THE ASYMPTOTIC SOLUTION OF A FAMILY OF BOUNDARY-VALUE-PROBLEMS INVOLVING EXPONENTIALLY SMALL TERMS

In this paper, the authors consider the family of boundary value problems [GRAPHICS] in the limit \epsilon\ --> 0. This problem has recently appeared as a model for magnetic field annihilation but the equation itself, with variously different boundary conditions, has an extensive literature. Using a combination of asymptotic and numerical analyses, the paper gives a comprehensive treatment of the small \epsilon\ problem, paying particular attention to the question of duality of solutions. For epsilon > 0, this is intimately connected with the occurrence of exponentially small terms in the asymptotic solution. When alpha = O(1) these terms are forced by the boundary layer at y = 1, and the techniques used to deal with this case are well known from previous work on the equation. However, for small \alpha\, a case which reveals the true nature of the duality properties of the asymptotic solution, these well-known methods are not applicable, and a new approach via the initial value formulation of (*) is used. The approach is based on a scaling method which enables the problem to be reduced to a one-parameter family of problems of initial value type. This considerably simplifies the search for and construction of numerical solutions that are used to support the asymptotic analysis. For epsilon < 0, it is shown that convergence to the epsilon = 0 solution only takes place for a restricted range of values of alpha and that, for sufficiently small \epsilon\, there is only one solution to the given boundary value problem.