The Airy equation and its alternative analytic solution

This paper is concerned with the alternative forms of the analytic solution of the Airy equation. Instead of the traditional Taylor series or asymptotic methods, a homotopy analysis technique is employed, which does not require a small perturbation parameter or large asymptotic parameter. It is shown that a proper choice of linear operator during implementation of the homotopy method can yield uniformly valid solutions, representing the exact decaying/growing Airy functions. Convergence of the homotopy series found is proved mathematically. The obtained explicit analytical expressions for the solution generate results that compare excellently with the numerically computed and large asymptotic ones.

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