Bernstein modal basis: application to the spectral Petrov-Galerkin method for fractional partial differential equations

TL;DR

This paper introduces a Bernstein modal basis for spectral Petrov-Galerkin methods applied to fractional PDEs, enabling efficient solutions with sparse linear systems and demonstrating spectral accuracy through numerical examples.

Contribution

It extends the modal basis concept to dual Bernstein polynomials and develops a spectral Petrov-Galerkin method for fractional PDEs with proven efficiency.

Findings

Produces banded sparse linear systems for constant coefficient problems

Demonstrates spectral accuracy in numerical tests

Shows efficiency of the proposed method

Abstract

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.