Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations

TL;DR

This paper introduces a unified approach to find explicit meromorphic solutions of a broad class of third order boundary layer ODEs, explaining the scarcity of closed-form solutions for equations like Falkner-Skan.

Contribution

It develops a general framework using complex analysis to classify and explicitly find all meromorphic solutions of a wide class of third order boundary layer equations.

Findings

All generic solutions are rational or rational in one exponential.

Explicit solutions are obtained for generic cases.

Some meromorphic or single-valued solutions are found in non-generic cases.

Abstract

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.