Summary: The Bagley-Torvik equation, which has an important role in fractional calculus, is solved by generalizing the Taylor collocation method [cf. R. L. Bagley
and P. J. Torvik
[J. Appl. Mech. 51, 294–298 (1984; Zbl 1203.74022
)]. The proposed method has a new algorithm for solving fractional differential equations. This new method has many advantages over variety of numerical approximations for solving fractional differential equations. To assess the effectiveness and preciseness of the method, results are compared with other numerical approaches. Since the Bagley-Torvik equation represents a general form of the fractional problems, its solution can give many ideas about the solution of similar problems in fractional differential equations.