Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations. (11th August 2017)

• Record Type: Journal Article
• Authors: Jani, M.; Babolian, E.; Javadi, S.
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 18(2018)
• Page Start: 7663
• Page End: 7672
• Publication Date: 2017-08-11
• Subjects: Bernstein polynomials -- Petrov‐Galerkin -- dual Bernstein polynomials -- fractional partial differential equations -- modal basis; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4551
• Languages: English
• ISSNs: 0170-4214

Abstract

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 nonorthogonal 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 coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.