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

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Book cover Integral Methods in Science and Engineering, Volume 1

Abstract

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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Acknowledgment

This work was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER) under the partnership agreement PT2020. The third author was supported by CAPES, Coordination of Superior Level Staff Improvement - Brazil.

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Authors and Affiliations

  1. Center of Mathematics, University of Porto, Porto, Portugal

    P. B. Vasconcelos, J. Matos & M. S. Trindade

  2. Economics Faculty, University of Porto, Porto, Portugal

    P. B. Vasconcelos

  3. Polytechnic School of Engineering Porto, Porto, Portugal

    J. Matos

Authors
  1. P. B. Vasconcelos
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  2. J. Matos
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  3. M. S. Trindade
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Corresponding author

Correspondence to P. B. Vasconcelos .

Editor information

Editors and Affiliations

  1. Department of Mathematics, The University of Tulsa, Tulsa, Oklahoma, USA

    Christian Constanda

  2. Department of Mathematics, The University of Tulsa, Tulsa, Oklahoma, USA

    Matteo Dalla Riva

  3. Department of Mathematics, University of Padova, Padova, Italy

    Pier Domenico Lamberti

  4. Systems Analysis, Prognosis and Control, Fraunhofer Institute for Industrial Math, Kaiserslautern, Germany

    Paolo Musolino

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Vasconcelos, P.B., Matos, J., Trindade, M.S. (2017). Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations. In: Constanda, C., Dalla Riva, M., Lamberti, P., Musolino, P. (eds) Integral Methods in Science and Engineering, Volume 1. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59384-5_27

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