Bernstein dual-Petrov-Galerkin method: application to 2D time fractional diffusion equation

TL;DR

This paper introduces a Bernstein dual-Petrov-Galerkin spectral method for efficiently solving 2D fractional diffusion equations, demonstrating stability, convergence, and high accuracy through numerical examples.

Contribution

The paper develops a novel spectral discretization using dual Bernstein polynomials and derives an exact sparse operational matrix for the fractional diffusion problem.

Findings

The method produces banded linear systems that are computationally efficient.

Numerical results confirm the stability and high accuracy of the approach.

The approach is effective for 2D fractional diffusion equations.

Abstract

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.