Abstract
In this paper, we present a collocation method for the approximate solution of two-dimensional Hammerstein integral equations over polygonal regions in \({\mathbb R}^{2}\). For a reformulation of the equation, we triangulate the polygonal region and use a special type of quadratic interpolation on each triangle. This produces a quadrature formula with a higher degree of precision than expected, leading to a numerical method that is superconvergent at the collocation nodes. We prove the convergence of the method and give error estimates. The implementation is also discussed and the applicability of the proposed scheme is presented on numerical examples. The paper concludes with a discussion on the efficiency of the proposed method, also highlighting some ideas for future research in this area.
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References
Assari, P.: The numerical solution of Fredholm-Hammerstein integral equations by combining the collocation method and radial basis functions. Filomat 33(3), 667–682 (2019). https://doi.org/10.2298/FIL1903667A
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind Cambridge Monographs on Applied and Computational Mathematics, vol. 4. Cambridge University Press, Cambridge (1997)
Atkinson, K.E., Graham, I., Sloan, I.: Piecewise continuous collocation for boundary integral equations, SIAM. J. Numer. Anal. 20(1), 172–186 (1983)
Aziz, I., Islam, S.: New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets. J. Comput. Appl. Math. 239, 333–345 (2013). https://doi.org/10.1016/j.cam.2012.08.031
Bazm, S.: Numerical solution of a class of nonlinear two-dimensional integral equations using bernoulli polynomials. Sahand Comm. Math. Anal. 3 (1), 37–51 (2016). https://scma.maragheh.ac.ir/article_15994_6d676e68a2b7a882c1334cdc50d1acf4.pdf
Brezinski, C., Redivo-Zaglia, M.: Extrapolation methods for the numerical solution of nonlinear Fredholm integral equations. J. Integral Equ. Appl. 31(1), 29–57 (2019). https://doi.org/10.1216/JIE-2019-31-1-29
Fallahzadeh, A.: Solution of two-dimensional Fredholm integral equation via RBF-triangular method. J. Interp. Approx. Sci. Comput. 2012, 1–5 (2012). https://doi.org/10.5899/2012/jiasc-00002
Hungerford, T.: Algebra. Springer, New York (1974)
Kumar, S., Sloan, I.H.: A new collocation-type method for Hammerstein integral equations. Math. of Comp. 48, 585–593 (1987)
Micula, S.: A numerical method for two-dimensional Hammerstein integral equations. Stud. Univ. Babeş-Bolyai Math. 66(2), 267–277 (2021). https://doi.org/10.24193/subbmath.2021.2.03
Micula, S.: A spline collocation method for Fredholm-Hammerstein integral equations of the second kind in two variables. Appl. Math. Comp. 265, 352–357 (2015). https://doi.org/10.1016/j.amc.2015.05.017
Mosa, G.A., Abdou, M.A., Rahby, A.S.: Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag. AIMS Math. 6(8), 8525–8543 (2021). https://doi.org/10.3934/math.2021495
Neamprem, K., Klangrak, A., Kaneko, H.: Taylor-series expansion methods for multivariate Hammerstein integral equations. Int. J. Appl. Math. 47(4). http://www.iaeng.org/IJAM/issues_v47/issue_4/IJAM_47_4_10.pdf (2017)
Sobolev, S.L.: Cubature formulas on the sphere invariant under finite groups of rotations. Sov. Math. 3, 1307–1310 (1962)
Tufa, A.R., Zegeye, H., Thuto, M.: Iterative solutions of nonlinear integral equations of Hammerstein type. Int. J. Anal. Appl. 9(2), 129–141 (2015). http://www.etamaths.com
Wazwaz, A.M.: Linear and Nonlinear Integral Equations, Methods and Applications. Higher Education Press, Beijing. Springer, New York (2011)
Xie, J., Huang, Q., Zhao, F.: Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations in two-dimensional spaces based on Block Pulse functions. J. Comput. Appl. Math. 317, 565–572 (2017). https://doi.org/10.1016/j.cam.2016.12.028
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Micula, S. Numerical solution of two-dimensional Hammerstein integral equations via quadratic spline collocation. Numer Algor 93, 1225–1241 (2023). https://doi.org/10.1007/s11075-022-01465-x
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DOI: https://doi.org/10.1007/s11075-022-01465-x