Due October 17, 2002.

1. Find the roots of
   

   x2-2.0004x+0.9998 = 0

   by means of a perturbation method. Calculate two terms in the appropriate perturbation series. Compare with the exact solutions.
   

2. 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(x³+x²)+4x²-3x-1 = 0

   (b)

   ex³+(x-2)² = 0

   (c)

   
   

   
   x³+(3-2e)x²+(3+e)x+1-2e = 0

3. Find an asymptotic approximation, for small e, of each solution x of the equation
   

   x2+e   ___ Ö2+x   = cose

   calculate the first three (nonzero) terms [Holmes, p.23, 1(h)].

   
   

   
   

4. Find a two-term asymptotic approximation, for small e, of each solution x of the transcendental equation
   

   eex2 = 1+ e 1+x2

   [Holmes, p.23, 1(m)].

   
   

   
   

5. 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
   

   ecoshq-q = 0

   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.

   (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.

   
   

   
   

6. Determine the eigenvalues of the matrix     A + e B     for small e, where
   

   
   

   A   = æ ç è a 0 0 b ö ÷ ø           B   = æ ç è 0 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.

   
   

   
   

7. 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
   

   R2(t) = 1-Kt+e   é ê ë 1 2 arcsin   __ ÖKt   -2   __ ÖKt      æ ç è 1- 3 4   ____ Ö1-Kt   ö ÷ ø ù ú û +O(e2)

   where e is a parameter that depends on the physico-chemical variables and K is the evaporation rate constant.
   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(e²).