Second order linear differential equations with generalized trapezoidal intuitionistic Fuzzy boundary value

Subject Areas : Ordinary differential equations

S. P. Mondal ¹ , T. K. Roy ²

1 - Department of Mathematics, National Institute of Technology, Agartala, Jirania-799046, Tripura, India
2 - Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, West Bengal, India

Received: 2015-11-16 Accepted : 2015-11-16 Published : 2015-05-01

Keywords: fuzzy set, fuzzy differential equation, generalized trapezoidal intutionistic fuzzy number,

Abstract :

In this paper the solution of a second order linear differential equations with intuitionistic fuzzy boundary value is described. It is discussed for two different cases: coefficientis positive crisp number and coefficient is negative crisp number. Here fuzzy numbers aretaken as generalized trapezoidal intutionistic fuzzy numbers (GTrIFNs). Further a numerical example is illustrated.

References:

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

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

[3] K. T. Atanassov, Intuitionistic fuzzy sets, VII ITKRs Session, Soa, Bulgarian, 19-83.

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

[5] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) an outline, Information Sciences 172, (2005) 1-40.

[6] S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Transaction on Systems Man Cybernetics 2, (1972) 30 34.

[7] D. Dubois, H. Prade, Towards fuzzy dierential calculus: Part 3, Dierentiation, Fuzzy Sets and Systems 8, (1982) 225-233.

[8] M. L. Puri, D. A. Ralescu, Dierentials of fuzzy functions, Journal of Mathematical Analysis and Application
91, (1983) 552558.

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

[10] O. Kaleva, Fuzzy dierential equations, Fuzzy Sets and Systems 24, (1987) 301-317.

[11] B. Bede, A note on two-point boundary value problems associated with non-linear fuzzy differential equations, Fuzzy Sets. Syst.157, (2006) 986-989.

[12] B. Bede,S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets Syst.151, (2005) 581-599.

[13] Y. Chalco-Cano, H. Romn-Flores, On the new solution of fuzzy dierential equations, Chaos Solitons Fractals 38, (2008) 112-119.

[14] B. Bede,I. J. Rudas and A. L. Bencsik, First order linear fuzzy dierential equations under generalized differentiability, Inf. Sci. 177, (2007) 1648-1662.

[15] Y. Chalco-Cano, M.A.Rojas-Medar,H.Romn-Flores, Sobre ecuaciones differencial esdifusas, Bol. Soc. Esp. Mat. Apl. 41, (2007) 91-99.

[16] Y. Chalco-Cano, H. Romn-Flores and M. A. Rojas-Medar, Fuzzy dierential equations with generalized derivative, in:Proceedings of the 27th North American Fuzzy Information Processing Society International Conference, IEEE, 2008.

[17] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71, (2009) 1311-1328.

[18] A. Khastan,J. J. Nieto, A boundary value problem for second-order fuzzy dierential equations, Nonlinear Anal. 72, (2010) 3583-3593.

[19] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientic, Singapore, 1994.

[20] B. Bede, S. G. Gal, Generalizations of the dierentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Set Systems, 151 (2005) 581-599.

[21] L. Stefanini, A generalization of Hukuhara difference for interval and fuzzy arithmetic, in: D. Dubois, M.A. Lubiano, H. Prade, M. A. Gil, P. Grzegorzewski, O. Hryniewicz (Eds.), Soft Methods for Handling Variability and Imprecision, in: Series on Advances in Soft Computing, 48 (2008).

[22] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis, 71 (2009) 1311-1328.

[23] A. Armand, Z. Gouyandeh, Solving two-point fuzzy boundary value problem using the variational iteration method, Communications on Advanced Computational Science with Applications, Vol. 2013, (2013) 1-10.

[24] N. Gasilov, S. E. Amrahov, A. G. Fatullayev, Solution of linear dierential equations with fuzzy boundary values, Fuzzy Sets and Systems 257, (2014) 169-183.

[25] N. Gasilov, S. E. Amrahov, A. G. Fatullayev, A. Khastan, A new approach to fuzzy initial value problem ,18(2), (2014) 217-225.

[26] B. Bede, L. Stefanini, Generalized dierentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013) 119-141.

[27] R. Rodrguez-Lpez, On the existence of solutions to periodic boundary value problems for fuzzy linear differential equations, Fuzzy Sets and Systems, 219, (2013) 1-26.

[28] S. Melliani, L. S. Chadli, Introduction to intuitionistic fuzzy partial differential Equations, Fifth Int. Conf. on IFSs, Sofia, 22-23 Sept. 2001.

[29] S. Abbasbandy, T. Allahviranloo, Numerical Solution of Fuzzy Differential Equations by Runge-Kutta and the Intuitionistic Treatment, Journal of Notes on Intuitionistic Fuzzy Sets, Vol. 8, No. 3, (2002) 43-53.

[30] S. Lata, A.Kumar, A new method to solve time-dependent intuitionistic fuzzy differential equation and its application to analyze the intutionistic fuzzy reliability of industrial system, Concurrent Engineering: Research and Applications, (2012) 1-8.

[31] S. P. Mondal and T. K. Roy, First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number, Journal of Uncertainty in Mathematics Science (2014) 1-17.

[32] S. P. Mondal. and T. K. Roy, System of Differential Equation with Initial Value as Triangular Intuitionistic Fuzzy Number and its Application, Int. J. Appl. Comput. Math, (2010).

[33] L. C. Barros, L. T. Gomes, P. A Tonelli, Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems, 230, (2013) 39-52.

[34] M.R.Seikh, P.K.Nayak and M.Pal, Generalized Triangular Fuzzy Numbers In Intuitionistic Fuzzy Environment, International Journal of Engineering Research and Development, Volume 5, Issue 1 (2012) 08-13.

[35] H.J.Zimmerman, Fuzzy set theory and its applications, Kluwer Academi Publishers, Dordrecht (1991).