New implementation of the Tau method for PDEs

Matos, José; Rodrigues, Maria João; Vasconcelos, Paulo B.

doi:10.1016/j.cam.2003.09.054

Cited by 12 publications

(6 citation statements)

References 13 publications

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“…, where S F T by (8), making use of the Tau Toolbox fred function. The independent vector is b T = 1, 1, 0.5(e −2 − 1), 0, .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…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%

See 1 more Smart Citation

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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“…, where S F T by (8), making use of the Tau Toolbox fred function. The independent vector is b T = 1, 1, 0.5(e −2 − 1), 0, .…”

Section: Numerical Resultsmentioning

confidence: 99%

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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“…Moreover, Ortiz and Samara [20] proposed an approximation technique for solving nonlinear ordinary differential equations using the operational Tau method. In recent years, the operational Tau method has been extended for the numerical solution of different kinds of ordinary differential equations (ODEs) [21,22] and partial differential equations (PDEs) [23,24]. In addition, various authors have implemented the Tau method for solving integral and integrodifferential equations [25,26].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Heat Transfer Process in PCM Storage Using Tau Method

Heydari¹,

Talati²

2015

Mathematical Problems in Engineering

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Thermal energy storage units that utilize phase change materials have been widely employed to balance temporary temperature alternations and store energy in many engineering systems. In the present paper, an operational approach is proposed to the Tau method with standard polynomial bases to simulate the phase change problems in latent heat thermal storage systems, that is, the two-dimensional solidification process in rectangular finned storage with a constant end-wall temperature. In order to illustrate the efficiency and accuracy of the present method, the solid-liquid interface location and the temperature distribution of the fin for three test cases with different geometries are obtained and compared to simplified analytical results in the published literature. The results indicate that using a two-dimensional numerical approach can predict the solid-liquid interface location more accurately than the simplified analytical model in all cases, especially at the corners.

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“…Tau method was invented by Lanczos [17] for solving ordinary differential equations (ODEs) and then the expansion of the method were carried out for many different problems such as partial differential equations (PDEs) [18][19][20][21][22], integral equations (IEs) [23], integro-differential equations (IDEs) [24], and so forth. [25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

An extension of the Tau numerical algorithm for the solution of linear and nonlinear Lane–Emden equations

Vanani

Soleymani

2012

Math Methods in App Sciences

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The purpose of this paper is to present a numerical algorithm for solving the Lane–Emden equations as singular initial value problems. The proposed algorithm is based on an operational Tau method (OTM). The main idea behind the OTM is to convert the desired problem to some operational matrices. Firstly, we use a special integral operator and convert the Lane–Emden equations to integral equations. Then, we use OTM to linearize the integral equations to some operational matrices and convert the problem to an algebraic system. The concepts, properties, and advantages of OTM and its application for solving Lane–Emden equations are presented. Some orthogonal polynomials are also used to reduce the volume of computations. Finally, several experiments of Lane–Emden equations including linear and nonlinear terms are given to illustrate the validity and efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.

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New implementation of the Tau method for PDEs

Cited by 12 publications

References 13 publications

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

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

Numerical Solution of Heat Transfer Process in PCM Storage Using Tau Method

An extension of the Tau numerical algorithm for the solution of linear and nonlinear Lane–Emden equations

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