Ogilvie 



Define the n term outer expansion of y(t) as [ y| (t) + . . . 

 + yn(t)] ; def ine ~the _ m term inner expansion of y(t) as 

 [ Y| (t) +... + Ym{T)] . In the n term outer expansion, substitute 

 t = er and rearrange the result into a series ordered according to 

 €; truncate this expression after m terms, which gives the m 

 term inner expansion of the n term outer expansion . Similarly, 

 in the m term inner expansion, substitute t = t/ C and rearrange 

 the result into a series ordered according to €; truncate this 

 expression after n terms, which gives the n term outer expan- 

 sion of the m term inner expansion. The matching rule states 

 thati 



The m term inner expansion of the n term outer expansion 

 = the n term outer expansion of the m term inner expan- 

 sion. 



In the example discussed in the previous paragraphs, the 

 outer solution could not be obtained by a simple iteration scheme. 

 The matching principle can now be used to determine the constants 

 in the outer solution, and so an iteration scheme is now available, 

 requiring, however, that inner and outer expansions be obtained 

 simultaneously. In the example, the inner solution could be ob- 

 tained completely and independently of the outer, but this is an 

 accident which occurred because of the siinple nature of the prob- 

 lem above. Ordinarily, in cases in which one might consider using 

 the method of matched asymptotic expansions, one must proceed 

 step-by-step to find first a termi in one expansion, then a term in 

 the other expansion, and so on. 



It is worthwhile to be fairly precise about certain definitions. 

 We use the equivalence sign, "~," frequently. For example, we 

 write: 



N 



<(>(x,y,z;e) ~ ) (^n(x,y,z;€) 



n=0 



This means that: 

 N 



^ - y ^nl = o(^ 3-s e -^ for fixed values of (x,y,z) 



rt=0 



Also, it implies that t^n+l = o(«^n) as e -* 0. The qualification that 

 (x,y,z) should be fixed is very important. In the example above, 

 we would have the equivalent statement for the outer expansion: 



N 

 |y(t;e) - ) yn(t;e) | = o^y^ as e^O for fixed t, 

 n=l 

 and, for the Inner expansion: 



672 



