Singular Perturbation Problems in Ship Hydrodynamias 



If we consider that t = 0(1) as € "♦ 0, then the following approxi- 

 mation is valid for y(t): 



y(t)~ e {1 +|(2 - t) +J2 (6 - 3t + Zt^) +...}. 

 This approximation could be obtained step-by-step, iteratively: 



where y(t) ~ [j yr,(t). However, it is not unifornaly valid at t = 0, 

 and the constants cannot be determined. On the other hand, we 

 could consider that t = 0(e) as e -*• and rearrange the exact 

 solution accordingly. This is most easily done if we set t = er 

 and rewrite everything in terms of t. The approximation for y(t) 

 is then: 



y(t)~i +f(i-I)fc^(^-l+:!i)+... 



This approximation could be obtained completely fronn the di^ 

 ential equation by an iteration scheme in which we let y{t) ~ 

 the individual terms satisfying the equation: 



Y;{t) + 2Y' (T) = - cY„., (T) [ Y'„ H dY/dr] 



and the conditions: 



Y,(0) = 1; Y^(0)=0, n>l; Y;(0)=0, n>l 



However, this solution is not uniformly valid for t -*• oo; in fact, 

 one would hardly suspect that it represents a solution decaying ex- 

 ponentially with time. 



The difficulty arises because the problem is characterized 

 by two time scales, l/p. and l/pg* and the two are grossly differ- 

 ent. One of the two exponentials in the exact solution decays very 

 rapidly and the other decays at a moderate rate. The contrast in 

 these two time scales, along with the fact that each has its dominant 

 effect in a distinct range of time, allows us to apply the method of 

 matched asymptotic expansions to this problem. The Van Dyke 

 prescription for doing this is as follows: 



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