# Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations

## Abstract

The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate.

- Authors:

- Cairo University, Department of Mathematics, Faculty of Science (Egypt)

- Publication Date:

- OSTI Identifier:
- 22769289

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 3; 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; ALGORITHMS; CONVERGENCE; DIFFERENTIAL EQUATIONS; LIOUVILLE THEOREM; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; POLYNOMIALS; QUADRATURES; SYMMETRY

### Citation Formats

```
Abd-Elhameed, W. M., and Youssri, Y. H., E-mail: youssri@sci.cu.edu.eg.
```*Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0488-Z.

```
Abd-Elhameed, W. M., & Youssri, Y. H., E-mail: youssri@sci.cu.edu.eg.
```*Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations*. United States. doi:10.1007/S40314-017-0488-Z.

```
Abd-Elhameed, W. M., and Youssri, Y. H., E-mail: youssri@sci.cu.edu.eg. Sun .
"Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations". United States. doi:10.1007/S40314-017-0488-Z.
```

```
@article{osti_22769289,
```

title = {Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations},

author = {Abd-Elhameed, W. M. and Youssri, Y. H., E-mail: youssri@sci.cu.edu.eg},

abstractNote = {The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate.},

doi = {10.1007/S40314-017-0488-Z},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 3,

volume = 37,

place = {United States},

year = {2018},

month = {7}

}