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Based on the fractional-order thermoelastic theory, the dynamic response of a finite quasicrystal (QC) rod fixed at both ends and subjected to a moving heat source is analyzed. The governing equations of one-dimensional (1D) hexagonal QC rod are derived. And then the numerical solution is obtained by using the Laplace transform and its numerical inversion. The effects of time; the velocity of heat source; and fractional-order parameter on displacement, stress, and temperature of the QCs are discussed and compared with classical materials in this paper.