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

## 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.

- Authors:

- Kharazmi University, Department of Mathematics, Faculty of Mathematical Sciences and Computer (Iran, Islamic Republic of)
- The University of Texas Rio Grande Valley, School of Mathematical and Statistical Sciences (United States)

- Publication Date:

- OSTI Identifier:
- 22769311

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; COMPUTERIZED SIMULATION; DIFFUSION EQUATIONS; MATRICES; POLYNOMIALS; TWO-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL SYSTEMS

### Citation Formats

```
Jani, M., E-mail: mostafa.jani@gmail.com, Javadi, S., E-mail: javadi@khu.ac.ir, Babolian, E., E-mail: babolian@khu.ac.ir, and Bhatta, D., E-mail: dambaru.bhatta@utrgv.edu.
```*Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0455-8.

```
Jani, M., E-mail: mostafa.jani@gmail.com, Javadi, S., E-mail: javadi@khu.ac.ir, Babolian, E., E-mail: babolian@khu.ac.ir, & Bhatta, D., E-mail: dambaru.bhatta@utrgv.edu.
```*Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation*. United States. doi:10.1007/S40314-017-0455-8.

```
Jani, M., E-mail: mostafa.jani@gmail.com, Javadi, S., E-mail: javadi@khu.ac.ir, Babolian, E., E-mail: babolian@khu.ac.ir, and Bhatta, D., E-mail: dambaru.bhatta@utrgv.edu. Tue .
"Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation". United States. doi:10.1007/S40314-017-0455-8.
```

```
@article{osti_22769311,
```

title = {Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation},

author = {Jani, M., E-mail: mostafa.jani@gmail.com and Javadi, S., E-mail: javadi@khu.ac.ir and Babolian, E., E-mail: babolian@khu.ac.ir and Bhatta, D., E-mail: dambaru.bhatta@utrgv.edu},

abstractNote = {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.},

doi = {10.1007/S40314-017-0455-8},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}