  In the spectral Petrov-Galerkin methods, the trial and test functions are
required to satisfy particular boundary conditions. By a suitable linear
combination of orthogonal polynomials, a basis, that is called the modal basis,
is obtained. In this paper, we extend this idea to the non-orthogonal dual
Bernstein polynomials. A compact general formula is derived for the modal basis
functions based on dual Bernstein polynomials. Then, we present a
Bernstein-spectral Petrov-Galerkin method for a class of time fractional
partial differential equations with Caputo derivative. It is shown that the
method leads to banded sparse linear systems for problems with constant
coefficient. Some numerical examples are provided to show the efficiency and
the spectral accuracy of the method.
