Bernstein modal basis: application to spectral Petrov-Galerkin method for higher-order differential equations

In spectral Petrov-Galerkin methods, the trial (basis) and test functions are required to satisfy particular boundary conditions. So, by a suitable linear combination of orthogonal polynomials, a basis, called the modal basis, is built. In this paper, we extend this idea to the non-orthogonal dual Bernstein polynomials (DBP). A compact formula is derived for the modal basis functions made by DBPs. Next, we present a Bernstein-spectral Petrov-Galerkin method for a class of time fractional PDEs. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Numerical examples are provided to show the efficiency and spectral accuracy of the method.