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Volume 16, Issue 4
Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer

Alexander Zadorin & Svetlana Tikhovskaya

Int. J. Numer. Anal. Mod., 16 (2019), pp. 590-608.

Published online: 2019-02

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  • Abstract

It is known that the solution of a singularly perturbed problem corresponds to the function with large gradients in a boundary layer. The application of Lagrange polynomial on a uniform mesh to interpolate such functions leads to large errors. To achieve the error estimates uniform with respect to a small parameter, we can use either a polynomial interpolation on a mesh which condenses in a boundary layer or we can use special interpolation formulas which are exact on a boundary layer component of the interpolating function. In this paper, we construct and study the formulas of numerical differentiation based on the interpolation formulas which are exact on a boundary layer component. We obtained the error estimates which are uniform with respect to a small parameter. Some numerical results validating the theoretical estimates are discussed.

  • Keywords

Function of one variable, exponential boundary layer, formulas of numerical differentiation, an error estimate.

  • AMS Subject Headings

65D25, 41A30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zadorin@ofim.oscsbras.ru (Alexander Zadorin)

s.tihovskaya@yandex.ru (Svetlana Tikhovskaya)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-16-590, author = {Alexander and Zadorin and zadorin@ofim.oscsbras.ru and 13109 and Laboratory of Mathematical Modeling in Mechanics of Omsk brunch, Sobolev Institute of Mathematics, Omsk, Pevtsova street, 13, 644099, Russia and Alexander Zadorin and Svetlana and Tikhovskaya and s.tihovskaya@yandex.ru and 13110 and Laboratory of Mathematical Modeling in Mechanics of Omsk brunch, Sobolev Institute of Mathematics, Omsk, Pevtsova street, 13, 644099, Russia and Svetlana Tikhovskaya}, title = {Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {4}, pages = {590--608}, abstract = {

It is known that the solution of a singularly perturbed problem corresponds to the function with large gradients in a boundary layer. The application of Lagrange polynomial on a uniform mesh to interpolate such functions leads to large errors. To achieve the error estimates uniform with respect to a small parameter, we can use either a polynomial interpolation on a mesh which condenses in a boundary layer or we can use special interpolation formulas which are exact on a boundary layer component of the interpolating function. In this paper, we construct and study the formulas of numerical differentiation based on the interpolation formulas which are exact on a boundary layer component. We obtained the error estimates which are uniform with respect to a small parameter. Some numerical results validating the theoretical estimates are discussed.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13016.html} }
TY - JOUR T1 - Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer AU - Zadorin , Alexander AU - Tikhovskaya , Svetlana JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 590 EP - 608 PY - 2019 DA - 2019/02 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13016.html KW - Function of one variable, exponential boundary layer, formulas of numerical differentiation, an error estimate. AB -

It is known that the solution of a singularly perturbed problem corresponds to the function with large gradients in a boundary layer. The application of Lagrange polynomial on a uniform mesh to interpolate such functions leads to large errors. To achieve the error estimates uniform with respect to a small parameter, we can use either a polynomial interpolation on a mesh which condenses in a boundary layer or we can use special interpolation formulas which are exact on a boundary layer component of the interpolating function. In this paper, we construct and study the formulas of numerical differentiation based on the interpolation formulas which are exact on a boundary layer component. We obtained the error estimates which are uniform with respect to a small parameter. Some numerical results validating the theoretical estimates are discussed.

Alexander Zadorin & Svetlana Tikhovskaya. (2019). Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer. International Journal of Numerical Analysis and Modeling. 16 (4). 590-608. doi:
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