Geodesic
On the method of solving nonlinear Fredholm integral equation of the second kind with piecewise-smooth kernels
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 27 (2025) no. 1, pp. 11-24.

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The work is devoted to the development of iterative methods for solving nonlinear Fredholm integral equations of the second kind with piecewise-smooth kernels. A new approach to constructing their solutions is proposed which combines the method of successive approximations with polynomial interpolations of functions on the segment $[-1,\,1]$. In this case, the original integral equation is reduced to a Volterra-type equation where the unknown function is defined on the segment mentioned. The free term of the equation is chosen as the initial approximation. At each iteration of successive approximations method, the kernel of the integral equation is represented as a partial sum of a series in Chebyshev polynomials orthogonal on the segment $[-1,\,1]$. The coefficients in this expansion are found using the orthogonality of vectors formed by the values of these polynomials at the zeros of the polynomial whose degree is equal to the number of unknown coefficients. An approximation of the solution is made by interpolation of the obtained values of the solution at the Chebyshev nodes at each iteration. The work also constructs a solution to an integral equation whose free term has a discontinuity point of the first kind. The results of the computational experiments are presented, which demonstrate the effectiveness of the proposed approach.
Keywords: nonlinear Fredholm integral equations, method of successive approximations, Chebyshev polynomials, Chebyshev nodes 
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O. V. Germider; V. N. Popov. On the method of solving nonlinear Fredholm integral equation of the second kind with piecewise-smooth kernels. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 27 (2025) no. 1, pp. 11-24. http://geodesic.mathdoc.fr/item/SVMO_2025_27_1_a0/

[1] A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Springer-Verlag, Beijing, 2011, 658 pp. | MR

[2] N. Karamollahi, M. Heydari, Gh.B. Loghmani, “Approximate solution of nonlinear Fredholm integral equations of the second kind using a class of Hermite”, Mathematics and Computers in Simulation, 187 (2021), 414–432 | DOI | MR

[3] S. Amiri, M. Hajipour, D. Baleanu, “On accurate solution of the Fredholm integral equations of the second kind”, Applied Numerical Mathematics, 150 (2020), 478-490 | DOI | MR

[4] Y. Ordokhani, M. Razzaghi, “Solution of nonlinear Volterra-Fredholm–Hammerstein integral equations via a collocation method and rationalized Haar functions”, Applied Mathematics Letters, 21 (2008), 4-9 | DOI | MR

[5] S. Islam, I. Aziz, A. Al-Fhaid, “An improved method based on Haar wavelets for numerical solution of nonlinear integral and”, Journal of Computational and Applied Mathematics, 260 (2014), 449-469 | DOI | MR

[6] S. Bazm, A. Hosseini, J.S. Azevedo, F. Pahlevani, “Existence, uniqueness, and numerical approximation of solutions of a nonlinear functional integral”, Journal of Computational and Applied Mathematics, 439 (2024), 115602 | DOI | MR

[7] A.N. Tynda, D.N. Sidorov, I.R. Muftakhov, “Chislennyi metod resheniya sistem nelineinykh integralnykh uravnenii Volterra I roda s razryvnymi yadrami”, Zhurnal Srednevolzhskogo matematicheskogo obschestva, 20:1 (2018), 55-63 | DOI

[8] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962, 566 pp. | MR

[9] N.S. Bakhvalov, N.P. Zhidkov, G.M. Kobelkov, Chislennye metody, Laboratoriya znanii, Moskva, 2020, 636 pp. | MR

[10] O.V. Germider, V.N. Popov, “Otsenka konstanty Lebega dlya Chebyshevskogo raspredeleniya uzlov”, Zhurnal Srednevolzhskogo matematicheskogo obschestva, 25:4 (2023), 242-254 | DOI | MR

[11] Bellman R., Kalaba R., Quasilinearization and Nonlinear Boundary Value Problems, Elsevier, New York, 1965, 62 pp. | MR

[12] S. Bazm, P. Limaand, S. Nemati, “Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type”, Journal of Computational and Applied Mathematics, 398 (2021), 113628 | DOI | MR

[13] J. Mason, D. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, New York, 2002, 360 pp. | DOI | MR

[14] S. Liu, G. Trenkler, “Hadamard, Khatri-Rao, Kronecker and other matrix products”, International Journal of Information and Systems Sciences, 4:1 (2008), 160-177 | MR