Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers

In this paper we construct three new test problems, called Models A, B and C, whose solutions have two-dimensional boundary layers. Approximate analytic solutions are found for these problems, which converge rapidly as the number of terms in their expansion increases. The approximations are valid for =10−8 in practical computations. Surprisingly, the algorithm for Model A can be carried out even for →∞. Model C has a simple exact solution. These three new accurate and approximate analytic solutions with two-dimensional boundary layers may be more useful for testing numerical methods than those in [Z.C. Li, H.Y. Hu, C.H. Hsu, S. Wang, Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains, I. Convergence analysis, J. Comput. Appl. Math. 166 (2004) 181–208] in the sense that the series solutions from the former converge much faster than those of the latter when is small.