Solution of εY″+YY′−Y=0 by a nonasymptotic method

Abstract

The solution of εy″(x)+y(x)y′(x)−y(x)=0, with the boundary conditionsy(0)=α,y(1)=β, is obtained by a nonasymptotic method. It is shown that the nature of the inner solution for both left-hand and right-hand boundary layers depends on the roots of a transcendental equation. From sketches of this function, the location of the roots can be found. For the left-hand boundary layer, depending on the relative size and signs of α and β, 13 cases exist for the possible solution of the transcendental equation. Of these cases, only five correspond to acceptable solutions. Similar remarks apply to the right-hand boundary layer solutions. Numerical experience with the method is also reported to confirm the theoretical analysis.