Numerical Solution of Fuzzy Differential Equations by Extended Runge-Kutta Method and the Dependency Problem

  IJMTT-book-cover
 
International Journal of Mathematical Trends and Technology (IJMTT)          
 
© 2014 by IJMTT Journal
Volume-6                          
Year of Publication : 2014
Authors : K. Kanagarajan , S. Muthukumar , S. Indrakumar
  10.14445/22315373/IJMTT-V6P511

MLA

K. Kanagarajan , S. Muthukumar , S. Indrakumar. "Numerical Solution of Fuzzy Differential Equations by Extended Runge-Kutta Method and the Dependency Problem", International Journal of Mathematical Trends and Technology (IJMTT). V6:113-122 February 2014. ISSN:2231-5373. www.ijmttjournal.org. Published by Seventh Sense Research Group.

Abstract
In this paper we use extended Runge-Kutta-like formulae of order four (ERK4) and of order five (ERK5) by taking into account the dependency problem that arises in fuzzy setting. This method is adopted to solve the dependency problem in fuzzy computation. Examples are presented to illustrate the theory.

References

[1] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential equations by Taylor Method, Computational Methods in Applied Mathematics, 2 (2002), 113-124.
[2] S. Abbasbandy, T. Allahviranloo, Numerical solution of Fuzzy differential equation by Runge- Kutta method, Nonlinear Studies, 11, (2004), 117-129.
[3] M. Ahamad, M. Hasen, A new approach to incorporate uncertainity into Euler method, Applied Mathematical Sciences , 4(51), (2010), 2509-2520.
[4] M. Ahamed, M. Hasan, A new fuzzy version of Euler's method for solving diffrential equations with fuzzy initial values, Sians Malaysiana, 40, (2011), 651-657.
[5] M. Ahmad, M. Hasan, Incorporating optimization technique into Zadeh's extension principle for computing non-monotone functions with fuzzy variable, Sains Malaysiana, 40, (2011) 643-650.
[6] N. Z. Ahmad, H. K. Hasan, B. De Baets, A new method for computing continuous function with fuzzy variable, Journal of Applied Sciences, 11(7) ,(2011), 1143-1149.
[7] A. H. Alsonosi Omar, Y. Abu Hasan, Numerical solution of fuzzy differential equations and the dependency problem, Applied Mathematics and Computation, 219, (2012), 1263-1272.
[8] A. Bonarini, G. Bontempi, A Qualitative simulation approach for fuzzy dynamical models, ACM Trans. Model. Comput. Simulat, 4, (1994), 285-313.
[9] R. Brent, Algorithms for Minimization without Derivatives, Dover Pubns, 2002.
[10] J. J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems,110, (2000) 43- 54.
[11] D. Dubois, H. Prade, Towards fuzzy differential calculus part 3: differentiation, Fuzzy Sets and Systems, 8, (1982), 225-233.
[12] B. Ghazanfari, A. Shakerami, Numerical solutions of fuzzy differential equations by extended Runge-Kutta like formulae of order 4, Fuzzy Sets and Systems, 189, (2011), 74-91.
[13] Kaleva Osmo, Fuzzy differential equations, Fuzzy Sets and Systems, 24, (1987), 301-317.
[14] A. Karimi Dizicheh, S. Salahshour, Fudzaih Bt. Ismail, A note on ``Numerical solutions of fuzzy differential equations by extended Runge-Kutta-like formulae of order 4", Fuzzy ets and systems, (23), (2013), 96-100.
[15] M. Ma, M. Friedman, A. Kandel, Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105, (1999), 133-138.
[16] S. Palligkinis, G. Papageorgiou, I. Famelis, Runge-Kutta methods for fuzzy differential equations, Applied Mathematics and Computation, 209, (2009), 97-105.
[17] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24(3) (1987) 319-330.
[18] Xinyuan Wu, Jianlin Xia, Extended Runge-Kutta-like formula, Applied Numerical Mathematics 56 (2006) 1584-1605.
[19] L. A. Zadeh, Fuzzy Sets, Information and Control 8(1965) 338-353.

Keywords
The Extended Runge-Kutta method, Fuzzy initial value problem, Dependency problem in fuzzy computation.