Proceedings of International Conference on Applied Innovation in IT
2025/06/27, Volume 13, Issue 2, pp.289-295
A Novel Numerical Approach for Solving Initial Value Problems in Heat Equations Using Variational Regularization and Intelligent Particle Swarm Optimization
H.K. Al-Mahdawi, Kilan M. Hussien, Ghassan K. Ali, Fadhil Kadhem Zaidan, Ali Subhi Alhumaima and Hussein Alkattan Abstract: In this article, we use the variational approach as a regularization tool to solve the initial value problem which appear in a heat partial differential equation. Although the temperature which obtained in time t=T>0 is known, the initial temperature distribution remains unknown. By using the separation of variables method, the partial differential equation is transformed into a Fredholm integral equation of the first kind. We then apply a discretization algorithm to reduce the integral equation to a system of linear algebraic equations, commonly referred to as an inverse linear operator problem. The variational regularization method is employed to obtain a regularized solution. We also present a fundamental analysis of this method for solving inverse problems. Furthermore, we describe the application of the Intelligent Particle Swarm Optimization (IPSO) technique to determine the optimal regularization parameter. Our results demonstrate that integrating particle swarm optimization with variational optimization is both effective and computationally feasible.
Keywords: Intelligent Particle Swarm Optimization, Regularization Parameter, Variational Regularization, Heat Equation.
DOI: 10.25673/120448
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References:
- F. John, “Numerical solution of problems which are not well posed in the sense of Hadamard,” in Fritz John, Springer, 1985, pp. 411–424.
- G. Huang, Y. Liu, and F. Yin, “Tikhonov regularization with MTRSVD method for solving large-scale discrete ill-posed problems,” J. Comput. Appl. Math., vol. 405, p. 113969, 2022.
- S. I. Kabanikhin, “Inverse and ill-posed problems,” Sib. Elektron. Mat. Izv., vol. 7, 2010.
- A. I. Sidikova, “A study of an inverse boundary value problem for the heat conduction equation,” Numer. Anal. Appl., vol. 12, no. 1, pp. 70–86, 2019, doi: 10.1134/S1995423919010063.
- A. I. Sidikova and H. K. Al-Mahdawi, “The solution of inverse boundary problem for the heat exchange for the hollow cylinder,” in AIP Conference Proceedings, 2022, vol. 2398, no. 1, p. 060050.
- V. P. Tanana and A. I. Sidikova, “Solving the inverse heat conduction boundary problem for composite materials,” in Inverse Problems and Applications, Springer, 2020, pp. 133–143, doi: 10.1007/978-3-030-42831-0_13.
- H. Montegranario, “Inverse problems and variational regularization,” in An Introduction to Variational Calculus: Applications in Image Processing, Springer, 2024, pp. 89–105.
- F. Margotti and J. Rabelo, “Tikhonov-like methods with inexact minimization for solving linear ill-posed problems,” Inverse Probl., vol. 36, no. 12, p. 125013, 2020.
- B. Hofmann and P. Mathé, “A priori parameter choice in Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems,” in Inverse Problems and Related Topics: Shanghai, China, October 12–14, 2018, Springer, 2020, pp. 169–176.
- V. P. Tanana, “The classification of ill-posed problems and optimal methods for their solution,” Izv. Vyss. Uchebnykh Zaved. Mat., no. 11, pp. 106–112, 1977.
- A. N. Tikhonov and V. B. Glasko, “The approximate solution of Fredholm integral equations of the first kind,” USSR Comput. Math. Math. Phys., vol. 4, no. 3, pp. 236–247, 1964.
- L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math., vol. 73, no. 3, pp. 615–624, 1951, doi: 10.2307/2372313.
- S. A. Noaman, H. K. Al-Mahdawi, B. T. Al-Nuaimi, and A. I. Sidikova, “Iterative method for solving linear operator equation of the first kind,” MethodsX, vol. 10, p. 102210, 2023.
- D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. ACM, vol. 9, no. 1, pp. 84–97, 1962.
- R. Zhang, F. Xie, and X. Luo, “A discretizing Tikhonov regularization method via modified parameter choice rules,” J. Inverse Ill-posed Probl., 2024.
