### Abstract

Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.

Original language | English |
---|---|

Pages (from-to) | 1552-1573 |

Number of pages | 22 |

Journal | International Journal of Computer Mathematics |

Volume | 91 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2014 |

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### Keywords

- infinite-horizon
- nonclassical pseudospectral method
- nonlinear programming
- variational problems
- weighted interpolation

### ASJC Scopus subject areas

- Applied Mathematics
- Computer Science Applications
- Computational Theory and Mathematics

### Cite this

*International Journal of Computer Mathematics*,

*91*(7), 1552-1573. https://doi.org/10.1080/00207160.2013.854338

**Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems.** / Maleki, Mohammad; Hashim, Ishak; Abbasbandy, Saeid.

Research output: Contribution to journal › Article

*International Journal of Computer Mathematics*, vol. 91, no. 7, pp. 1552-1573. https://doi.org/10.1080/00207160.2013.854338

}

TY - JOUR

T1 - Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems

AU - Maleki, Mohammad

AU - Hashim, Ishak

AU - Abbasbandy, Saeid

PY - 2014

Y1 - 2014

N2 - Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.

AB - Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.

KW - infinite-horizon

KW - nonclassical pseudospectral method

KW - nonlinear programming

KW - variational problems

KW - weighted interpolation

UR - http://www.scopus.com/inward/record.url?scp=84906786960&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906786960&partnerID=8YFLogxK

U2 - 10.1080/00207160.2013.854338

DO - 10.1080/00207160.2013.854338

M3 - Article

VL - 91

SP - 1552

EP - 1573

JO - International Journal of Computer Mathematics

JF - International Journal of Computer Mathematics

SN - 0020-7160

IS - 7

ER -