Modification of fourier approximation for solving boundary value problems having singularities of boundary layer type

Boris Semisalov, Georgy Kuzmin

Research output: Contribution to journalConference articlepeer-review

Abstract

A method for approximating smooth functions has been developed using non-polynomial basis obtained by mapping of Fourier series domain to the segment [1, 1]. High rate of convergence and stability of the method is justified theoretically for four types of coordinate mappings, the dependencies of approximation error on values of derivatives of approximated functions are obtained. Algorithms for expanding of functions into series with coupled basis composed of Chebyshev polynomials and designed non-polynomial functions are implemented. It was shown that for functions having high order of smoothness and extremely steep gradients in the vicinity of bounds of segment the accuracy of proposed method cardinally exceeds that of Chebyshev's approximation. For such functions method allows to reach an acceptable accuracy using only = 10 basis elements (relative error does not exceed 1 per cent).

Original languageEnglish
Pages (from-to)406-422
Number of pages17
JournalCEUR Workshop Proceedings
Volume1839
Publication statusPublished - 1 Jan 2017

Keywords

  • Boundary value problem
  • Chebyshev polynomial
  • Collocation method
  • Coordinate mapping
  • Estimate of convergence rate
  • Fourier series
  • Non-polynomial basis
  • Singular perturbation
  • Small parameter

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