Solution and Analysis of Coupled Homogeneous Linear Intuitionistic Fuzzy Difference Equation

Abdul Alamin ¹ ( Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, Haringhata, Nadia-741249, West Bengal, India. )
Mostafijur Rahaman ² ( Department of Mathematics, School of Liberal Arts & Sciences, Mohan Babu University, Tirupati, Andhra Pradesh 517102, India. )
Kamal Hossain Gazi ³ ( Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, Haringhata, Nadia-741249, West Bengal, India. )
Shariful Alam ⁴ ( Department of Mathematics, Indian Institute of Engineering Science and Technology, Howrah-711103, West Bengal, India. )
Sankar Prasad Mondal ⁵ ( Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, Haringhata, Nadia-741249, West Bengal, India. )

Submited date : 2024-11-28 Accepted date : 2025-02-01

Keywords: Intuitionistic fuzzy set, Intuitionistic fuzzy difference equation, Stability analysis.,

Abstract :

In this paper, we have considered the initial valued coupled homogeneous linear difference equation in an intuitionistic fuzzy environment. We have given an outline of the general and analytical solutions at some fixed iteration levels and discussed the stability condition near the trivial intuitionistic equilibrium point of the system through a lemma. The proposed model is applied to check a predator-prey model and the graphical solutions are taken with taking different intuitionistic initial predator and prey population sizes to observe the dynamics in a short-sighted plan. Also, the graphical outcomes in an intuitionistic environment validate the theoretical result and the interrelated dynamical nature to fathom the significance of this work.

References:

