An extension of Ortiz’ recursive formulation of the tau method to certain linear systems of ordinary differential equations

Crisci, M. R.; Russo, E.

doi:10.1090/s0025-5718-1983-0701622-9

Cited by 10 publications

(2 citation statements)

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“…The recursive approach of Tau method was firstly introduced in 1969 by Ortiz [7] to solve a class of ordinary differential equations, and was extended for solving a system of ODE's in [24,33]. This method firstly considered the approximate solution as a linear combination of some suitable functions called canonical polynomials which are calculated recursively, and secondly transformed the underlying problem to an equivalent system of algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

Shahmorad¹,

Mokhtary²,

Talaei³

et al. 2021

Preprint

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This paper provides an efficient recursive approach of the spectral Tau method, to approximate the solution of system of generalized Abel-Volterra integral equations. In this regards, we first investigate the existence, uniqueness as well as smoothness of the solutions under various assumptions on the given data. Next, from a numerical perspective, we express approximated solution as a linear combination of suitable canonical polynomials which are constructed by an easy to use recursive formula. Mostly, the unknown parameters are calculated by solving a low dimensional algebraic systems independent of degree of approximation which prevent from high computational costs. Obviously, due to singular behavior of the exact solution, using classical polynomials to construct canonical polynomials, leads to low accuracy results. In this regards, we develop a new fractional order canonical polynomials using Müntz-Legendre polynomials which have a same asymptotic behavior with the solution of underlying problem. The convergence analysis is discussed, and the familiar spectral accuracy is achieved in L ∞ -norm. Finally, the reliability of the method is evaluated using various problems.Keywords The recursive approach of the Tau method • System of generalized Abel-Volterra integral equations • Müntz-Legendre polynomials • Fractional vector canonical polynomials • convergence analysis.

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Section: Introductionmentioning

confidence: 99%

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

Shahmorad¹,

Mokhtary²,

Talaei³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Initially developed for linear differential problems with polynomial coefficients, it has been used to solve broader mathematical formulations: functional coefficients, nonlinear differential and integro-differential equations. Several studies applying the tau method have been performed to approximate the solution of differential linear and non-linear equations [2,7], partial differential equations [8,9] and integro-differential equations [1,11], among others. Nevertheless, in all these works the tau method is tuned for the approximation of specific problems and not offered as a general purpose numerical tool.…”

Section: Introductionmentioning

confidence: 99%

Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations

Vasconcelos

Matos

Trindade

2017

Integral Methods in Science and Engineering, Volume 1

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In this paper an extension of the spectral Lanczos' tau method to systems of nonlinear integro-differential equations is proposed. This extension includes (i) linearization coefficients of orthogonal polynomials products issued from nonlinear terms and (ii) recursive relations to implement matrix inversion whenever a polynomial change of basis is required and (iii) orthogonal polynomial evaluations directly on the orthogonal basis. All these improvements ensure numerical stability and accuracy in the approximate solution. Exposed in detail, this novel approach is able to significantly outperform numerical approximations with other methods as well as different tau implementations. Numerical results on a set of problems illustrate the impact of the mathematical techniques introduced.

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A survey of recent applications of the Tau Method to problems in mathematical modelling

Ortiz

1988

Mathematical and Computer Modelling

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An extension of Ortiz’ recursive formulation of the tau method to certain linear systems of ordinary differential equations

Abstract: Abstract. Ortiz' step-by-step recursive formulation of the Lanczos tau method is extended to the numerical solution of linear systems of differential equations with polynomial coefficients.Numerical comparisons are made with Gear's and Enright's methods.

Cited by 10 publications

References 14 publications

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations

A survey of recent applications of the Tau Method to problems in mathematical modelling

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