Adaptive strategies for system of fuzzy differential equation: application of arms race model

Volume 18, Issue 2, pp 192--205 http://dx.doi.org/10.22436/jmcs.018.02.07

Publication Date: January 28, 2018 Submission Date: December 13, 2016

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Authors

Sankar Prasad Mondal - Department of Mathematics, Midnapore College (Autonomous), Midnapore, West Midnapore-721101, West Bengal, India Najeeb Alam Khan - Department of Mathematics, University of Karachi, Karachi 75270, Pakistan Oyoon Abdul Razzaq - Department of Humanities and Natural Sciences, Bahria University, Karachi 75260, Pakistan Tapan Kumar Roy - Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-03, West Bengal, India

Abstract

The paper presents adaptive stratagems to scrutinize the system of first order fuzzy differential equations (SFDE) in two modes, fuzzy and in crisp sense. Its fuzzy solutions are carried out using two approaches, namely, Zadeh's extension principle and generalized Hukuhara derivative (gH-derivative). While, different defuzzification techniques; central of area method (COA), bisector of area method (BOA), largest of maxima (LOM), smallest of maxima (SOM), mean of maxima (MOM), regular weighted point method (RWPM), graded mean integration value (GMIV), and center of approximated interval (COAI), are employed to discuss the crisp solutions. Moreover, the arms race model (ARM), which have a significant implication in international military planning, are pragmatic examples of system of first order differential equations, but not studied in fuzzy sense, hitherto. Therefore, ARM is re-established and studied here with fuzzy numbers to estimate its uncertain parameters, as a practical utilization of SFDE. Additionally, an illustrative example of ARM is undertaken to clarify the appropriateness of the proposed approaches.

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Keywords

• Fuzzy differential equation
• defuzzification
• Hukuhara derivative
• extension principle

MSC

•  34A07

References

• [1] T. Allahviranloo, Z. Gouyandeh, A. Armand, A. Hasanoglu, On fuzzy solutions for heat equation based on generalized Hukuhara differentiability, Fuzzy Sets and Systems, 265 (2015), 1–23.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] L. C. Barros, R. C. Bassanezi, P. A. Tonelli, Fuzzy modelling in population dynamics, Ecol. Model., 128 (2000), 27–33.
  • View Article
  • Google Scholar


• [3] B. Bede, I. J. Rudas, J. Fodor, Friction model by using fuzzy differential equations, Lecture Notes in Comput. Sci., 4529 (2007), 23–32.
  • View Article
  • Google Scholar
  • MATH


• [4] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119– 141.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] A. Bencsik, B. Bede, J. Tar, J. Fodor, Fuzzy differential equations in modeling hydraulic differential servo cylinders, Third Romanian-Hungarian joint symposium on applied computational intelligence (SACI), Timisoara, Romania (2006)
  • Google Scholar


• [6] J. J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 43–54.
  • View Article
  • Google Scholar


• [7] J. Casasnovas, F. Rosselló , Averaging fuzzy biopolymers, Fuzzy Sets and Systems, 152 (2005), 139–158.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] S. H. Chen, S. T. Wang, S. M. Chang, Some properties of graded mean integration representation of L-R type fuzzy numbers, Tamsui Oxf. J. Math. Sci., 22 (2006), 185–208.
  • MathSciNet
  • Google Scholar
  • MATH


• [9] G. L. Diniz, J. F. R. Fernandes, J. F. C. A. Meyer, L. C. Barros, A fuzzy Cauchy problem modelling the decay of the biochemical oxygen demand in water, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference, Vancouver, BC, Canada, 1 (2001), 512–516.
  • View Article
  • Google Scholar


• [10] M. S. El Naschie, From experimental quantum optics to quantum gravity via a fuzzy Kähler manifold, Chaos Solitons Fractals, 25 (2005), 969–977.
  • View Article
  • Google Scholar
  • MATH


• [11] O. S. Fard, An iterative scheme for the solution of generalized system of linear fuzzy differential equations, World Appl. Sci. J., 7 (2009), 1597–1604.
  • Google Scholar


• [12] N. Gasilov, Ş. E. Amrahov, A. G. Fatullayev, A geometric approach to solve fuzzy linear systems of differential equations, Appl. Math. Inf. Sci., 5 (2011), 484–499.
  • MathSciNet
  • Google Scholar


• [13] W. W. Hill, Several sequential augmentations of Richardson’s arms race model, Math. Comput. Model., 16 (1992), 201– 212.
  • View Article
  • Google Scholar
  • MATH


• [14] E. Hüllermeier, An approach to modelling and simulation of uncertain dynamical systems, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, Fuzziness Knowledge-Based Systems, 5 (1997), 117–137.
  • View Article
  • MathSciNet
  • Google Scholar


• [15] N. A. Khan, O. A. Razzaq, A. Ara, F. Riaz, Numerical solution of system of fractional differential equations in imprecise environment, Numerical Simulation-From Brain Imaging to Turbulent Flow, InTech (2016)


• [16] N. A. Khan, O. A. Razzaq, M. Ayyaz, On the solution of fuzzy differential equations by Fuzzy Sumudu Transform, Nonlinear Eng., 4 (2015), 49–60.
  • View Article
  • Google Scholar


• [17] A. Khastan, I. Perfilieva, Z. Alijani, A new fuzzy approximation method to Cauchy problems by fuzzy transform, Fuzzy Sets and Systems, 288 (2016), 75–95.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] M. J. McLean, An introduction to mathematical models in the social and life sciences, M. Olinick Addison-Wesley Inc., London, (1978), Appl. Math. Modell., 3 (1979), 238–239.


• [19] M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Román-Flores, R. C. Bassanezi , Fuzzy differential equations and the extension principle, Inform. Sci., 177 (2007), 3627–3635.
  • View Article
  • MathSciNet
  • Google Scholar


• [20] S. P. Mondal, T. K. Roy, First order linear non homogeneous ordinary differential equation in fuzzy environment, Math. Theory Model., 3 (2013), 85–95.


• [21] S. Naaz, A. Alam, R. Biswas, Effect of different defuzzification methods in a fuzzy based load balancing application, Int. J. Comput. Sci., 8 (2011), 261–267.


• [22] R. Saneifard, Another method for defuzzification based on regular weighted point, Int. J. Ind. Math., 4 (2011), 147–152.


• [23] R. Saneifard, R. Saneifard, A method for defuzzification based on centroid point, Turkish J. Fuzzy Sys., 2 (2011), 36–44.
  • Google Scholar


• [24] H. Zarei, A. V. Kamyad, A. A. Heydari, Fuzzy modeling and control of HIV infection, Comput. Math. Methods Med., 2012 (2012 ), 17 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] Y. Zhang, J.-H. He, S.-Q. Wang, P. Wang, A dye removal model with a fuzzy initial condition, Therm. Sci., 20 (2016), 867–870.
  • Google Scholar