The asymptotic solution of a family of boundary value problems involving exponentially small terms

In this paper, the authors consider the family of boundary valueproblems in the limit || 0. This problem has recently appeared as a modelfor magnetic field annihilation but the equation itself, withvariously different boundary conditions, has an extensive literature.Using a combination of asymptotic and numerical analyses, thepaper gives a comprehensive treatment of the small || problem,paying particular attention to the question of duality of solutions.For |0, this is intimately connected with the occurrence ofexponentially small terms in the asymptotic solution. When =0(1) these termsz are forced by the boundary layer at y = 1,and the techniques used to deal with this case are well knownfrom previous work on the equation. However, for small ||, acase which reveals the true nature of the duality propertiesof the asymptotic solution, these well-known methods are notapplicable, and a new approach via the initial value formulationof (*) is used. The approach is based on a scaling method whichenables the problem to be reduced to a one-parameter familyof problems of initial value type. This considerably simplifiesthe search for and construction of numerical solutions thatare used to support the asymptotic analysis. For 0, it is shownthat convergence to the =0 solution only takes place for a restrictedrange of values of a and that, for sufficiently small || thereis only one solution to the given boundary value problem.