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A Highly Accurate Difference Method for Approximating the Solution and Its First Derivatives of the Dirichlet Problem for Laplace’s Equation on a Rectangle

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Abstract

A pointwise error estimation of the form \(O(\rho h^{8}),\) h is the mesh size, for the approximate solution of the Dirichlet problem for Laplace’s equation on a rectangular domain is obtained as a result of three-stage (9-point, 5-point and 5-point) finite difference method; here \(\rho \) \(=\rho (x,y)\) is the distance from the current grid point \((x,y)\in \Pi ^{h}\) to the boundary of the rectangle \(\Pi .\) Using this error estimation, it is proved that the proposed three-stage method constructed to find an approximate value of the first derivatives of the solution converges uniformly with an order of \(O(h^{8})\). The illustrated results of the numerical experiment support the analysis made.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, NAS of Azerbaijan, Baku, Azerbaijan

    Adiguzel A. Dosiyev

  2. Department of Mathematics, Near East University, Nicosia, Turkish Republic of Northern Cyprus

    Hediye Sarikaya

Authors
  1. Adiguzel A. Dosiyev
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  2. Hediye Sarikaya
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Correspondence to Adiguzel A. Dosiyev.

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Dosiyev, A.A., Sarikaya, H. A Highly Accurate Difference Method for Approximating the Solution and Its First Derivatives of the Dirichlet Problem for Laplace’s Equation on a Rectangle. Mediterr. J. Math. 18, 252 (2021). https://doi.org/10.1007/s00009-021-01900-8

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  • DOI: https://doi.org/10.1007/s00009-021-01900-8

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