By Ali Hasan Nayfeh
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2. 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.
Some of the sources of nonuniformities are: infinite domain, small parameter multiplying the highest derivative, type change of a partial differential equation, and presence of singularities. In the infinite domain case, the nonuniformity manifests itself in the presence of so-called secular terms such as xn cos z and zn sin z,which make fm(x)lfm-l(z)unbounded as x approaches infinity. In the case of the small parameter multiplying the highest derivative, the perturbation expansion cannot satisfy all the boundary and initial conditions, and the expansion thus is not valid in boundary and initial layers.
The modified expansion is of the form c2 - - (1 7 + €C1 + c2c1 + . 35) and equating coefficients of E and c2 on both sides. 40) is singular at this point. It can also be shown that this singularity is compounded in the higher approximation. Thus the first of the above expansions is valid for r < cz' and the second is valid for r > c;' both break down near r = cz'. 6 by using the method of matched asymptotic expansions. 4. 43) =0 for large A. For Izi < 1, the solutions of this equation are oscillatory, while for Is1 > 1 they behave as exponentials.