For a uniformly valid (T_{0}>0) for every (minmathbb{N}^), we need the estimates in next section.
Moreover, we show that the obtained approximate solutions are valid for the whole solution domain and the approximations are uniformly valid not only for small parameters, but also for very large parameters.
This solution resolves the difficulties associated with singular behavior in both space and time, and a space time matching scheme is used to form uniformly valid composite solutions for temperature or concentration and for the radius of the sphere.
The method of multiple scales is used to formulate a uniformly valid perturbation expansion for the response near the resonance, and a system of first order ordinary differential equations for the slowly varying amplitude and phase of the response results.
Accurate, uniformly valid expressions are derived for both the frequencies of free oscillation and the corresponding damping ratios.
Uniformly valid approximations are composed for the region 0 ⩽ x ⩽ 1 by synthesizing expansions pertaining to three separate regions; near x = 0, near x = 1, and the intermediate region in between.
These equations are then solved for uniformly valid solutions using a singular perturbation technique [19, 20, 21] that eliminates secular terms in the perturbed solution; the shock strength is assumed small, but the accompanying first-order jump discontinuity may be finite (i.e., O(1)) or small.
Furthermore, the mechanism for deriving the uniformly valid WKBJ solutions near smooth caustics strongly suggests that the uniform theory as derived for the two-way wave equation should be extendable to the one-way wave equation.
Hence, for obtaining the uniformly valid asymptotic solution, corrections must be performed at the regions of boundary layers.
Then for all and Also, is uniformly valid on.
