The method of matched asymptotic expansions is a powerful systematic analytical method for asymptotically calculating solutions to singularly perturbed PDE problems. It has been successfully used in a wide variety of applications (cf. Kevorkian and Cole (1993), Lagerstrom (1988), Dyke (1975)). However, there are certain special classes of problems where this method has some apparent limitations. In particular, for singular perturbation PDE problems leading to infinite logarithmic series in powers of v = -1/log ε, where ε is a small positive parameter, it is well-known that this method may be of only limited practical use in approximating the exact solution accurately. This difficulty stems from the fact that v → 0 very slowly as ε decreases. Therefore, unless many coefficients in the infinite logarithmic series can be obtained analytically, the resulting low order truncation of this series will typically not be very accurate unless ε is very small. Singular perturbation problems involving infinite logarithmic expansions arise in many areas of application in two-dimensional spatial domains including, low Reynolds number fluid flow past bodies of cylindrical cross-section, low Peclet number convection-diffusion problems with localized obstacles, and the calculation of the mean first passage time for Brownian motion in the presence of small traps, etc.
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