Shahnam Javadi

Mostafa Jani, Esmail Babolian, Shahnam Javadi

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.

Mostafa Jani, Shahnam Javadi, Esmail Babolian et al.

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.