The approximate solutions are established for a class of weakly singular Volterra integral equations derived from the boundary layer flow with a chemical surface reaction. Taking some orders of chemical reaction as examples, the fractional series expansions about zero and their Pad$\acute{\rm e}$ rational approximations are obtained. By interpreting the divergent integrals as the Hadamard finite part integrals, the asymptotic expansions with higher order logarithmic terms at infinity are derived with the aid of the numerical integration method. Practical calculation shows that the combination of the expansions at zero and infinity can efficiently solve this kind of boundary layer flow problem with chemical surface reaction on the whole half infinity interval.