A fast, spectrally accurate solver for the Falkner-Skan equation

We present a new numerical technique, the Gegenbauer homotopy analysis method, which allows for the construction of iterative solutions to nonlinear differential equations. This technique is a numerical extension of the semi-analytic homotopy analysis method that exhibits spectral convergence while performing sparse matrix operations in Gegenbauer space. This technique is used to present solutions to the Falkner–Skan equation, a well known problem in boundary layer fluid dynamics. These solutions are compared to previously published works, and the convergence properties exhibited by this new technique are considered.