Intuitionistic Fuzzy Differential Equations and its Applications: A Review

Authors

  • Soheil Salahshour * Departmant of Engineering and Natural Sciences, Istanbul Okan University, Istanbul, Turkey. https://orcid.org/0000-0002-9437-8797
  • Sankar Prasad Mondal Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, 741249, India.

https://doi.org/10.48314/jidcm.v1i1.62

Abstract

This paper presents a systematic brief review of the topic Intuitionistic Fuzzy Differential Equation (IFDEs) and its applications, which an extension of fuzzy differential equations. The fundamental ideas, mathematical constructions, and solution tactics of IFDEs are drawn. Various existed procedures for handling uncertainty or impreciseness in dynamic systems using IFDEs are deliberated. The paper also mentions recent progressions and key applications in economics engineering, and decision sciences models associated with IFDE. Future research guidelines and existing challenges are briefly addressed lastly.

Keywords:

Fuzzy set, Intuitionistic fuzzy sets, Fuzzy differential equation

References

  1. [1] Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
  2. [2] Zadeh, L. A., Klir, G. J., & Yuan, B. (1996). Fuzzy sets, fuzzy logic, and fuzzy systems: Selected papers (Vol. 6). World Scientific. https://B2n.ir/hq5948
  3. [3] Zadeh, L. A. (1969). Toward a theory of fuzzy systems. https://ntrs.nasa.gov/api/citations/19690027655/downloads/19690027655.pdf
  4. [4] Zadeh, L. A. (1972). A fuzzy-set-theoretic interpretation of linguistic hedges.
  5. [5] Journal of cybernetics, 2(3), 4–34. https://doi.org/10.1080/01969727208542910
  6. [6] Zadeh, L. A. (1971). Quantitative fuzzy semantics. Information sciences, 3(2), 159–176. https://doi.org/10.1016/S0020-0255(71)80004-X
  7. [7] Singh, P., Gazi, K. H., Rahaman, M., Basuri, T., & Mondal, S. P. (2024). Solution strategy and associated results for fuzzy mellin transformation. Franklin open, 7, 100112. https://doi.org/10.1016/j.fraope.2024.100112
  8. [8] Pamucar, D., & Djorovic, B. (2012). Fuzzy logic applied to organizational design of the administrative management. Metalurgia international, 17(5), 87. https://www.proquest.com/openview/c4b08b4bccb15d7845ab07b90d774dc9/1?pq-origsite=gscholar&cbl=886383
  9. [9] Alamin, A., Biswas, A., Gazi, K. H., & Sankar, S. P. M. (2024). Modelling with neutrosophic fuzzy sets for financial applications in discrete system. Spectrum of engineering and management sciences, 2(1), 263–280. https://sems-journal.org/index.php/sems/article/download/33/36
  10. [10] Singh, P., Gor, B., Gazi, K. H., Mukherjee, S., Mahata, A., & Mondal, S. P. (2023). Analysis and interpretation of Malaria disease model in crisp and fuzzy environment. Results in control and optimization, 12, 100257. https://doi.org/10.1016/j.rico.2023.100257
  11. [11] Gazi, K. H., Biswas, A., Singh, P., Rahaman, M., Maity, S., Mahata, A., & Mondal, S. P. (2025). A comprehensive literature review of fuzzy differential equations with applications. Journal of fuzzy extension and applications, 6(1), 134–161. https://doi.org/10.22105/jfea.2024.449970.1426
  12. [12] Nguyen, A. T., Taniguchi, T., Eciolaza, L., Campos, V., Palhares, R., & Sugeno, M. (2019). Fuzzy control systems: Past, present and future. IEEE computational intelligence magazine, 14(1), 56–68. https://doi.org/10.1109/MCI.2018.2881644
  13. [13] Luhandjula, M. K. (1989). Fuzzy optimization: An appraisal. Fuzzy sets and systems, 30(3), 257–282. https://doi.org/10.1016/0165-0114(89)90019-5
  14. [14] Archana, R., & Jeevaraj, P. S. E. (2024). Deep learning models for digital image processing: A review. Artificial intelligence review, 57(1), 11. https://doi.org/10.1007/s10462-023-10631-z
