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By Ali H. Nayfeh
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George E. P. field & Norman R. Draper
Evolutionary Operation: A Statistical strategy for approach Improvement
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Finite teams of Lie kind: Conjugacy Classes
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William G. Cochran & Gertrude M. Cox
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Sample text
7! -- -sin x - - cos x + .. as + E4- sin 5 + E5- cos 4! €-to 5! 3) Note that the coefficients of all powers of E are bounded for all values of hence a,(z) is no more singular than a,-,(x), and the expansion is uniformly valid as a consequence. 4) Each term of this expansion except the first is singular at z = 0 and is more singular than the preceding one, hence the expansion is not uniformly valid. It. breaks down near x = 0. 5) + We can see this if we look at [l (~/x)]l'~ whose Taylor series expansion converges only if I E / z ~ is less than unity.
14) where the Reynolds number R = U L / v with v the kinematic viscosity of the fluid. 14) needs to be supplemented by boundary conditions. 16) The second condition represents the vanishing of the velocity tangent t o the body (the so-called no-slip condition), while the first condition represents the vanishing of the normal velocity component. 19) which is a third-order rather than a fourth-order differential equation. 17), and one of them must be dropped. 16) must be dropped. Therefore the resulting solution for yo, although it approximates the exact solution very well as R -+ 03 away from the body, breaks down near the body.
In fact, the first 9 terms give a value for J0(2) correct to 11 significant figures. However, as x increases, the number of terms needed to yield the same accuracy increases rapidly. 13) yields the same accuracy. As x increases further, an accurate result is obtained with far less labor by using the asymptotic divergent series (1 513). 6. Nonuniform Expansions In parameter perturbations the quantities to be expanded can be functions of one or more variables besides the perturbation parameter. 1) m=O where the coefficients a,,, are functions of x only.