect to the boundary layer may be matched to obtain a uniformly valid approximation over the entire domain of interest. The singular perturbation method applied in this context is also called the method of matched asymptotic expansions.

For instance, let us consider the quadratic equation

\[ \varepsilon\, x^2 + x -1 = 0 , \qquad |\varepsilon | ⪡ 1 , \]
has one solution, x = 1, when ε = 0, but two solutions when ε ≠ 0. This is an example of a singular perturbation in which the solutions of the equation have a different character when ε = 0 than when ε ≠ 0.

In this example, if ε ≠ 0, the product of the roots is −1/ε, and if 0 < |ε| ⪡ 1, one roots is near 1 so the other must be approximately −1/ε.

The two roots are given by the usual expression,

RJB solves
\begin{align*} x_1 (\varepsilon ) &= \frac{2}{1 + \sqrt{1 + 4 \varepsilon}} = 1 - \varepsilon + 2 \varepsilon^2 - 5 \varepsilon^3 + 14 \varepsilon^4 + O(\varepsilon^5 ) , \\ x_2 (\varepsilon ) &= - \left( \frac{1 + \sqrt{1 + 4 \varepsilon}}{2\varepsilon} \right) = -\frac{1}{\varepsilon} -1 + \varepsilon - 2 \varepsilon^2 + 5 \varepsilon^3 + O\left( \varepsilon^4 \right) , \end{align*}
where the expansions converge for |ε| < ¼. The first root can be obtained from a direct application of perturbation theory and x₁(ε) → 1 as ε → 0. The second root is O*1/ε), and tends to infinitty as ε → 0and is not a root of the unperturbed equation and cannot therefore be given by a direct application of perturbation theory.

This example illustrates one aspect of singular perturbations, but it also shows that the distinction between regular and singular perturbations is not as clear cut as our definitions suggest. In this case, we can find a new variable in which the perturbation problem becomes regular, though we should emphasise that such tricks are not normally possible. If x ∼ 1/ε both the εx² and the x terms of equation εx² + x - 1 = 0 are O(ε−1), suggesting that we define a new variable y = εx to obtain the equation

\[ y^2 + y - \varepsilon = 0 , \]
the solution of which can be obtained using regular perturbation theory. Thus, by rescaling the variable a singular perturbation has been converted into a regular perturbation. It must be emphasised that this is rarely possible.

   
Example 1: Consider the following boundary value problem on interval 0 ≤ x ≤ 1 when 0 < ε ⪡ 1:
\[ \begin{split} \varepsilon \,y'' (x) + \left( a + \varepsilon \right) y' (x) + y(x) = 0 , \\ y(0) = 1, \quad y(1) = 1. \end{split} \tag{1.1} \]
By observing that the characteristic equation εr² + (1 + ε)r + 1 = 0, has two roots r₁ = −1 and r₂ = −1/ε, the boundary value problem (1.1) can be solve analystically :
RJB solves
\[ y(x) = \frac{e^{-x} - e^{-x/\varepsilon}}{e^{-1} - e^{-1/\varepsilon}} . \] However, if we assume a regular perturbation solution \[ y = y_0 (x) + \varepsilon\,y_1 (x) + \varepsilon^2 y_2 (x) + \cdots , \] its substitution into Eq.(1.1) leads to the system \begin{align*} y'_0 + y_0 &= 0, \\ y'_1 + y_1 &= - y''_0 - y'_0 , \end{align*} etc.

Since the general solution of the first equation y′ + y = 0 is just y𔔀 = C e−x, there is no way y𔔀 can satisfy both boundary consitions. Thus, contrary to the regular perturbation case, it is impossible for the leading order perturbation solution to satisfy both boundary conditions. Since there are two boundary conditions, two linearly independent solutions of (1.1) are required. There is no obvious way to get the second solution by regular perturbation. In particular, a rescaling of y does not help us at all. It turns out that what is needed is a rescaling of x.    ■

End of Example 1

 

 

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