Mathematics and Computational Sciences

Numerical solution of a nonlinear integral equation for the determination of the unknown time-dependent diffusivity of radioactive materials

Document Type : Original Article

Authors

Bahman Babayar-Razlighi 1
Mohammad Babayar-Razlighi 2

1 Department of Mathematics, Faculty of science, Qom University of Technology, Qom, Iran

2 Faculty of Mechanical Engineering, K.N. Toosi University of Technology, P. O. Box 19395-1999, Tehran, Iran.

https://doi.org/10.30511/mcs.2026.2067885.1430
Abstract
The thermal diffusion coefficient in radioactive materials is not a constant value, and this makes the heat transfer problem different in such materials and materials that are not homogeneous and have been destroyed or are disintegrating in some way. In the problem under discussion, the heat diffusion coefficient is time-dependent and satisfies a nonlinear integral equation. The existence and uniqueness of the solution to the integral equation in question are discussed in detail in Chapter 13 of the Book "Encyclopedia of the One-Dimensional Heat Equation" by Cannon, J. R. The integral equation in question is not a standard Volterra integral equation and therefore has not been studied much from a numerical perspective. For example, if we apply the fixed point method, which is a powerful tool in the analysis of existence and uniqueness, discussed in Chapter 13 of the aforementioned Book, to a numerical solution, we cannot go even one step forward with this method. Since the unknown function is located at the kernel of a nested integral, applying canonical methods becomes difficult. Therefore, in this paper, we have discussed a hybrid method of numerical integration and iterative methods that solves the problem with sufficient accuracy. In Section 5, we have extracted several sample problems using the properties of the heat equation in the case where the thermal diffusivity is Time-dependent. The numerical solution of these sample problems in the Section 6 demonstrates the efficiency and accuracy of the proposed method.

Keywords

Nonlinear integral equation
Heat equation
Numerical solution
Time-dependent thermal diffusivity

Subjects

[1] Barman, K. J., & Mishra, A. (2024). Analysis of metro network by applying graph theoretical notions.
[2] Begum, F. (2024). Colorability, chromatic number and its applications on graph theory. Anusandhanvallari, 1917–1930. 
[3] Biedl, T., Bose, P., & Murali, K. (2024). A parameterized algorithm for vertex and edge connectivity of embedded graphs. In Proceedings of the 32nd Annual European Symposium on Algorithms (ESA 2024) (pp. 24–1). Schloss Dagstuhl–Leibniz-Zentrum für Informatik. 
[4] Biondini, F., Ballio, F., di Prisco, M., Bianchi, S., DAngelo, M., Zani, G., ... Ferreira, K. F. (2022). Bridge vulnerability and hazard assessment for risk-based infrastructure management. In Bridge Safety, Maintenance, Management, Life-Cycle, Resilience and Sustainability (pp. 1864–1873). CRC Press.
[5] Conlon, D., & Lee, J. (2024). Domination inequalities and dominating graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 177(1), 167–184.
[6] De Silva, D., Gallo, M., De Falco, L., & Nigro, E. (2023). Fire risk assessment of bridges: From state of the art to structural vulnerability mitigation. Journal of Civil Structural Health Monitoring, 13(8), 1613–1632.
[7] Goranci, G., Henzinger, M., Nanongkai, D., Saranurak, T., Thorup, M., & Wulff-Nilsen, C. (2023). Fully dynamic exact edge connectivity in sublinear time. In Proceedings of the 2023 Annual ACM–SIAM Symposium on Discrete Algorithms (SODA) (pp. 70–86). Society for Industrial and Applied Mathematics.
[8] Jafari, H., Salehfard, S., & Aqababaei Dehkordi, D. (2024). Introducing a simple method for detecting the path between two different vertices in graphs. Mathematics and Computational Sciences, 4(4), 10–16.
[9] Karuppiah, K., Velusamy, S., Banu, M. N., Sunantha, D., Jenitha, G., & Brindha, R. (2025). Topological data analysis (TDA) as a framework for understanding deep learning behavior. In Proceedings of the IEEE 5th International Conference on ICT in Business Industry & Government (ICTBIG) (pp. 1–7). IEEE. 
[10] Kumar, J. S., Archana, B., Muralidharan, K., & Srija, R. (2025). Spectral graph theory: Eigenvalues, Laplacians, and graph connectivity. Metallurgical and Materials Engineering, 31(3), 78–84.
[11] Kungumaraj, E., Thivyarathi, R. C., Kenneth, C. R., Ramesh, R., Anand, M. C. J., & Naganathan, S. (2024). Efficiency enhancement in heptagonal fuzzy transportation problems. In Proceedings of the International Conference on Intelligent and Fuzzy Systems (pp. 578–585). Springer Nature Switzerland. 
[12] Kungumaraj, E., Clement Joe Anand, M., Saikia, U., Dabass, V., Ranjitha, B., & Tiwari, M. (2024). Topologized graphical method in solving fuzzy transportation problem with computational techniques. In Proceedings of the International Conference on Intelligent and Fuzzy Systems (pp. 513–521). Springer Nature Switzerland.
[13] Meybodi, M. A., Safari, M., & Davoodijam, E. (2024). Domination-based graph neural networks. International Journal of Computers and Applications, 46(11), 998–1005. 
[14] Pasham, S. D. (2023). Network topology optimization in cloud systems using advanced graph coloring algorithms. The Metascience, 1(1), 122–148. 
[15] Paulraj Jayasimman, I., Jenitha, G., & Kumaravel, A. (2018). The energy graph for minimum majority domination. International Journal of Engineering & Technology, 7(4.10), 562–565. 
[16] Raza, A., & Munir, M. M. (2025). Laplacian spectra and structural insights: Applications in chemistry and network science. Frontiers in Applied Mathematics and Statistics, 11, 1519577.
Mathematics and Computational Sciences
Volume 7, Issue 1
Winter 2026
Pages 162-176

  • Receive Date 07 August 2025
  • Revise Date 06 December 2025
  • Accept Date 02 January 2026

Home

Submit Manuscript

Reviewers

Contact Us