Adaptive Regularisation Method for Solving Nonlinear Fredholm Integral Equations of the First Kind
DOI:
https://doi.org/10.37134/jsml.vol13.2.2.2025Keywords:
Tikhonov approach, inverse visualisation, control theory, signal processing, noise distortionAbstract
This paper presented a comprehensive approach to the construction of a robust regularisation technique for solving the nonlinear Fredholm integral equation of the first kind, a class of problems frequently encountered in such areas of signal processing, inverse imaging, and control theory. The purpose of the study was to develop an efficient and reasonable procedure to regularise this type of equation, which improves the accuracy of solutions in conditions where standard methods are ineffective due to noise or nonlinear distortion. The study proposed a modification of Tikhonov’s method that uses nonlinear functionals that reflect the specific structure of the original problem. Furthermore, an algorithmic strategy for selecting the normative parameter was implemented, factoring in the a priori knowledge of the expected smoothness of the solution. This enabled the development of an efficient technique that adapts to diverse types of problems and provides stable performance even under challenging conditions. Numerous experiments were conducted on both synthetic and real datasets to verify the effectiveness of the method. The findings showed that the proposed approach considerably improves the decision accuracy and convergence rate compared to standard regulatory methods, even in the presence of strong noise in the data. The comparative analysis confirmed that the new method has advantages in terms of computational efficiency and ability to adapt to diverse types of kernels and functional settings. Furthermore, experimental results demonstrated a marked reduction of errors in the recovered functions as well as a stable convergence rate, even for high dimensional problems. The proposed scheme can automatically adapt to the different nature of noise and nonlinear distortion, which makes it a versatile tool for use in many applications that require high accuracy and efficiency in solving nonlinear integral equations.
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References
Ahues M, Dias d’Almeida F, Fernandes R, Vasconcelos PB. (2022). Two numerical approaches for nonlinear weakly singular integral equations. ArXiv (Cornell University), 1-25.
Al-Hawary T, Amourah A, Salah J, Yousef F. (2024). Two Inclusive Subfamilies of bi-univalent Functions. International Journal of Neutrosophic Science, 24(4), 315-323.
Alturk A, Coşgun T. (2019). The use of Lavrentiev regularization method in Fredholm integral equations of the first kind. International Journal of Advances in Applied Mathematics and Mechanics, 7(2), 70-79.
Alturk A. (2016). The regularization-homotopy method for the two-dimensional Fredholm integral equations of the first kind. Mathematical and Computational Applications, 21(2), 9.
Alybaev K, Murzabaeva A. (2018). Singularly perturbed first-order equations in complex domains that lose their uniqueness under degeneracy. AIP Conference Proceedings, 1997, 020076.
Amiraliyev GM, Durmaz ME, Kudu M. (2020). Fitted second order numerical method for a singularly perturbed Fredholm integro-differential equation. Bulletin of the Belgian Mathematical Society - Simon Stevin, 27(1), 71-88.
Amourah A, Al-Hawary T, Yousef F, Salah J. (2025). Collection of Bi-Univalent Functions Using Bell Distribution Associated With Jacobi Polynomials. International Journal of Neutrosophic Science, 25(1), 228-238.
Amourah A, Frasin BA, Salah J, Al-Hawary T. (2024). Fibonacci Numbers Related to Some Subclasses of Bi-Univalent Functions. International Journal of Mathematics and Mathematical Sciences, 2024, 8169496.
Asanov A, Orozmamatova J. (2019). About uniqueness of solutions of fredholm linear integral equations of the first kind in the axis. Filomat, 33(5), 1329-1333.
Ashirbaev BY, Yuldashev TK. (2024). Derivation of a Controllability Criteria for a Linear Singularly Perturbed Discrete System with Small Step. Lobachevskii Journal of Mathematics, 45(3), 938-948.
Avrunin OG, Tymkovych MY, Pavlov SV, Timchik SV, Kisała P, Orakbaev Y. (2015). Classification of CT-brain slices based on local histograms. Proceedings of SPIE - The International Society for Optical Engineering, 9816, 98161J.
Cherniha R, King JR, Kovalenko S. (2016). Lie symmetry properties of nonlinear reaction-diffusion equations with gradient-dependent diffusivity. Communications in Nonlinear Science and Numerical Simulation, 36, 98-108.
Cherniha R, Pliukhin O. (2013). New conditional symmetries and exact solutions of reaction-diffusion-convection equations with exponential nonlinearities. Journal of Mathematical Analysis and Applications, 403(1), 23-37.
Cherniha R, Serov M, Rassokha I. (2008). Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations. Journal of Mathematical Analysis and Applications, 342(2), 1363-1379.
Cherniha R, Serov M. (2006). Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, II. European Journal of Applied Mathematics, 17(5), 597-605.
Cimen E, Cakir M. (2021). A uniform numerical method for solving singularly perturbed Fredholm integro-differential problem. Computational and Applied Mathematics, 40, 42.
Crucinio FR, Doucet A, Johansen AM. (2021). A particle method for solving Fredholm equations of the first kind. Journal of the American Statistical Association, 118(542), 937-947.
De Micheli E, Magnoli N, Viano GA. (1998). On the regularization of Fredholm integral equations of the first kind. SIAM Journal on Mathematical Analysis, 29(4), 855-877.
Doll N. (2024). Orientation flow for skew-adjoint Fredholm operators with odd-dimensional kernel. Journal of Functional Analysis, 286(1), 110194.
Durmaz ME, Amiraliyev GM. (2021). A robust numerical method for a singularly perturbed Fredholm integro-differential equation. Mediterranean Journal of Mathematics, 18, 24.
Efendiev M. (2023). Linear and nonlinear non-Fredholm operators: Theory and applications. Singapore: Springer.