[1] Atanassov KT, Stoeva S. Intuitionistic fuzzy sets. Fuzzy sets and Systems. 1986; 20(1): 87-96. DOI: https://doi.org/10.1016/S0165-0114(86)80034-3
[2] Atanassov K. Intuitionistic Fuzzy Sets Theory and Applications. International Journal of Advanced Computer Science and Applications (Physica-Verlag, Heidelberg). 1999; 14-17. DOI: https://doi.org/10.1007/978-3-7908-1870-3 1
[3] Zadeh LA. Fuzzy sets. Information and control. 1965; 8(3): 338-353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X
[4] Dan S, Kar MB, Majumder S, Roy B, Kar S, Pamucar D. Intuitionistic type-2 fuzzy set and its properties. Symmetry. 2019; 11(6): 808. DOI: https://doi.org/10.3390/sym11060808
[5] Zulqarnain RM, Siddique I, Ali R, Pamucar D, Marinkovic D, Bozanic D. Robust aggregation operators for intuitionistic fuzzy hypersoft set with their application to solve MCDM problem. Entropy. 2021; 23(6): 688. DOI: https://doi.org/10.3390/e23060688
[6] Krishankumar R, Ravichandran KS, Aggarwal M, Pamucar D. An improved entropy function for the intuitionistic fuzzy sets with application to cloud vendor selection. Decision Analytics Journal. 2023; 7: 100262. DOI: https://doi.org/10.1016/j.dajour.2023.100262
[7] Alamin A, Mondal SP, Alam S, Goswami A. Solution and stability analysis of non-homogeneous difference equation followed by real life application in fuzzy environment. S˜adhan˜a. 2020; 45: 1-20. DOI: https://doi.org/10.1007/s12046-020-01422-1
[8] Alamin A, Rahaman M, Mondal SP, Chatterjee B, Alam S. Discrete system insights of logistic quota harvesting model: a fuzzy difference equation approach. Journal of uncertain systems. 2022; 15(02): 2250007. DOI: https://doi.org/10.1142/S1752890922500076
[9] Alamin A, Rahaman M, Prasad Mondal S, Alam S, Salimi M, Ahmadian A. Analysis on the behavior of the logistic fixed effort harvesting model through the difference equation under uncertainty. International Journal of Modelling and Simulation. 2023; 1-17. DOI: https://doi.org/10.1080/02286203.2023.2246830
[10] Maayah B, Arqub OA. Uncertain M-fractional differential problems: existence, uniqueness, and approximations using Hilbert reproducing technique provisioner with the case application: series resistorinductor circuit. Physica Scripta. 2024; 99(2): 025220. DOI: https://doi.org/10.1088/1402-4896/ad1738
[11] Abu Arqub O, Mezghiche R, Maayah B. Fuzzy M-fractional integrodifferential models: theoretical existence and uniqueness results, and approximate solutions utilizing the Hilbert reproducing kernel algorithm. Frontiers in Physics. 2023; 11: 1252919. DOI: https://doi.org/10.3389/fphy.2023.1252919
[12] Abu Arqub O, Singh J, Maayah B, Alhodaly M. Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the MittagLeffler kernel differential operator. Mathematical Methods in the Applied Sciences. 2023; 46(7): 7965-7986. DOI: https://doi.org/10.1002/mma.7305
[13] Abu Arqub O, Singh J, Alhodaly M. Adaptation of kernel functionsbased approach with AtanganaBaleanuCaputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations. Mathematical Methods in the Applied Sciences. 2023; 46(7): 7807-7834. DOI: https://doi.org/10.1002/mma.7228
[14] Melliani S, Chadli LS. Intuitionistic fuzzy differential equation. Notes on Intuitionistic Fuzzy Sets. 2000; 6(2): 37-41.
[15] Melliani S, Chadli LS. Introduction to intuitionistic fuzzy partial differential equations. Notes on intuitionistic Fuzzy sets. 2001; 7(3): 39-42.
[16] Tudu S, Gazi KH, Rahaman M, Mondal SP, Chatterjee B, Alam S. Type-2 fuzzy differential inclusion for solving type-2 fuzzy differential equation. Annals of Fuzzy Mathematics and Informatics. 2023; 25(1): 33-53. DOI: https://doi.org/10.30948/afmi.2023.25.1.33
[17] Abbasbandy S, Allahviranloo T. Numerical solution of fuzzy differential equation by Runge-Kutta method and intuitionistic treatment. Notes on IFS. 2002; 8(3): 45-53. DOI: https://doi.org/10.3390/mca16040935
[18] Mondal SP, Roy TK. System of differential equation with initial value as triangular intuitionistic fuzzy number and its application. International Journal of Applied and Computational Mathematics. 2015; 1: 449-474. DOI: https://doi.org/10.1007/s40819-015-0026-x
[19] Ettoussi R, Melliani S, Elomari M, Chadli LS. Solution of intuitionistic fuzzy differential equations by successive approximations method. Notes on Intuitionistic Fuzzy Sets. 2015; 21(2): 51-62.
[20] Nirmala V, Pandian SC. Numerical approach for solving intuitionistic fuzzy differential equation under generalised differentiability concept. Applied Mathematical Sciences. 2015; 9(67): 3337-3346. DOI: https://doi.org/10.12988/ams.2015.54320
[21] Alamin A, Mondal SP, Alam S, Goswami A. Solution and stability analysis of non-homogeneous difference equation followed by real life application in fuzzy environment. S˜adhan˜a. 2020; 45: 1-20. DOI: https://doi.org/10.1007/s12046-020-01422-1
[22] Singh P, Gor B, Gazi KH, Mukherjee S, Mahata A, Mondal SP. Analysis and interpretation of Malaria disease model in crisp and fuzzy environment. Results in Control and Optimization. 2023; 100257. DOI: https://doi.org/10.1016/j.rico.2023.100257
[23] Melliani S, Elomari M, Chadli LS, Ettoussi R. Extension of Hukuhara difference in intuitionistic fuzzy set theory. Notes on Intuitionistic Fuzzy Sets. 2015; 21(4): 34-47.
[24] Mondal SP, Vishwakarma DK, Saha AK. Intutionistic Fuzzy Difference Equation. In Emerging Research on Applied Fuzzy Sets and Intuitionistic Fuzzy Matrices, IGI Global. 2017; 112-131. DOI: https://doi.org/10.4018/978-1-5225-0914-1.ch005
[25] Biswas A, Gazi KH, Bhaduri P, Mondal SP. Site selection for girls hostel in a university campus by mcdm based strategy. Spectrum of Decision Making and Applications. 2025; 2(1): 68-93. DOI: https://doi.org/10.31181/sdmap21202511
[26] Biswas A, Gazi KH, Bhaduri P, Mondal SP. Neutrosophic fuzzy decision-making framework for site selection. Journal of Decision Analytics and Intelligent Computing. 2024; 4(1): 187-215. DOI: https://doi.org/10.31181/jdaic10004122024b
[27] Van Hoa N, Allahviranloo T, Pedrycz W. A new approach to the fractional Abel k- integral equations and linear fractional differential equations in a fuzzy environment. Fuzzy Sets and Systems. 2024; 481: 108895. DOI: https://doi.org/10.1016/j.fss.2024.108895
[28] Alamin A, Gazi KH, Mondal SP. Solution of second order linear homogeneous fuzzy difference equation with constant coefficients by geometric approach. Journal of Decision Analytics and Intelligent Computing. 2024; 4(1): 241-252. DOI: https://doi.org/10.31181/jdaic10021122024a
[29] Alamin A, Biswas A, Gazi KH, Sankar SPM. Modelling with Neutrosophic Fuzzy Sets for Financial Applications in Discrete System. Spectrum of Engineering and Management Sciences. 2024; 2(1): 263- 280. DOI: https://doi.org/10.31181/sems21202433a
[30] Atanassov KT. Intuitionistic fuzzy sets, VII ITKRs Session, Sofia deposed in Central Sci. Technical Library of Bulg. Acad. of Sci. 1983; 1697: 84. DOI: http://dx.doi.org/10.1016/S0165-0114(86)80034-3
[31] Keyanpour M, Akbarian T. Solving intuitionistic fuzzy nonlinear equations. Journal of Fuzzy Set Valued Analysis. 2014; 2014: 1-6. DOI: https://doi.org/10.5899/2014/jfsva-00142
[32] Silvester JR. Determinants of block matrices. The Mathematical Gazette. 2000; 84(501): 460-467. DOI: https://doi.org/10.2307/3620776
[33] Marotto FR. Introduction to mathematical modeling using discrete dynamical systems. Thomson Brooks/Cole. 2006.
[34] Murray JD. Models for interacting populations. In Mathematical Biology, Springer, New York, NY. 1993; 79-118.
[35] LaSalle J. The Stability and Control of Discrete Processes. Springer-Verlag. 1986. DOI: https://doi.org/10.1007/978-1-4612-1076-4