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

@article{Jani2016BandedOM, title={Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation}, author={Mostafa Jani and Esmail Babolian and Shahnam Javadi and Dambaru D. Bhatta}, journal={Numerical Algorithms}, year={2016}, volume={75}, pages={1041-1063} }

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… Expand

#### 6 Citations

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

- Mathematics
- 2016

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… Expand

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

- Mathematics, Physics
- 2016

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… Expand

Numerical resolution of large deflections in cantilever beams by Bernstein spectral method and a convolution quadrature

- Mathematics
- 2018

The mathematical modeling of the large deflections for the cantilever beams leads to a nonlinear differential equation with the mixed boundary conditions. Different numerical methods have been… Expand

A numerical scheme for space-time fractional advection-dispersion equation

- Computer Science, Mathematics
- ArXiv
- 2015

A numerical resolution of the space-time fractional advection-dispersion equation using Bernstein polynomials as basis and a product integration method in order to simplify the evaluation of the terms involving spatial fractional order derivatives is developed. Expand

#### References

SHOWING 1-10 OF 34 REFERENCES

A new operational matrix for solving fractional-order differential equations

- Computer Science, Mathematics
- Comput. Math. Appl.
- 2010

The main aim is to generalize the Legendre operational matrix to the fractional calculus and reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. Expand

The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

- Computer Science, Mathematics
- J. Comput. Appl. Math.
- 2011

In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. Expand

Application of the Exact Operational Matrices Based on the Bernstein Polynomials

- Mathematics
- 2013

This paper aims to develop a new category of operational matrices. Exact operational matrices (EOMs) are matrices which integrate, differentiate and product the vector(s) of basis functions without… Expand

A Bernstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations

- Computer Science, Mathematics
- Math. Comput. Model.
- 2012

A new approach implementing a collocation method in combination with operational matrices of Bernstein polynomials for the numerical solution of VFIDEs is introduced, which reduces such problems to ones of solving systems of algebraic equations. Expand

Finite difference/spectral approximations for the time-fractional diffusion equation

- Mathematics, Computer Science
- J. Comput. Phys.
- 2007

It is proved that the full discretization is unconditionally stable, and the numerical solution converges to the exact one with order O(@Dt^2^-^@a+N^- ^m), where @Dt,N and m are the time step size, polynomial degree, and regularity of the exact solution respectively. Expand

A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations

- Mathematics, Computer Science
- Appl. Math. Comput.
- 2016

A collocation method that based on Bernstein polynomials is presented for nonlinear Fredholm-Volterra integro-differential equations (NFVIDEs) and error analysis is applied for the Bernstein series solutions. Expand

Algorithms for polynomials in Bernstein form

- Mathematics, Computer Science
- Comput. Aided Geom. Des.
- 1988

Bernstein forms for various basic polynomial procedures are developed, and are found to be of similar complexity to their customary power forms, establishing the viability of systematic computation with the Bernstein form, avoiding the need for (numerically unstable) basis conversions. Expand

DMLPG solution of the fractional advection–diffusion problem

- Mathematics
- 2015

Abstract The aim of this work is application of the direct meshless local Petrov–Galerkin (DMLPG) method for solving a two-dimensional time fractional advection–diffusion equation. This method is… Expand

On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations

- Mathematics
- 2011

A new formula expressing explicitly the derivatives of Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves is proved, and a formula expressing the… Expand

Bernstein operational matrix of fractional derivatives and its applications

- Mathematics
- 2014

Abstract In this paper, Bernstein operational matrix of fractional derivative of order α in the Caputo sense is derived. We also apply this matrix to the collocation method for solving multi-order… Expand