  15. [15] Momena, A. F., Gazi, K. H., Rahaman, M., Sobczak, A., Salahshour, S., Mondal, S. P., & Ghosh, A. (2024). Ranking and challenges of supply chain companies using mcdm methodology. Logistics, 8(3), 87. https://doi.org/10.3390/logistics8030087
  16. [16] Qin, J., Zeng, M., Wei, X., & Pedrycz, W. (2024). Ranking products through online reviews: A novel data-driven method based on interval type-2 fuzzy sets and sentiment analysis. Journal of the operational research society, 75(5), 860–873. https://doi.org/10.1080/01605682.2023.2215823
  17. [17] Arifuddin, R., Dirgantara, W., Sumarahinsih, A., Hafsari, R. P. I., Maulana, F. I., Nugraha, A. T., & Sobhita, R. A. (2024). Baby room temperature and humidity control system using fuzzy logic. Emitor: Jurnal teknik elektro, 24(3), 275–280. https://doi.org/10.23917/emitor.v24i3.6403
  18. [18] Shukur, F., Mosa, S. J., & Raheem, K. M. H. (2024). Optimization of fuzzy-PD control for a 3-DOF robotics manipulator using a back-propagation neural network. Mathematical modelling of engineering problems, 11(1), 199. 10.18280/mmep.110122
  19. [19] Momena, A. F., Mandal, S., Gazi, K. H., Giri, B. C., & Mondal, S. P. (2023). Prediagnosis of disease based on symptoms by generalized dual hesitant hexagonal fuzzy multi-criteria decision-making techniques. Systems, 11(5), 231. https://doi.org/10.3390/systems11050231
  20. [20] Gazi, K. H., Mondal, S. P., Chatterjee, B., Ghorui, N., Ghosh, A., & De, D. (2023). A new synergistic strategy for ranking restaurant locations: A decision-making approach based on the hexagonal fuzzy numbers. RAIRO-operations research, 57(2), 571–608. https://doi.org/10.1051/ro/2023025
  21. [21] Li, M., Zhang, X., Jiang, H., & Liu, J. (2025). Some novel fuzzy logic operators with applications in Fuzzy neural networks. Information sciences, 702, 121897. https://doi.org/10.1016/j.ins.2025.121897
  22. [22] Mayengo, M. M., Kgosimore, M., & Chakraverty, S. (2020). Fuzzy modeling for the dynamics of alcohol-related health risks with changing behaviors via cultural beliefs. Journal of applied mathematics, 2020(1), 8470681. https://doi.org/10.1155/2020/8470681
  23. [23] Rahaman, M., Gazi, K. H., Rabih, M., Alamin, A., Razzaq, O. A., Khan, N. A., Alam, S. (2025). Solutions of linear homogeneous fuzzy fractional differential equations using the mittag-leffler function. Journal of uncertain systems, 2550013. https://doi.org/10.1142/S1752890925500138
  24. [24] Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy sets and systems, 20(1), 87–96. http://dx.doi.org/10.1016/S0165-0114(86)80034-3
  25. [25] Alamin, A., Rahaman, M., Gazi, K. H., Alam, S., & Mondal, P. (2026). Solution and analysis of coupled homogeneous linear intuitionistic fuzzy difference equation. Transactions on fuzzy sets and systems, 9(1), 1. https://doi.org/10.71602/tfss.2026.1191802
  26. [26] Mandal, S., Gazi, K. H., Salahshour, S., Mondal, S. P., Bhattacharya, P., & Saha, A. K. (2024). Application of interval valued intuitionistic fuzzy uncertain mcdm methodology for ph.d supervisor selection problem. Results in control and optimization, 15, 100411. https://doi.org/10.1016/j.rico.2024.100411
  27. [27] Zeng, W. Y., Cui, H. S., Liu, Y. Q., Yin, Q., & Xu, Z. S. (2022). Novel distance measure between intuitionistic fuzzy sets and its application in pattern recognition. Iranian journal of fuzzy systems, 19(3), 127–137. https://ijfs.usb.ac.ir/article_6947_95418044d4226792a36e9f0b8122ce45.pdf
  28. [28] Tao, R., Liu, Z., Cai, R., & Cheong, K. H. (2021). A dynamic group MCDM model with intuitionistic fuzzy set: Perspective of alternative queuing method. Information sciences, 555, 85–103. https://doi.org/10.1016/j.ins.2020.12.033
  29. [29] Rasoulzadeh, M., Edalatpanah, S. A., Fallah, M., & Najafi, S. E. (2024). A hybrid model for choosing the optimal stock portfolio under intuitionistic fuzzy sets. Iranian journal of fuzzy systems, 21(2), 161–179. https://ijfs.usb.ac.ir/article_8340_581a8b63a9848686d8f52e898b17193d.pdf
