Bernstein dual-Petrov-Galerkin method: application to 2D time fractional diffusion equation
This work provides a spectral method for fractional PDEs, but it is an incremental extension of existing Petrov-Galerkin techniques to a specific problem class.
The authors developed a Bernstein dual-Petrov-Galerkin method for solving 2D time fractional diffusion equations, achieving efficient banded linear systems and demonstrating accuracy through numerical examples.
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.