Article ID: MTJPAM-D-24-00121

Title: Analytic methods to solve Fredholm integral equations

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-24-00121; Volume 7 / Issue 2 / Year 2025, Pages 1-19

Document Type: Research Paper

Author(s): ‪Raghad Sahib Shamsah ^a, Shilpi Jain ^b, Praveen Agarwal ^c, Shaher Momani ^d

^aDepartment of Petroleum, Faculty of Engineering, University of Kerbala, P.O.Box 1125, Freaha, Karbala, Iraq

^bDepartment of Mathematics, Poornima College of Engineering, Jaipur-302022, India

^cNonlinear Dynamics Research Centre (NDRC), Ajman University, Ajman, UAE – Anand International College of Engineering, Jaipur-303012, India

^dNonlinear Dynamics Research Centre (NDRC), Ajman University, Ajman, UAE – Department of Mathematics, The University of Jordan, Amman, Jordan

Received: 3 August 2024, Accepted: 27 April 2025, Published: 1 November 2025

Corresponding Author: Raghad Sahib Shamsah (Email address: raghad.sahib@uokerbala.edu.iq)

Full Text: PDF

---

Abstract

This manuscript presents an innovative semi-analytical approach for solving linear Fredholm integral equations (FIEs) of the first as well as second kinds. Utilizing the properties of Fourier and Mellin transformations, we derive analytical solutions that substantially improve the comprehension and resolution of these equations. A key innovation of our approach is the ability to effectively manage non-smooth kernels through the degeneration of kernel functions, facilitating their separation and simplification. Empirical examples illustrate the method’s effectiveness, demonstrating superior numerical stability and convergence rates compared to existing techniques. This work not only fills a critical gap in the literature but also provides a robust framework for future research in integral equations, paving the way for advancements in various scientific and engineering applications.

Keywords: Integral equation, Fredholm equations, Fourier transform, Mellin transform, kernel function, convolution theory

References:

1. S. Abbasbandy, Numerical solutions of the integral equations: Homotopy perturbation method and Adomian’s decomposition method, Appl. Math. Comput. 173, 493–500, 2006.
2. N. Aliev and S. M. Hosseini, A regularization of Fredholm type singular integral equations, Int. J. Math. Math. Sci. 26 (2), 123–128, 2001.
3. N. Aliev and S. M. Hosseini, Multidimensional singular Fredholm integral equations in a finite domain and their regularization, Southeast Asian Bull. Math. 27, 395–408, 2003.
4. Y. E. Amikohove, Multidimensional first kind integral equations; illposed problems of mathematical physics and analysis, Novosibirsk, Russian, 1984.
5. A. Arikoglu and I. Ozkol, Solutions of integral and integro-differential equation systems by using differential transform method, Comput. Math. Appl. 56, 2411–2417, 2008.
6. E. Babolian, J. Biazar and A. R. Vahidi, The decomposition method applied to systems of Fredholm integral equations of the second kind, Appl. Math. Comput. 148, 443–452, 2004.
7. J. Biazar and H. Ghazvini, Numerical solution for special nonlinear Fredholm integral equation by HPM, Appl. Math. Comput. 195, 681–687, 2008.
8. P. Bräuer, M. Fassler and M. Jaroniec, Numerical solutions of the adsorption integral equation utilizing the spline functions, Thin Solid Films 123, 245–272, 1985.
9. G. Germano, C. E. Phelan, D. Marazzina and G.Fusai, Solution of Wiener-Hopf and Fredholm integral equations by fast Hilbert and Fourier transforms, 2021; ArXiv: 2106.05326.
10. A. Golbabai and S. Seifollahi, Numerical solution of the second kind integral equations using radial basis function networks, Appl. Math. Comput. 174, 877–883, 2006.
11. A. Golbabai, M. Mammadov and S. Seifollahi, Solving a system of nonlinear integral equations by an RBF network, Comput. Math. Appl. 57, 1651–1658, 2009.
12. A. Gorguis, Charpit and Adomian for solving integral equations, Appl. Math. Comput. 193, 446–454, 2007.
13. M. Inoue, M. Kuroumaru and S. Yamaguchi, A solution of Fredholm integral equation by means of the spline fit approximation, Comput. Fluids 7, 33–46, 1979.
14. A. Jeffrey and H. H. Dai, Fourier and Mellin transforms: Handbook of mathematical functions with formulas, graphs and mathematical tables, CRC Press, 2019.
15. A. Karoui, Wavelets: Properties and approximate solution of a second kind integral equation, Comput. Math. Appl. 46, 263–277, 2003.
16. X. Z. Liang, M. C. Liu and X. J. Che, Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets, J. Comput. Appl. Math. 136, 149–161, 2001.
17. W. V. Lovitt, Linear integral equations, Dover Publications Inc., New York, 1924.
18. K. Maleknejad and H. Derili, Numerical solution of Hammerstein integral equations by using combination of spline-collocation method and Lagrange interpolation, Appl. Math. Comput. 190, 1557–1562, 2007.
19. K. Maleknejad and M. Tavassoli Kajani, Solving second kind integral equations by Galerkin methods with hybrid Legendre and Block-Pulse functions, Appl. Math. Comput. 145, 623–629, 2003.
20. K. Maleknejad, N. Aghazadeh and R. Mollapourasl, Numerical solution of Fredholm integral equation of the first kind with collocation method and estimation of error bound, Appl. Math. Comput. 179, 352–359, 2006.
21. S. G. Mikhlin, Linear integral equations, Hindustan Pub. Corp., Delhi, 1960.
22. J. S. Nadjafi and A. Ghorbani, He’s homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Comput. Math. Appl. 58, 2379–2390, 2009.
23. R. Piessens, Computing integral transforms and solving integral equations using Chebyshev polynomial approximations, J. Comput. Appl. Math. 121, 113–124, 2000.
24. D. Porter and D. S. G. Stirling, Integral equations: A practical treatment, from spectral theory to applications, Cambridge University Press, Cambridge, 1990.
25. L. Pretorius and D. Eyre, Spline-Gauss rules and the Nyström method for solving integral equations in quantum scattering, J. Comput. Appl. Math. 18, 235–247, 1987.
26. J. Rashidinia and M. Zarebnia, Numerical solution of linear integral equations by using Sinc-collocation method, Appl. Math. Comput. 168, 806–822, 2005.
27. R. Saadati, M. Dehghan, S. M. Vaezpour and M. Saravi, The convergence of He’s variational iteration method for solving integral equations, Comput. Math. Appl. 58, 2167–2171, 2009.
28. E. M. Stein and R. Shakarchi, Fourier analysis: An introduction, Princeton University Press, Princeton, New Jersey, 2003.
29. C. J. Tranter, Integral transforms in mathematical physics, John Wiley, New York, 1951.
30. R. J. Vaccaro, Convolution and Fourier transforms for communications engineers, Artech House, 2014.
31. G. Vainikko, A. Kivinukk and J. Lippus, Fast solvers of integral equations of the second kind: wavelet methods, J. Complexity 21, 243–273, 2005.
32. A. Wazwaz, A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equations, Appl. Math. Comput. 181 (2), 1703–1712, 2006.
33. J. Y. Xiao, L. H. Wen and D. Zhang, Solving second kind Fredholm integral equations by periodic wavelet Galerkin method, Appl. Math. Comput. 175, 508–518, 2006.

---

Cite this article

---