A recursive formulation of collocation in terms of canonical polynomials

El-Daou, M.K.; Ortiz, Eduardo L.

doi:10.1007/bf02238075

Cited by 18 publications

(11 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Following the same arguments we can derive analogous results when H N is of the form (11). It is worth to note that the Tau method with a perturbation of the form (11) is equivalent to the spectral collocation method at the zeros of V N (x), see (El-Daou & Ortiz, 1994). This equivalence permits to solve nonlinear differential equations using the spectral approach more effectively than the recursive Tau.…”

Section: Corollary 4 the Assumptions Of The Previous Theorem Hold Ifmentioning

confidence: 99%

“…Its basic idea is to perturb the right hand side of the differential equation in a way that an exact polynomial solution of the new equation can be found analytically. This method was devised in (Lanczos, 1956) to find polynomial approximations for simple linear ordinary differential equations (ODE) and it was extended later on to treat differential equations with different level of complexities, (see (Ortiz, 1969), (Ortiz & Samara, 1984), (El-Daou & Ortiz, 1992-1994, and (Liu & Pan, 1999)). …”

Section: Introductionmentioning

confidence: 99%

“…The recursive approach, proposed in (Ortiz, 1969), permits to obtain an approximate polynomial solution expressed in terms of a special polynomials basis called canonical polynomials. This technique has been thoroughly investigated in a series of papers (see for example (Crisci & Russo, 1983), (Freilich & Ortiz, 1982) and (El-Daou & Ortiz, 1994-1998). In the operational Tau, (see (Ortiz & Samara, 1981)), the ODE is transformed to a system of linear algebraic equations using some simple elementary matrices.…”

Section: Introductionmentioning

confidence: 99%

“…Although the three approaches of the Tau Method explained above appear to be different, it was shown in (El-Daou & Ortiz, 1992-1994) that they are equivalent in the sense that they yield the same approximate solution. However, the suitability of those approaches is judged by the problem under consideration.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multivariate Canonical Polynomials in the Tau Method with Applications to Optimal Control Problems

Daou¹,

Alhamad²,

Zadeh³

2015

JMR

View full text Add to dashboard Cite

The Tau method is a highly accurate technique that approximates differential equations efficiently. This paper discusses two approaches of the Tau Method: recursive and spectral. In the recursive Tau, the approximate solution of the differential equation is obtained in terms of a special polynomial basis called canonical polynomials. The present paper extends this concept to the multivariate canonical polynomial vectors and proposes a self starting algorithm to generate those vectors. In the spectral Tau, the approximate solution is obtained as a truncated series expansions in terms of a set of orthogonal polynomials where the coefficients of the expansions are obtained by forcing the defect of the differential equation to vanish at the some selected points. In this paper we use the spectral Tau to solve a class of optimal control problems associated with a nonlinear system of differential equations. Some numerical examples that confirm our method are given.

show abstract

Section: Corollary 4 the Assumptions Of The Previous Theorem Hold Ifmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multivariate Canonical Polynomials in the Tau Method with Applications to Optimal Control Problems

Daou¹,

Alhamad²,

Zadeh³

2015

JMR

View full text Add to dashboard Cite

show abstract

“…In general, the collocation approach does not require basis functions to be orthogonal. For example, splines [4], canonical polynomials [5], or even mesh-free formulations, based on specially constructed functions [6], had been proposed. So far, none of the non-orthogonal approaches could compete with the standard pseudospectral methods in popularity because there is no clear and decisive advantage in those alternatives.…”

Section: Introductionmentioning

confidence: 99%

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

Chen¹

2016

View full text Add to dashboard Cite

In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.

show abstract

The Tau Method as an analytic tool in the discussion of equivalence results across numerical methods

El-Daou

Ortiz

1998

Computing

View full text Add to dashboard Cite

A recursive formulation of collocation in terms of canonical polynomials

Cited by 18 publications

References 16 publications

Multivariate Canonical Polynomials in the Tau Method with Applications to Optimal Control Problems

Multivariate Canonical Polynomials in the Tau Method with Applications to Optimal Control Problems

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

The Tau Method as an analytic tool in the discussion of equivalence results across numerical methods

Contact Info

Product

Resources

About