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

Mohammad Maleki, Ishak Hashim, Saeid Abbasbandy

Research output: Contribution to journalArticle

3 Citations (Scopus)

### 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 1552-1573 22 International Journal of Computer Mathematics 91 7 https://doi.org/10.1080/00207160.2013.854338 Published - 2014

### Fingerprint

Pseudospectral Method
Finite Horizon
Nonlinear programming
Variational Problem
Orthogonal Polynomials
Infinite-horizon Problems
Nonlinear Problem
Interpolation
Polynomials
Nonlinear Programming
Interpolate
Gauss Points
Infinite Domain
Infinite Horizon
Interpolants
Direct Method
Weight Function
Monotonic

### 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

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

In: International Journal of Computer Mathematics, Vol. 91, No. 7, 2014, p. 1552-1573.

Research output: Contribution to journalArticle

title = "Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems",
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.",
keywords = "infinite-horizon, nonclassical pseudospectral method, nonlinear programming, variational problems, weighted interpolation",
author = "Mohammad Maleki and Ishak Hashim and Saeid Abbasbandy",
year = "2014",
doi = "10.1080/00207160.2013.854338",
language = "English",
volume = "91",
pages = "1552--1573",
journal = "International Journal of Computer Mathematics",
issn = "0020-7160",
publisher = "Taylor and Francis Ltd.",
number = "7",

}

TY - JOUR

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

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

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JO - International Journal of Computer Mathematics

JF - International Journal of Computer Mathematics

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