Volume 26 , Issue 3 , PP: 259-272, 2025 | Cite this article as | XML | Html | PDF | Full Length Article
Hamzeh Zureigat 1 *
Doi: https://doi.org/10.54216/IJNS.260318
In this paper, two numerical methods that are method of successive approximations and Fredholm’s first fundamental theorem are developed, reformatted, and applied to solve fuzzy second kind Fredholm integral equations with a separable kernel. The fuzziness in the equations is represented utilizing convex normalized triangular fuzzy numbers, which are based on a single and double parametric form of fuzzy numbers. A comparative analysis study between the proposed schemes are discussed through numerical experiment. It was found that Fredholm's first fundamental theorem is more efficient and effective than method of successive approximations. Furthermore, the double parametric form of fuzzy number is a general and more reliable than single parametric form since it reduced the computational cost and provides more certain fuzzy cases.
Fuzzy Fredholm integral equations , Successive approximations , Fredholm&rsquo , s first fundamental method , Numerical methods
[1] L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, vol. 1, no. 1, pp. 3-28, 1978.
[2] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965.
[3] O. S. Fard and M. Sanchooli, “Two successive schemes for numerical solution of linear fuzzy Fredholm integral equations of the second kind,” Australian Journal of Basic and Applied Sciences, vol. 4, no. 5, pp. 817-825, 2010.
[4] A. Ahmad et al., “A review on therapeutic potential of Nigella sativa: A miracle herb,” Asian Pacific Journal of Tropical Biomedicine, vol. 3, no. 5, pp. 337-352, 2013.
[5] M. Almutairi, H. Zureigat, A. Izani Ismail, and A. Fareed Jameel, “Fuzzy numerical solution via finite difference scheme of wave equation in double parametrical fuzzy number form,” Mathematics, vol. 9, no. 6, p. 667, 2021.
[6] A. E. Ghazouani, M. H. Elomari, and S. Melliani, “A semi-analytical solution approach for fuzzy fractional acoustic waves equations using the Atangana Baleanu Caputo fractional operator,” Soft Computing, vol. 28, no. 17, pp. 9307-9315, 2024.
[7] S. A. Altaie, “Numerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials,” Journal of Al-Nahrain University, vol. 15, no. 1, pp. 133-139, 2012.
[8] H. Zureigat et al., “Numerical Solution for Fuzzy Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells,” International Journal of Environmental Research and Public Health, vol. 20, no. 4, p. 3766, 2023.
[9] M. S. Y. Amawi, “Fuzzy Fredholm Integral Equation of the Second Kind,” Master’s thesis, An-Najah National University, Faculty of Graduate Studies, Nablus, Palestine, 2014.
[10] A. Fallatah, M. O. Massa'deh, and A. U. Alkouri, “Normal and Cosets of (š¾,)-Fuzzy Hx-Subgroups,” Journal of Applied Mathematics & Informatics, vol. 40, no. 3-4, pp. 719-727, 2022.
[11] S. Bodjanova, “Median alpha-levels of a fuzzy number,” Fuzzy Sets and Systems, vol. 157, no. 7, pp. 879-891, 2006.
[12] J. J. Buckley and T. Feuring, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 110, no. 1, pp. 43-54, 2000.
[13] S. Chakraverty and S. Tapaswini, “Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations,” Chinese Physics B, vol. 23, no. 12, p. 120202, 2014.
[14] J. Dong, G. Wan, and Z. Liang, “Accumulation of salicylic acid-induced phenolic compounds and raised activities of secondary metabolic and antioxidative enzymes in Salvia,” (details missing).
[15] D. Jawad Hashim et al., “New Series Approach Implementation for Solving Fuzzy Fractional TwoāPoint Boundary Value Problems Applications,” Mathematical Problems in Engineering, vol. 2022, p. 7666571, 2022.
[16] H. H. Fadravi, R. Buzhabadi, and H. S. Nik, “Solving linear Fredholm fuzzy integral equations of the second kind by artificial neural networks,” Alexandria Engineering Journal, vol. 53, no. 1, pp. 249-257, 2014.
[17] D. Dubois and H. Prade, “Towards fuzzy differential calculus part 1: Integration of fuzzy mappings,” Fuzzy Sets and Systems, vol. 8, no. 1, pp. 1-17, 1982.