  30. [30] Yazdi, M., Kabir, S., Kumar, M., Ghafir, I., & Islam, F. (2023). Reliability analysis of process systems using intuitionistic fuzzy set theory. In Advances in reliability, failure and risk analysis (pp. 215–250). Springer. https://doi.org/10.1007/978-981-19-9909-3_10
  31. [31] Vlachos, I. K., & Sergiadis, G. D. (2007). Intuitionistic fuzzy information--applications to pattern recognition. Pattern recognition letters, 28(2), 197–206. https://doi.org/10.1016/j.patrec.2006.07.004
  32. [32] Akram, M., Ashraf, A., & Sarwar, M. (2014). Novel applications of intuitionistic fuzzy digraphs in decision support systems. The scientific world journal, 2014(1), 904606. https://doi.org/10.1155/2014/904606
  33. [33] Das, S., Guha, D., & Dutta, B. (2016). Medical diagnosis with the aid of using fuzzy logic and intuitionistic fuzzy logic. Applied intelligence, 45, 850–867. https://doi.org/10.1007/s10489-016-0792-0
  34. [34] Çitil, M. (2019). Application of the intuitionistic fuzzy logic in education. Communications in mathematics and applications, 10(1), 131–143. 10.26713/cma.v10i1.964
  35. [35] Jeevaraj, S., Rajesh, R., & Lakshmana Gomathi Nayagam, V. (2023). A complete ranking of trapezoidal-valued intuitionistic fuzzy number: An application in evaluating social sustainability. Neural computing and applications, 35(8), 5939–5962. https://doi.org/10.1007/s00521-022-07983-y
  36. [36] Liu, P. (2016). Special issue “Intuitionistic fuzzy theory and its application in economy, technology and management.” Technological and economic development of economy, 22(3), 327–335. https://doi.org/10.3846/20294913.2016.1185047
  37. [37] Nirmala, V., & Pandian, S. C. (2015). Numerical approach for solving intuitionistic fuzzy differential equation under generalised differentiability concept. Applied mathematical sciences, 9(67), 3337–3346. http://dx.doi.org/10.12988/ams.2015.54320
  38. [38] Ettoussi, R., Melliani, S., Elomari, M., & Chadli, L. S. (2015). Solution of intuitionistic fuzzy differential equations by successive approximations method. Notes on intuitionistic fuzzy sets, 21(2), 51–62. https://B2n.ir/qg4396
  39. [39] Wang, L., & Guo, S. (2016). New results on multiple solutions for intuitionistic fuzzy differential equations. Journal of systems science and information, 4(6), 560–573. https://doi.org/10.21078/JSSI-2016-560-14
  40. [40] Amma, B. Ben, & Chadli, L. S. (2016). Numerical solution of intuitionistic fuzzy differential equations by Runge-Kutta method of order four. Notes on intuitionistic fuzzy sets, 22(4), 42–52. https://ifigenia.org/images/a/aa/NIFS-22-4-42-52.pdf
  41. [41] Biswas, S., Banerjee, S., & Roy, T. K. (2016). Solving intuitionistic fuzzy differential equations with linear differential operator by Adomian decomposition method, 3rd Int. Notes on intuitionistic fuzzy sets, 22(4), 25–41. https://ifigenia.org/images/e/eb/NIFS-22-4-25-41.pdf
  42. [42] Nirmala, V., Parimala, V., & Rajarajeswari, P. (2018). Application of Runge-Kutta method for finding multiple numerical solutions to intuitionistic fuzzy differential equations. Journal of physics: Conference series, 1139(1), 12012. http://dx.doi.org/10.1088/1742-6596/1139/1/012012
  43. [43] Akin, O., & Bayeg, S. (2018). System of intuitionistic fuzzy differential equations with intuitionistic fuzzy initial values. Notes intuitionistic fuzzy sets, 24(4), 141–171. https://ifigenia.org/images/archive/7/7f/20181215112708!NIFS-24-4-141-171.pdf
  44. [44] Geetha, S. P., & Sangeetha, K. K. (2018). Initial value problem of second order intuitionistic fuzzy ordinary differential equations. ScieXplore: International journal of research in science, 5(1), 1–10. http://dx.doi.org/10.15613/sijrs/2018/v5i1/188670
