  In the papers dealing with derivation and applications of operational
matrices of Bernstein polynomials, a basis transformation, commonly a
transformation to power basis, is used. The main disadvantage of this method is
that the transformation may be ill-conditioned. Moreover, when applied to the
numerical simulation of a functional differential equation, it leads to dense
operational matrices and so a dense coefficient matrix is obtained. In this
paper, we present a new property for Bernstein polynomials. Using this
property, we build exact banded operational matrices for derivatives of
Bernstein polynomials. Next, as an application, we propose a new numerical
method based on a Petrov-Galerkin variational formulation and the new
operational matrices utilizing the dual Bernstein basis for the time-fractional
advection-dispersion equation. Finally, we show that the proposed method leads
to a narrow-banded linear system and so less computational effort is required
to obtain the desired accuracy for the approximate solution. Some numerical
examples are provided to demonstrate the efficiency of the method.
