Geodesic
Solution of two-dimensional Fredholm integral equations of the second kind by the method of collocation and least squares with polynomial approximation
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 18 (2025) no. 1, pp. 118-129

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A new numerical algorithm based on the collocation and least squares method is proposed for the numerical solution of a two-dimensional Fredholm integral equation of the second kind. The solution is sought in the form of a polynomial approximant with undetermined coefficients, after substituting which into the equation, obtains an approximate equation with respect to the undetermined coefficients. The collocation method is used to solve that equation, and the number of collocation points is usually taken to be greater than the number of coefficients of the sought approximant. Overdetermined system of linear algebraic equations (SLAE) with respect to the sought coefficients are obtained by collocations of the approximate equation. The proposed algorithm is implemented in a computer program. Presented the results of numerical experiments on solving several equations for which are known results obtained by other methods cited in well-known publications. By comparing the results obtained by the new proposed algorithm with results achieved by other methods shown it's advantage in accuracy of the approximate solution over the compared methods. In numerical experiments were investigated the influence of the method parameters on the condition number of SLAE matrix, the solution of which is used to find polynomial approximation of the solution of integral equation. The tables of the numerical results show the values of the algorithm parameters with which were obtained specific solutions: the degree of the approximating polynomial, the number of cells and nodes of the Gauss quadrature, the degree of the SLAE overdetermination and condition number of it's matrix.
Keywords: two-dimensional Fredholm integral equations of the second kind, direct method, collocation method, linear least squares problem, SLAE condition number.
Mots-clés : Gauss quadratures 
@article{VYURU_2025_18_1_a8,
     author = {V. P. Shapeev and P. I. Kirillov},
     title = {Solution of two-dimensional {Fredholm} integral equations of the second kind by the method of collocation and least squares with polynomial approximation},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {118--129},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2025},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2025_18_1_a8/}
}
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V. P. Shapeev; P. I. Kirillov. Solution of two-dimensional Fredholm integral equations of the second kind by the method of collocation and least squares with polynomial approximation. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 18 (2025) no. 1, pp. 118-129. http://geodesic.mathdoc.fr/item/VYURU_2025_18_1_a8/