MA3AM/MA4AM ASYMPTOTIC METHODS
PROBLEM SHEET 1
1. Find the first three terms in the expansion u(x;ϵ) = u0 (x) + ϵu1 (x) + ϵ2u2 (x) + ... for the solution of
where u(0) = 1, and 0 < ϵ ≪ 1. Determine the exact solution, and compare.
2. Show that the Van der Pol equation
has the approximate solution u ≈ u0 + ϵu1 , where u0 = c0 cos(t + α0 ) and
for constants c0 , c1 ,α0 , and α 1 . [Hint: sin A cos2 A = 4/1(sin A + sin(3A)).]
3. Determine the first three terms in the expansion u = u0 + ϵu1 + ϵ2u2 + ... for the solution of the quadratic equation
u = 1 + ϵu2 ,
where 0 < ϵ ≪ 1. Determine the exact solution, and expand both roots in terms of powers of ϵ . What do you notice?
4. Find the first two terms in the expansion u(t;ϵ) = u0 (t)+ϵu1 (t)+ ... for the solution of
where u(0) = 1, u′ (0) = 0, and 0 < ϵ ≪ 1. [Hint: cos3 t = 4/1(cos 3t + 3 cos t).]
5. Determine the order (O) of the following expressions as ϵ → 0:
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