calculate that is the entire asymptotic series for the outer region whereas the correction to the inner solution is and the constant of integration must be obtained from inner-outer matching. Notice, the intuitive idea for matching of taking the limits i.e. doesn't apply at this level. This is simply because the underlined term doesn't converge to a limit. The methods to follow in these types of cases are either to go for a) method of an intermediate variable or using b) the Van-Dyke matching rule. The former method is cumbersome and works always whereas the Van-Dyke matching rule is easy to implement but with limited applicability. A concrete boundary value problem having all the essential ingredients is the following. Consider the boundary value problem The conventional outer expansion gives , where must be obtained from matching. The problem has boundary layers both on the left and on the right. The left boundary layer near has a thickness whereas the right boundary layer near has thickness . Let us first calculate the solution on the left boundary layer by rescaling , then the differential equation to satisfy on the left is and accordingly, we assume an expansion . The inhomogeneous condition on the left provides us the reason to start the expansion at . The leading order solution is . This with van-Dyke matching gives . Let us now calculate the solution on the right rescaling , then the differential equation to satisfy on the right is and accordingly, we assume an expansion The inhomogeneous condition on the right provides us the reason to start the expansion at . The leading order solution is . This with van-Dyke matching gives . Proceeding in a similar fashion if we calculate the higher order-corrections we get the solutions as Composite solution To obtain our final, matched, composite solution, valid on the whole domain, one popular method is the uniform method. In this method, we add the inner and outer approximations and subtract their overlapping value, , which would otherwise be counted twice. The overlapping value is the outer limit of the inner boundary layer solution, and the inner limit of the outer solution; these limits were above found to equal . Therefore, the final approximate solution to this boundary value problem is, Note that this expression correctly reduces to the expressions for and when is and O(1), respectively. Accuracy This final solution satisfies