Asymptotic and Perturbation Methods in Applied
Asymptotic and Perturbation Methods in Applied Mathematics ESAM 420-1
Problem Set 1
Due October 17, 2002.
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.
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.
e(x3+x2)+4x2-3x-1 = 0
ex3+(x-2)2 = 0
x3+(3-2e)x2+(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)].
Find a two-term asymptotic approximation, for small e, of
each solution x of the transcendental equation
eex2 = 1+
[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
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.
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.
Find an approximation for the second root.
Determine the eigenvalues of the matrix A
+ e B for small e, where
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 ?
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
R2(t) = 1-Kt+e
where e is a parameter that depends on the physico-chemical
variables and K is the evaporation rate constant.
The droplet lifetime tL is the time at which R vanishes; i.e. R(tL) = 0. Write down an asymptotic approximation for tL, with a
remainder ~ O(e2).
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