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A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains

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Abstract

This paper investigates a numerical method for solving two-dimensional nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The scheme utilizes the shape functions of the moving least squares (MLS) approximation constructed on scattered points as a basis in the discrete collocation method. The MLS methodology is an effective technique for approximating unknown functions which involves a locally weighted least square polynomial fitting. The proposed method is meshless, since it does not need any background mesh or cell structures and so it is independent of the geometry of the domain. The scheme reduces the solution of two-dimensional nonlinear integral equations to the solution of nonlinear systems of algebraic equations. The error analysis of the proposed method is provided. The efficiency and accuracy of the new technique are illustrated by several numerical examples.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, 15914, Iran

    Pouria Assari, Hojatollah Adibi & Mehdi Dehghan

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  1. Pouria Assari
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  2. Hojatollah Adibi
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  3. Mehdi Dehghan
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Correspondence to Mehdi Dehghan.

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Assari, P., Adibi, H. & Dehghan, M. A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains. Numer Algor 67, 423–455 (2014). https://doi.org/10.1007/s11075-013-9800-1

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