We propose a non-polynomial collocation method for solving fractional di®erential equations. The construction of such a scheme is based on the classical equivalence between certain fractional di®erential equations and corresponding Volterra integral equations. Usually, we cannot expect the solution of a fractional di®erential equation to be smooth and this poses a challenge to the convergence analysis of numerical schemes. In this paper, the approach we present takes into account the potential non-regularity of the solution, and so we are able to obtain a result on optimal order of con- vergence without the need to impose inconvenient smoothness conditions on the solution. An error analysis is provided for the linear case and several examples are presented and discussed.