  45. [45] Ettoussi, R., Melliani, S., & Chadli, L. S. (2018). On intuitionistic fuzzy laplace transforms for second order intuitionistic fuzzy differential equations. In Recent advances in intuitionistic fuzzy logic systems: Theoretical aspects and applications (pp. 153–168). Springer. https://doi.org/10.1007/978-3-030-02155-9_13
  46. [46] Melliani, S., Belhallaj, Z., Elomari, M., & Chadli, L. S. (2021). Approximate solution of intuitionistic fuzzy differential equations with the linear differential operator by the homotopy analysis method. Advances in fuzzy systems, 2021(1), 5579669. https://doi.org/10.1155/2021/5579669
  47. [47] Man, S., Saw, B. C., Bairagi, A., & Hazra, S. B. (2023). Finite difference method for intuitionistic fuzzy partial differential equations. Computer sciences & mathematics forum, 7,(1), 48. https://doi.org/10.3390/IOCMA2023-14406
  48. [48] Rahaman, M., Alam, S., Alamin, A., Mondal, S. P., & Singh, P. (2023). An application of intuitionistic fuzzy differential equation to the inventory model. In Fuzzy optimization, decision-making and operations research: Theory and applications (pp. 679–702). Springer. https://doi.org/10.1007/978-3-031-35668-1_30
  49. [49] Harir, A., Melliani, S., & Chadli, L. S. (2024). Intuitionistic fuzzy generalized conformable fractional derivative. Progress in fractional differentiation and applications, 10(1), 63–72. http://dx.doi.org/10.18576/pfda/100106
  50. [50] Acharya, A., Mahato, S., Sil, N., Mahata, A., Mukherjee, S., Mahato, S. K., & Roy, B. (2024). An intuitionistic fuzzy differential equation approach for the lake water and sediment phosphorus model. Healthcare analytics, 5, 100294. https://doi.org/10.1016/j.health.2023.100294
  51. [51] Ceylan, T. (2024). Intuitionistic fuzzy eigenvalue problem. An international journal of optimization and control: Theories & applications (IJOCTA), 14(3), 220–228. https://doi.org/10.11121/ijocta.1471
  52. [52] Singh, R. M., Singh, D., & Gourh, R. (2024). Approach to fuzzy differential equations in Intuitionistic fuzzy metric spaces using generalized contraction theorems. Journal of hyperstructures, 13(1), 109–123. https://doi.org/10.22098/jhs.2024.14825.1011
  53. [53] Ben Amma, B., Melliani, S., & Chadli, L. S. (2024). New results for some intuitionistic fuzzy partial functional differential equations with state-dependent delay. Sahand communications in mathematical analysis, 21(2), 251–274. https://doi.org/10.22130/scma.2023.1999306.1285
  54. [54] Sadiki, H., Oufkir, K., Elomari, M., & Bakhadach, I. (2024). The solution of the intuitionistic fuzzy linear fractional partial differential equations using the homotopy analysis method. International journal of engineering & technology, 13(2), 252–261. https://B2n.ir/en8810
  55. [55] Amma, B. Ben, Melliani, S., & Chadli, L. S. (2024). Numerical solution of intuitionistic fuzzy differential equations by Nyström method. 2024 10th international conference on optimization and applications (ICOA) (pp. 1–9). IEEE. https://doi.org/10.1007/978-3-030-35445-9_11
  56. [56] Aslaoui, T., Amma, B. Ben, Melliani, S., & Chadli, L. S. (2025). Solving higher order intuitionistic fuzzy differential equations. TWMS journal of applied and engineering mathematics, 15(4), 840–852. https://jaem.isikun.edu.tr/web/index.php/current/130-vol15no4/1372

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Published

2025-03-16

Issue

Vol. 1 No. 1 (2025): Journal of Intelligent Decision and Computational Modelling

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Articles

How to Cite

Salahshour, S. ., & Mondal, S. P. . (2025). Intuitionistic Fuzzy Differential Equations and its Applications: A Review. Journal of Intelligent Decision and Computational Modelling, 1(1), 57-64. https://doi.org/10.48314/jidcm.v1i1.62