The paper considers a numerical method for solving the Fredholm integral equation of the first kind. The essence of the method is to replace the original equation with the corresponding regularized equation of the second kind, which is then solved by the modified spline collocation method. The solution in this case is represented as a linear combination of minimal splines. The coefficients at the splines are computed using local approximation (in some cases, quasi-interpolation) methods. Results of numerical experiments are presented and show that on model problems, the method proposed yields sufficiently accurate approximations, and the approximation accuracy can be improved by using minimal nonpolynomial splines and related functionals.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics [in Russian], Moscow (1976).
Yu. S. Zav’yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions [in Russian], Moscow (1980).
C. de Boor, A Practical Guide to Splines (Revised ed.), Springer-Verlag, New York (2001).
I. G. Burova and Yu. K. Dem’yanovich, Minimal Splines and Applications [in Russian], St. Petersburg Univ. Press, St. Petersburg (2010).
L. L. Schumaker, Spline Functions: Computational Methods, Society for Industrial and Applied Mathematics (2015).
M. Buhmann and J. Jäger, Quasi-Interpolation, Cambridge University Press (2022).
C. Allouch, P. Sablonniere, and D. Sbibih, “A modified Kulkarni’s method based on a discrete spline quasi-interpolant,” Math. Comput. Simul., 81, 1991–2000 (2011).
C. Dagnino, S. Remogna, and P. Sablonniere, “On the solution of Fredholm integral equation based on spline quasi-interpolating projectors,” BIT, 54, No. 4, 979–1008 (2014).
I. Sloan, “Improvement by iteration for compact operator equations,” Math. Comput., 30, 758–764 (1976).
R. Kulkarni, “On improvement of the iterated Galerkin solution of the second kind integral equations,” J. Numer. Math., 13, 205–218 (2005).
D. Černá and V. Finěk, “Galerkin method with new quadratic spline wavelets for integral and integro-differential equations,” J. Comput. Appl. Math., 363, 426–443 (2020).
O. Kosogorov and A. Makarov, “On some piecewise quadratic spline functions,” Lect. Notes Comput. Sci., 10187, 448–455 (2017).
E. K. Kulikov and A. A. Makarov, “Construction of approximation functionals for minimal splines,” Zap. Nauchn. Semin. POMI, 504, 136–156 (2021); English transl., J. Math. Sci., 262, No. 1, 84–89 (2022).
E. K. Kulikov and A. A. Makarov, “On modified spline collocations method for solving the Fredholm integral equation,” Diff. Uravn. Prots. Upr., No. 4, 211–223 (2021).
A. N. Tikhonov, “On the solution of incorrectly posed problems and the method of regularization,” Soviet Math., 4, 1035–1038 (1963).
I. G. Burova and V. M. Ryabov, “On the solution of Fredholm integral equation of the first kind,” WSEAS Trans. Math., 19, 699–708 (2021).
A. V. Lebedeva and V. M. Ryabov, “On regularization of the solution of integral equations of the first kind using quadrature formulas,” Vestn. St. Petersburg Univ., Ser. 1, 54, 361–365 (2021).
A.Wazwaz, “The regularization method for Fredholm integral equations of the first kind,” Comput. Math. Appl., 61, No. 10, 2981–2986 (2011).
D. Yuan and X. Zhang, “An overview of numerical methods for the first kind Fredholm integral equation,” SN Appl. Sci., 10, 1178–1190 (2019).
A. A. Makarov, “Construction of splines of maximal smoothness,” Probl. Mat. Anal., 60, 25–38 (2011); English transl., J. Math. Sci., 178, No. 6, 589–604 (2011).
E. K. Kulikov and A. A. Makarov, “Quadratic minimal splines with multiple nodes,” Zap. Nauchn. Semin. POMI, 482, 220–230 (2019); English transl., J. Math. Sci., 249, No. 2, 256–262 (2020).
E. K. Kulikov and A. A. Makarov, “On de Boor–Fix type functionals for minimal splines,” in: Topics in Classical and Modern Analysis (Applied and Numerical Harmonic Analysis) (2019), pp. 211–225.
P. Sablonniere, “Quadratic spline quasi-interpolants on bounded domains of ℝd, d = 1, 2, 3,” Rend. Semin. Mat., 61, No. 3, 229–246 (2003).
E. K. Kulikov and A. A. Makarov, “On approximation by hyperbolic splines,” Zap. Nauchn. Semin. POMI, 472, 179–194 (2018); English transl., J. Math. Sci., 240, No. 6, 822–832 (2019).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 514, 2022, pp. 113–125.
Translated by the authors.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kulikov, E.K., Makarov, A.A. A Method for Solving the Fredholm Integral Equation of the First Kind. J Math Sci 272, 558–565 (2023). https://doi.org/10.1007/s10958-023-06449-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06449-3