Fall 2002

Hermann Riecke

Problem Set 1 |

Due October 17, 2002.

- Find the roots of

by means of a perturbation method. Calculate two terms in the appropriate perturbation series. Compare with the exact solutions.x ^{2}-2.0004x+0.9998 = 0

- Find an approximation for the roots (including the complex one if they
exist) of the following equations, valid for small e. Write down
the first two nonzero terms in the expansion.
**(a)**- e(x
^{3}+x^{2})+4x^{2}-3x-1 = 0 **(b)**- ex
^{3}+(x-2)^{2}= 0 **(c)**-

x^{3}+(3-2e)x^{2}+(3+e)x+1-2e = 0

- Find an asymptotic approximation, for small e, of
__each__solution x of the equation

calculate the first three (nonzero) terms [Holmes, p.23, 1(h)].x ^{2}+e___

Ö2+x

= cose

- Find a two-term asymptotic approximation, for small e, of
__each__solution x of the transcendental equationee ^{x2}= 1+e

1+x^{2}[Holmes, p.23, 1(m)].

- In solving the problem describing thermal explosion in a chemically
reactive stick, it is found that the maximum temperature in the stick, q, is determined implicitly by

where e is a parameter that represents the ratio of the heat generated by the chemical reaction to the heat drained away by conduction. For a weakly exothermic reaction e is small.ecoshq-q = 0 **(a)**- Verify that the equation possesses two solutions with q > 0. The physically relevant root is the one with the smallest value q.
Find a explicit expression for q, valid for small e, and
write down the first two nonzero terms in the expansion. Compare your result
with the exact solution (which you could find numerically) for e = 0.05, 0.1, 0.15, 0.2.
**(b)**- Find an approximation for the second root.

- Determine the eigenvalues of the matrix
**A**+ e**B**for small e, where

**A**=æ

ç

èa 0 0 b ö

÷

ø**B**=æ

ç

è0 w w 0 ö

÷

ø**(a)**- Consider first the case of a ¹ b. Calculate the first three
nonzero terms in the appropriate expansions for each eigenvalue and show
that a
*O*(e) perturbation of a matrix need not result in a*O*(e) perturbation of the eigenvalues. What happen as a® b ? **(b)**- Consider the case a = b and calculate the first three nonzero
terms in the appropriate expansions for each eigenvalue. How close should a
be to b for this expansion to take over the one found above.

- The time history of an evaporating liquid droplet is described
mathematically by a relation that shows the change of the droplet radius R(t) with time t. An asymptotic expansion for R, valid for small e, is

where e is a parameter that depends on the physico-chemical variables and K is the evaporation rate constant.R ^{2}(t) = 1-Kt+eé

ê

ë1

2arcsin __

ÖKt

-2 __

ÖKt

æ

ç

è1- 3

4____

Ö1-Kt

ö

÷

øù

ú

û+ *O*(e^{2})

The droplet lifetime t_{L}is the time at which R vanishes; i.e. R(t_{L}) = 0. Write down an asymptotic approximation for t_{L}, with a remainder ~*O*(e^{2}).

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On 3 Oct 2002, 22:35.