A numerical method for two-dimensional Hammerstein integral equations
DOI:
https://doi.org/10.24193/subbmath.2021.2.03Keywords:
Hammerstein integral equations, spline collocation, interpolation.Abstract
In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R², which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area.
Mathematics Subject Classification (2010): 41A15, 45B05, 47G10, 65D07, 65R20.
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