- S. Dziri, S. Bouallègue, M. Mazen Alhato, and P. Siarry, “Artificial bee colony optimization–based enhancement of output power generation in grid-connected photovoltaic systems,” Trans. Inst. Meas. Control, 2025, doi: 10.1177/01423312241287755.
- J. Li and H. Cheng, “Determine source of Cahn–Hilliard equation with time-fractional derivative by iterative variational regularization method,” Math. Methods Appl. Sci., vol. 47, no. 4, pp. 2434–2451, 2024.
- H. K. Al-Mahdawi, F. H. Khorsheed, A. S. Alhumaima, A. J. Ramadhan, K. M. Hussien, and H. Alkattan, “Intelligent particle swarm optimization method for parameter selecting in regularization method for integral equation,” in BIO Web of Conferences, 2024, vol. 97, p. 39.
- H. K. Al-Mahdawi and A. S. Alhumaima, “Particle swarm optimization method for parameter selecting in Tikhonov regularization method for solving inverse problems,” e-Prime–Advances Electr. Eng. Electron. Energy, vol. 10, p. 100750, 2024.
- A. M. Gavgani and Y. S. Dogrusoz, “Use of genetic algorithm for selection of regularization parameters in multiple constraint inverse ECG problem,” in Proc. 2011 Annual Int. Conf. IEEE Eng. Med. Biol. Soc., 2011, pp. 985–988.
- J. L. G. Pallero et al., “Particle swarm optimization and uncertainty assessment in inverse problems,” Entropy, vol. 20, no. 2, p. 96, 2018.
- I. Mishkhal, S. A. A. L. Kareem, H. H. Saleh, A. Alqayyar, I. Hussein, and I. A. Jassim, “Solving course timetabling problem based on the edge coloring methodology by using Jedite,” in Proc. 2019 1st AL-Noor Int. Conf. Sci. Technol. (NICST), 2019.
- M. Hajihassani, D. J. Armaghani, and R. Kalatehjari, “Applications of particle swarm optimization in geotechnical engineering: A comprehensive review,” Geotech. Geol. Eng., vol. 36, pp. 705–722, 2018.
- O. U. Rehman, S. U. Rehman, S. Tu, S. Khan, M. Waqas, and S. Yang, “A quantum particle swarm optimization method with fitness selection methodology for electromagnetic inverse problems,” IEEE Access, vol. 6, pp. 63155–63163, 2018.
- G. V. Alekseev and D. A. Tereshko, “Particle swarm optimization-based algorithms for solving inverse problems of designing thermal cloaking and shielding devices,” Int. J. Heat Mass Transf., vol. 135, pp. 1269–1277, 2019.
- A. Cintas-Canto, J. Kaur, M. Mozaffari-Kermani, and R. Azarderakhsh, “ChatGPT vs. lightweight security: First work implementing the NIST cryptographic standard ASCON,” arXiv preprint arXiv:2306.08178, 2023.
- A. C. Canto, J. Kaur, M. M. Kermani, and R. Azarderakhsh, “Algorithmic security is insufficient: A comprehensive survey on implementation attacks haunting post-quantum security,” arXiv preprint arXiv:2305.13544, 2023.
- K. Hameed, “A state space approach via discrete Hartley transform for the solution of the state model of linear time-invariant systems,” IJApSc, vol. 2, no. 1, pp. 53–60, Mar. 2025, doi: 10.69923/k1qp1d65.
- L. A. Ogundele, F. E. Ayo, A. M. Adeleye, and S. O. Hassan, “A hybrid network intrusion detection framework using neural network-based decision tree model,” IJApSc, vol. 2, no. 1, pp. 74–93, Mar. 2025, doi: 10.69923/95zt9v71.
- I. Mishkhal, N. Abdullah, H. H. Saleh, N. I. R. Ruhaiyem, and F. H. Hassan, “Facial swap detection based on deep learning: Comprehensive analysis and evaluation,” Iraqi J. Comput. Sci. Math., vol. 6, no. 1, pp. 109–123, 2025.
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