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Generalized operational matrices and error bounds for polynomial basis

Author

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  • Guimarães, O.
  • Labecca, W.
  • Piqueira, José Roberto C.

Abstract

This work presents a direct way to obtain operational matrices for all complete polynomial basis, considering limited intervals, by using similarity relations. The algebraic procedure can be applied to any linear operator, particularly to the integration and derivative operations. Direct and spectral representations of functions are shown to be equivalent by using the compactness of Dirac notation, permitting the simultaneous use of the numerical solution techniques developed for both cases. The similarity methodology is computer oriented, emphasizing aspects concerning matrices operations, compacting the notation and lowering the computational costs and code elaboration times. To illustrate the operational aspects, some integral and differential equations are solved, including non-orthogonal basis examples, showing the generality of the method and the compatibility of the results with those obtained by other recent research works. As the Dirac notation is used, the projector operator allows to calculate the precision of the obtained solutions and the error superior limit, even if the exact solution is not available.

Suggested Citation

  • Guimarães, O. & Labecca, W. & Piqueira, José Roberto C., 2020. "Generalized operational matrices and error bounds for polynomial basis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 258-272.
    • Handle: RePEc:eee:matcom:v:172:y:2020:i:c:p:258-272
    DOI: 10.1016/j.matcom.2019.12.003
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    References listed on IDEAS

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    1. Doha, E.H. & Abd-Elhameed, W.M., 2009. "Efficient spectral ultraspherical-dual-Petrov–Galerkin algorithms for the direct solution of (2n+1)th-order linear differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(11), pages 3221-3242.
    2. F. Toutounian & Emran Tohidi & A. Kilicman, 2013. "Fourier Operational Matrices of Differentiation and Transmission: Introduction and Applications," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, April.
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    Cited by:

    1. Guimarães, Osvaldo & Labecca, William & Piqueira, José R.C., 2021. "Tensor solutions in irregular domains: Eigenvalue problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 110-130.

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