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A general class of functionally-fitted explicit pseudo two-stepRunge-Kutta-Nystr\"{o}m (FEPTRKN) methods for solving second-order initialvalue problems has been studied. These methods can be considered generalizedexplicit pseudo two-step Runge-Kutta-Nystr\"{o}m (EPTRKN) methods. We provedthat an $s$-stage FEPTRKN method has step order $p = s$ and stage order $r = s$for any set of distinct collocation parameters $(c_i)_{i=1}^s$.Supperconvergence for the accuracy orders of these methods can be obtained ifthe collocation parameters $(c_i)_{i=1}^s$ satisfy some orthogonalityconditions. We proved that an $s$-stage FEPTRKN method can attain accuracyorder $p = s + 3$. Numerical experiments have shown that the new FEPTRKNmethods work better than do EPTRKN methods on problems whose solutions can bewell approximated by the functions in bases on which these FEPTRKN methods aredeveloped.

Functionally-fitted explicit pseudo two-step Runge-Kutta-Nystr\"om methods