Frolov, O. (2022). Construction of canal surfaces based on a specified flat curvature line. Development Management, 21(3), 36-43.
Groetsch CW. (2007). Integral equations of the first kind, inverse problems and regularization: A crash course. Journal of Physics: Conference Series, 73, 012001.
Kal’chuk IV, Kravets VI, Hrabova UZ. (2020). Approximation of the classes WβrHα by three-harmonic Poisson integrals. Journal of Mathematical Sciences (United States), 246(1), 39-50.
Kerimkhulle S, Aitkozha Z. (2017). A criterion for correct solvability of a first order difference equation. AIP Conference Proceedings, 1880, 040016.
Kondratenko Y, Kondratenko V. (2014). Soft computing algorithm for arithmetic multiplication of fuzzy sets based on universal analytic models. Communications in Computer and Information Science, 469, 49-77.
Kress R. (1995). Integral equation methods in inverse obstacle scattering. Engineering Analysis with Boundary Elements, 15(2), 171-179.
Kryzhniy VV. (2023). Exponential relaxation data analysis by parametrized regularization of severely ill-posed Fredholm integral equations of the first kind. ArXiv (Cornell University), 1-10.
Lu F, Ou MJY. (2023). An adaptive RKHS regularization for Fredholm integral equations. Mathematical Methods in the Applied Sciences, 48(11), 11124-11140.
Manzhula V, Divak N, Melnik A. (2024). Structural identification method of nonlinear models of static systems based on interval data. Information Technologies and Computer Engineering, 21(1), 94-104.
Maripov A. (1994). Slitless and lensless rainbow holography. Journal of Optics, 25(4), 131-134.
Maripov AR, Ismanov Y. (1994). The Talbot effect (a self-imaging phenomenon) in holography. Journal of Optics, 25(1), 3-8.
Matoog RT, Mahdy AMS, Abdou MA, Mohamed DS. (2024). A computational method for solving nonlinear fractional integral equations. Fractal and Fractional, 8(11), 663.
Molabahrami A. (2013). An algorithm based on the regularization and integral mean value methods for the Fredholm integral equations of the first kind. Applied Mathematical Modelling, 37(23), 9634-9642.
Nabiei M, Yousefi SA. (2016). Newton type method for nonlinear Fredholm integral equations. ArXiv (Cornell University), 1-10.
Piskunov VG, Gorik AV, Cherednikov VN. (2000). Modeling of transverse shears of piecewise homogeneous composite bars using an iterative process with account of tangential loads 2. Resolving equations and results. Mechanics of Composite Materials, 36(6), 445-452.
Qiu R, Xu M, Qu W. (2024). Minimal-norm solution to the Fredholm integral equations of the first kind via the H-HK formulation. In: 14th International Conference on Information Technology in Medicine and Education (ITME). Guiyang: Institute of Electrical and Electronics Engineers.
Radi A, Elgasim Msis MEA. (2023). The numerical methods for solving nonlinear integral equations. IJRDO - Journal of Mathematics, 9(3), 1-12.
Rahimi MY, Shahmorad S, Talati F, Tari A. (2010). An operational method for the numerical solution of two-dimensional linear Fredholm integral equations with an error estimation. Bulletin of the Iranian Mathematical Society, 36(2), 119-132.
Rasekh M, Fakhri N. (2023). The use of homotopy regularization method for linear and nonlinear Fredholm integral equations of the first kind. Journal of Mathematics and Statistics Studies, 4(1), 19-25.
Saadabaev A, Usenov I. (2023). Regularization of the solution of a nonlinear integral equation of the first kind of Fredholm type in the space of continuous functions. Bulletin of Osh State University, 1(2), 187-193.
Salah J. (2016). Note on the modified caputo’s fractional calculus derivative operator. Far East Journal of Mathematical Sciences, 100(4), 609-615.
Salah J. (2024). On Uniformly Starlike Functions with Respect to Symmetrical Points Involving the Mittag-Leffler Function and the Lambert Series. Symmetry, 16(5), 580.
Srazhidinov A, Abdraeva N. (2023). Regularization of convolutional Volterra integral equations of the first kind. Bulletin of Osh State University, 4, 97-105.
Vaneeva O, Kuriksha O, Sophocleous C. (2015). Enhanced group classification of Gardner equations with time-dependent coefficients. Communications in Nonlinear Science and Numerical Simulation, 22(1-3), 1243-1251.
Voronin, A., Lebedeva, I., & Lebedev, S. (2022). A nonlinear mathematical model of dynamics of production and economic objects. Development Management, 21(2), 8-15.
Vovchok I. (2024). Mathematical models of individualised learning based on decision theory. Information Technologies and Computer Engineering, 21(3), 96-107.
Wazwaz AM. (2011). Nonlinear Fredholm integral equations. In: A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications. Heidelberg: Springer.
Wazwaz AM. (2011). The regularization method for Fredholm integral equations of the first kind. Computers & Mathematics with Applications, 61(10), 2981-2986.
Yahya K, Biazar J, Azari H, Fard PR. (2010). Homotopy perturbation method for image restoration and denoising. ArXiv (Cornell University), 1-5.
Yuan D, Zhang X. (2019). An overview of numerical methods for the first kind Fredholm integral equations. SN Applied Sciences, 1, 1228.
Yuldashev TK, Eshkuvatov ZK, Nik Long NMA. (2022). Nonlinear the first kind Fredholm integro-differential first-order equation with degenerate kernel and nonlinear maxima. Mathematical Modeling and Computing, 9(1), 74-82.
Yuldashev TK, Saburov KK. (2021). On Fredholm integral equations of the first kind with nonlinear deviation. Azerbaijan Journal of Mathematics, 11(2), 137-152.
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