This paper concerns the numerical approximation of Fractional Initial Value Problems (FIVPs). This is achieved by constructing k-step continuous BDFs. These continuous schemes are developed via the interpolation and collocation approach and are used to obtain the discrete k-step BDF and (k − 1) additional methods which are applied as numerical integrators in a block-by-block mode for the integration of FIVP. The properties of the methods are established and regions of absolute stability of the methods are plotted in the complex plane. Numerical tests including large systems arising form the semidiscretization of one-dimensional fractional Burger’s equation show that the methods are highly accurate and efficient.

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