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By employing a simplified nonlinear memory function proposed recently by thepresent author, a universal equation for a collective-intermediate scatteringfunction derived based on the time-convolutionless mode-coupling theory isnumerically solved to study the dynamics of glass-forming liquids. Thenumerical calculation is based on the simulation results performed on two typesof liquids, fragile liquids and strong liquids. The numerical solutions arethen shown to be uniquely determined by the long-time collective diffusioncoefficient $D(q_m)$, where $q_m$ is a first peak position of a staticstructure factor for a whole system. It is confirmed based on four differentsimulation results that the supercooled state consists of two substates, aweakly supercooled state in which the nonlinear parameter $\mu$ in the memoryfunction increases as $D$ decreases and a deeply supercooled state in which$\mu$ becomes constant up to the glass transition. Here $\mu$ is shown to beconstant in a liquid state, while it is shown to grow rapidly in a glass state.The value of $\mu$ in each state is shown to be uniquely determined by $D$ andalso to depend on a type of liquids. Hence there exists such a universalitythat there is only one solution for different liquids of a same type at a givenvalue of $D$. This may be consistent with the fact that strong liquids arestructurally quite different from fragile liquids. Thus, it is emphasized thatsuch a universality must be helpful to predict $q_m$ from experimental data.

An existence of universality in the dynamics of two types of glass-forming liquids -- fragile liquids and strong liquids