Perturbation Methods by Ali H. Nayfeh

By Ali H. Nayfeh

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George E. P. field & Norman R. Draper
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58) r2 Here again difficulty with the straightforward expansion arose because of the infinite domain. 59) satisfies the surface boundary conditions, but it does not satisfy the boundary condition at infinity. Thus, this expansion breaks down for large r and this breakdown is called Whitehead's paradox because Whitehead (1 889) was the first to obtain this solution, although by iteration, and the first to point out its nonuniformity. 2. 1. 2) where e is a small positive number. 2. 5) Y:, Y, = - Y L It can be seen that at any level of approximation n, ynV1is known, hence yn for any n is given by a first-order differential equation.

N,. = = n = constant, n,, = 0). 3. 26) where a is assumed to be small so that sin a m a, and b = r2/rl with r2 the radius of the rigid inclusion. 30). Hence two boundary conditions have to be dropped. 31). 32) is not valid for all r in [ b , 11. 5 by using the method of matched asymptotic expansions. 3. Type Change of a Partial Differential Equation Since the boundary and initial conditions required to make a well-posed problem depend on the type of the partial differential equation, difficulties might arise if the original equation changes, say, from elliptic to parabolic or hyperbolic as the small parameter vanishes.

Conditions under which these operations are justified are discussed by van der Corput (1956), ErdClyi (1956), and de Bruijn (1958). Addition and subtraction are justified in general. 3) EXERCISES Iff&; c) and q,,(e) are integrable functions of E, 19 then Multiplication is not justified in general because in the formal product of Ia,(z)p,,(~) and Zb,,(z)p,,(~)all products q,,(c)p,(e) occur, and it is generally not possible to arrange them so as to obtain an asymptotic sequence. In other words, multiplication is justified if the result is an asymptotic expansion.

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