This paper presents a family of computational schemes for the solution of the Bagley-Torvik equation. The schemes are based on the reformulation of the original problem into a system of fractional differential equations of order $1/2$. Then, suitable exponential integrators are devised to solve the resulting system accurately. The attainable order of convergence of exponential integrators for solving the fractional problem is studied. Theoretical findings are validated by means of some numerical examples. The advantages of the proposed method are illustrated by comparing several of the existing methods.
