Journal of Applied Nonlinear Dynamics
Fourth Order RungeKutta Method for Solving Firstorder Fully Fuzzy Differential Equations Under Strongly Generalized Hdifferentiability
Journal of Applied Nonlinear Dynamics 6(3) (2017) 387406  DOI:10.5890/JAND.2017.09.007
D. Vivek; K. Kanagarajan; S. Indirakumar
Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore  641 020, Tamilnadu, India
Download Full Text PDF
Abstract
In this paper we use fourth order RungeKutta method for solving fully fuzzy differential equations of the form y'(t) = a⊗y(t), y(0)= y0, t ∈ [0,T] under strongly generalized Hdifferentiability. The algorithm used here are based on cross product of two fuzzy numbers. Using cross product we can divide fully fuzzy differential equation (FFDE) into four different cases. We apply the results to a particular case of FFDE. The Convergence of this method is discussed and numerical examples are given to verify the reliability of this method.
Acknowledgments
The authors are greatful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from editor too.
References

[1]  Abbasbandy, S., Ahmady, N., and Ahmady, E. (2007), Numerical solutions of fuzzy differential equations by predictor corrector method, Inf. Sci., 177, 16331647. 

[2]  Kandel, A. and Byatt,WJ. (1978), Fuzzy differential equations, In: Proceedings of the international conference on cybernetics and society, Tokyo, 12131216. 

[3]  Abbasbandy, S. and Allahviranloo, T. (2001), Numerical solutions of fuzzy differential equations by Rungekutta method, Comput. Methods Appl. Math., 2, 113. 

[4]  Dubois, D. and Prade, H. (1982), Towards fuzzy differential calculus: Part 3, differentation, Fuzzy Sets Syst., 8, 225233. 

[5]  Kaleva, O. (1987), Fuzzy differential equations, Fuzzy Sets Syst., 24, 301317. 

[6]  Bede, B. and Gal, SG. (2005), Generalizations of differentiability of fuzzy number valued fuction with application to fuzzy differential equations, Fuzzy Sets Syst. 151, 581599. 

[7]  Bede, B., Rudas, IJ., and Bencsik, Al. (2007), First order linear fuzzy differential equations under generalized differentiability, Inf. Sci., 177, 16481662. 

[8]  Abbasbandy, S., Allahviranloo, T., Lopezpouso oscar, and Nieto, J.J. (2004), Numerical solutions of fuzzy differential equation inclusions, Comput. Methods Appl. Math., 48, 16331641. 

[9]  Gal, SG. Approximation theory in fuzzy setting. In: Anastassious GA (ed) Hand book of analyticcomputational methods in applied mahematics, Chapman Hall C R C Press, New york, 617666. 

[10]  Wu, C. and Gong, Z. (2001), On Henstock integral of fuzzynumbervalued fuctions I, Fuzzy Sets and Systems, 120, 523532. 

[11]  Friedman, M., Ming, M., and Kandel, A. (1999), Numerical solution of fuzzy differential and integral equations, Fuzzy Sets Systems, 106, 3548. 

[12]  Khastan, A., Bahrami, F., and Ivaz, K. (2009), New results on multiple solutions for Nthorder fuzzy differential under generalized differentiability, Boundary value problem, 113. 

[13]  Georgiou, DN., Nieto, JJ., and Rodriguez, R. (2005), Initial value problem for higherorder fuzzy differential equations, Nonlinear Anal., 63, 587600. 

[14]  Puri, ML. and Ralescu, DA. (1983), Differentials of fuzzy fuctions, Journal of Math. Anal. and appl., 91, 552558. 

[15]  Moloudzadeh, S., Allahviranloo, T., and Darabi, P. (2013), A new method for solving an arbitrary fully fuzzy system, Soft Compu., 17(9), 17251731. 

[16]  Darabi, P., Moloudzadeh, S., and Khandani, H. (2015), A numerical method for solving firstorder fully fuzzy differential equation under strongly generalized Hdifferentiability, Metho. and Appl.. 

[17]  Abbasbandy, S.,Kiani NA, T., and Barkhordari, M. (2009), Toward the existence and uniqueness of solution of secondorder fuzzy differential equations, Inf. Sci., 179, 12071215. 

[18]  Abbasbandy, S. and Allahviranloo, T. (2002), Numerical solutions of fuzzy differential equations by taylor method, Comput. Methods Appl. Math., 2, 113124. 

[19]  Nieto, JJ., Khastan, A., and Ivan, K. (2009), Numerical solution of fuzzy differential equations under generalized differentiability, Nonlinear Anal.: Hybrid System, 3, 700707. 

[20]  Ban, A. and Bede, B. (2006), Properties of the cross product of fuzzy numbers, Fuzzy Math., 14, 513531. 

[21]  Bede, B. and Fodor, J. (2006), Product type operations between fuzzy numbers and their applications in geology. Acta Polytec Hung, 3, 123139. 

[22]  Buckley, JJ. and Feuring, T. (2000), Fuzzy differential equation, Fuzzy Sets, 110, 4354. 

[23]  Buckley, JJ. and Feuring, T. (1999), Introduction to fuzzy partial differential equations, Fuzzy Sets Syst., 105, 241248. 

[24]  Congxin, W. and Shiji, S. (1998), Existence theorem to the Cauchy problem of fuzzy differential equation under compatnesstype conditions, Info. Sci., 108, 123134. 

[25]  Ghazanfari, B. and Shankerami, A. (2011), Numerical solutions of fuzzy differential equations by extended Rungekuttalike formulae of order four, Fuzzy sets and Systems, 189, 7491. 

[26]  He, O. and Yi, W. (1989), On fuzzy differential equations, Nonlinear Anal., 24, 321325. 

[27]  Jower, LJ., Buckley, JJ., and Reilly, KD. (2007), Simulating continuous fuzzy systems, Infom. Sci., 177, 436448. 

[28]  Kaleva, O. (1990), The Cauchy problem for fuzzy differential equations, Fuzzy Sets System, 35, 389396. 

[29]  Kanagarajan, K. and Sambath, M. (2010), Rungekutta Nystrom method of order three for solving fuzzy differential equations, Comput. methods, 2, 195203. 

[30]  Kloeden, P. (1999), Remarks on Peanolike theorems for fuzzy differential equations, Fuzzy Sets System, 105, 133138. 

[31]  Seikkala, S. (1987), On the fuzzy inital value problem, Fuzzy Sets and Systems, 24, 319330. 

[32]  Vasile Lupulescu. (2013), Hukuhara differentiability of intervalvalued functions and interval differential equations on time scales, Infom. Sci., 248, 5067. 