In this work, we specify some special Smarandache curves with reference to the Lorentzian Darboux frame of a curve $gamma$ on a spacelike surface in Minkowski 3-space R^3_1. We obtain the natural curvature functions of the special Smarandache curves and also give some properties to these curves when the curve $gamma$ is an asymptotic curve, a geodesic curve or principal curve. Finally, we shall give an example to illustrate these curves.

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