Solutions to Fuzzy Differential Equations using Pentagonal Intuitionistic Fuzzy Numbers

Mohammed Shapique, Jesura. j.


This paper proposes a method to solve first order differential equation considering the  initial condition as pentagonal intuitionistic fuzzy numbers(PIFNs). We described homogeneous first order ordinary differential equation (ODE) in fuzzy intuitionistic environment. The fuzzy intuitionistic measures both membership and non-membership function which is the main advantage of this paper .The solution is obtained and it is expressed in PIFNs. This is exemplified by two numerical examples. To illustrate the validity of the proposed paper we have solved two Numerical examples and graphs are plotted.

Full Text:



L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353.

K. T. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, Bulgarian, (1983).

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986) 87-96.

K. T. Atanassov, G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31 (3)(1989) 343-49.

K. T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33 (1) (1989) 37-45.

K. T. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, (1999).

K. T. Atanassov, Two theorems for Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 110 (2000) 267-269

K. Atanassov, G. Gargov, Elements of intuitionistic fuzzy logic, Part I, Fuzzy Sets and Systems, 95(1) (1998) 39-52.

S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Transactions on Systems Man Cybernetics, 2 (1972) 330-340.

Kaleva, O (1987), “Fuzzy differential equations”, Fuzzy sets and systems,24, pp.301-317.

Kaleva, O. (1990), “The Cauchy problem for Fuzzy differential equations”, Fuzzy sets and systems, 35, pp.389-396.

S. Seikkala, on the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987) 319-330.

D. Dubois, H. Parade, Operation on Fuzzy Number, International Journal of Fuzzy system, 9 (1978)613-626.

M. L. Puri, D. Ralescu, Differential for fuzzy function, J. Math. Anal. Appl, 91 (1983) 552-558.

R. Goetschel, W. Voxman, Elementary calculus, Fuzzy Sets and Systems, 18 (1986) 31-43.

J. J. Buckley, T. Feuring, Y. Hayashi, Linear systems of first order ordinary differential equations: Fuzzy initial conditions, Soft Computing, 6 (2002) 415-421

J. J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000) 43-54

C. Duraisamy, B. Usha, Another Approach to Solution of Fuzzy Differential Equations by Modified Euler’s Method, Proceedings of the International Conference on Communication and Computational Intelligence 2010,Kongu Engineering College, Perundurai, Erode, T.N.,India.27 – 29 December,2010.pp.52-55.

Barnabas Bede, Sorin G. Gal, Luciano Stefanini, Solutions of fuzzy differential equations with L-R fuzzy numbers, 4th International Workshop on Soft Computing Applications, 15-17 July, 2010 - Arad, Romania.

Sankar Prasad mondal and Tapan kumar roy, first order linear non homogeneous ordinary differential equation in fuzzy environment based on Laplace transform, J. Math. Computer. Sci. 3 (2013), No. 6, 1533-1564

Sankar Prasad Mondal and Tapan Kumar Roy, First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment, Mathematical theory and Modeling, Vol.3, No.1, 2013, 85-95

]Sankar Prasad Mondal, Sanhita Banerjee and Tapan Kumar Roy, First Order Linear Homogeneous Ordinary Differential Equation in Fuzzy Environment, Int. J. Pure Appl. Sci. Technol.14(1) (2013), pp. 16-26.

] Sankar Prasad Mondal and Tapan Kumar Roy, First Order Linear Homogeneous Ordinary Differential Equation with initial value as triangular intuitionistic fuzzy number, journal of uncertainty in mathematics sciences ,(2014),pp1-17

Bongju Lee and Yong Sik Yun, The pentagonal fuzzy numbers, journal of the chungcheong mathematical society, volume 27(2), May 2014

Ponnivalavan.k and Pathinathan, intuitionistic pentagonal fuzzy number, ARPN journal of engineering and applied sciences, Vol 10(2), 2015

Casasnovas, F. Rossell, Averaging fuzzy biopolymers, Fuzzy Sets and Systems, 152 (2005) 139- 158.

M. S. El Naschie, from experimental quantum optics to quantum gravity via a fuzzy Khler manifold, Chaos, Solitons and Fractals, 25 (2005) 969-977.

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

Hassan Zarei, Ali Vahidian Kamyad, Ali Akbar Heydari, Fuzzy Modeling and Control of HIV Infection, Computational and Mathematical Methods in Medicine, Article ID 893474, Vol 2012, 17 pages.

Muhammad Zaini Ahmad, Bernard De Baets, A Predator-Prey Model with Fuzzy Initial Populations, IFSA-EUSFLAT (2009).

L. C. Barros, R. C. Bassanezi, P. A. Tonelli, Fuzzy modeling in population dynamics, Ecol. Model, 128 (2000) 27-33.

Barnabas Bede, Imre J. Rudas, Janos Fodor, Friction Model by Fuzzy Differential Equations, IFSA 2007, LNAI 4529, pp.23-32, Springer-Verlag Berlin Heidelberg 2007.

B. Bede, S. G. Gal, Almost periodic fuzzy-number-value functions, Fuzzy Sets and Systems 147 (2004) 385-403.

Y. Chalco-Cano, H. Roman-Flores, on new solutions of fuzzy differential equations, Chaos, Solutions and Fractals 38 (2008) 112-119.

De Barros, Laécio Carvalho, and Francielle Santo Pedro. "Fuzzy differential equations with interactive derivative." Fuzzy Sets and Systems 309 (2017): 64-80.

Qiu, Dong, Yumei Xing, and Lihong Zhang. "Asymptotically stability of solutions of fuzzy differential equations in the quotient space of fuzzy numbers." Journal of Computational Analysis and Applications (2017): 1242.

Pakdaman, M., et al. "Solving differential equations of fractional order using an optimization technique based on training artificial neural network." Applied Mathematics and Computation 293 (2017): 81-95.

Darabi, P., S. Moloudzadeh, and H. Khandani. "A numerical method for solving first-order fully fuzzy differential equation under strongly generalized H-differentiability." Soft Computing 20.10 (2016): 4085-4098.

Khastan, Alireza, and Rosana Rodríguez-López. "On the solutions to first order linear fuzzy differential equations." Fuzzy Sets and Systems 295 (2016): 114-135.

Gomes, Luciana Takata, Laécio Carvalho de Barros, and Barnabas Bede. "Fuzzy Differential Equations." Fuzzy Differential Equations in Various Approaches. Springer International Publishing, 2015. 69-113.

Ahmad, M. Z., Mohammad Khatim Hasan, and S. Abbasbandy. "Solving fuzzy fractional differential equations using Zadeh's extension principle." The Scientific World Journal 2013 (2013).


  • There are currently no refbacks.

MAYFEB Journal of Mathematics 
Toronto, Ontario, Canada
ISSN 2371-6193