Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation

TL;DR

This paper introduces a novel approach to construct banded operational matrices for Bernstein polynomials, enabling efficient numerical solutions to fractional advection-dispersion equations with reduced computational effort.

Contribution

The paper develops exact banded operational matrices for Bernstein polynomial derivatives and applies them to a Petrov-Galerkin method for fractional PDEs, improving computational efficiency.

Findings

The new matrices are banded, reducing computational complexity.

The method effectively solves fractional advection-dispersion equations.

Numerical examples confirm the efficiency and accuracy of the approach.

Abstract

In the papers dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. Finally, we show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. Some numerical examples are provided to demonstrate the efficiency of the method.