  In this paper, we develop a Bernstein dual-Petrov-Galerkin method for the
numerical simulation of a two-dimensional fractional diffusion equation. A
spectral discretization is applied by introducing suitable combinations of dual
Bernstein polynomials as the test functions and the Bernstein polynomials as
the trial ones. We derive the exact sparse operational matrix of
differentiation for the dual Bernstein basis which provides a matrix based
approach for the spatial discretization. It is shown that the method leads to
banded linear systems that can be solved efficiently. The stability and
convergence of the proposed method is discussed. Finally, some numerical
examples are provided to support the theoretical claims and to show the
accuracy and efficiency of the method.
