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The main aim of this paper is to present new sets of non-perturbativefragmentation functions (FFs) for $D^0$ and $D^+$ mesons at next-to-leading(NLO) and, for the first time, at next-to-next-to-leading order (NNLO) in the$\overline{\mathrm{MS}}$ factorization scheme with five massless quark flavors.This new determination of FFs is based on the QCD fit to the {\tt OPAL}experimental data for hadron production in the electron-positronsingle-inclusive annihilation (SIA). We discuss in detail the novel aspects ofthe methodology used in our analysis and the validity of obtained FFs bycomparing with previous works in literature which have been carried out up toNLO accuracy. We will also incorporate the effect of charmed meson masscorrections into our QCD analysis and discuss the improvements upon inclusionof these effects. The uncertainties in the extracted FFs as well as in thecorresponding observables are estimated using the "Hessian" approach. For atypical application, we use our new FFs to make theoretical predictions for theenergy distributions of charmed mesons inclusively produced through the decayof unpolarized top quarks, to be measured at the CERN LHC. As a result of thisanalysis, suggestions are discussed for possible future studies on the currenttopic to consider any theory improvements and other available experimentalobservables.

NNLO charmed-meson fragmentation functions and their uncertainties in the presence of meson mass corrections