ON THE SOLUTIONS OF FUZZY FRACTIONAL DIFFERENTIAL EQUATION

Abstract

In this paper the exact and the approximate solutions of fuzzy fractional differential equation, in the sense of Caputo Hukuhara differentiability, with a fuzzy condition are constructed by using the fuzzy Laplace transform. The obtained solutions are expressed in the form of the fuzzy Mittag-Leffler function. The presented procedure is visualized and the graphs of the obtained approximate solutions are drawn by using the GeoGebra package.

Keywords

• Fractional,Fuzzy calculus,Laplace transform

References

1. Bede1 B., Rudas I.,Bencsik A., (2007), First order linear fuzzy differential equa- tions under generalized differentiability, Journal Information Sciences: an International Journal archive V. 177 I. 7, pp 1648- 1662.
2. Arshad S, Lupulescu V., (2011), Fractional differential equation with the fuzzy initial condition, Electronic Journal of Differential Equations, Vol. 2011 No. 34, pp. 1-8.
3. Caputo, M., (2008), Linear models of dissipation whose Q is almost frequency independent- II, Geo- phys. J. Royal Astronom. Soc., 13, No 5 (1967), 529-539.
4. Kadaka U., Basar,F., (2012), Power series of fuzzy numbers with real or fuzzy coefficients, Filomat 26:3, 519528.
5. V. Kiryakova, (2011), Fractional order differential and integral equations with Erd´elyi-Kober opera- tors: Explicit solutions by means of the transmutation method, American Institute of Physics - Conf. Proc. # 1410 (Proc. 37th Intern. Conf. AMEE’ 2011), 247-258; doi: 10.1063/1.3664376.
6. V. Kiryakova, (2012), Some operational tools for solving fractional and higher integer order differential equations: A survey on their mutual relations, American Institute of Physics - Conf. Proc. # 1497 (Proc. 38th Intern. Conf. AMEE’ 2012), 273-289; doi: 10.1063/1.4766795.
7. Mainardi, F., Yu. Luchko, Pagnini, G., (2001), The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and its Application, 4,2., 153-192.
8. Palash D., Hrishikesh B., Tazid A. (2011), Fuzzy Arithmetic with and without using -cut method: A Comparative Study, International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 99 Volume 2, Issue 1.

9. Salahshour S., Allahviranloo T., Abbasbandy S., (2012), Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun Nonlinear Sci Numer Simulat 17, 13721381.
10. Takaˇci, Dj., Takaˇci, A., (2007), On the approximate solution of mathematical model of a viscoelastic bar, Nonlinear Analysis, 67, 1560-1569.
11. Takaˇci, Dj., Takaˇci, A., ˇStrboja, M., (2010) On the character of operational solutions of the time- fractional diffusion equation, Nonlinear Analysis: 72, 5, 2367-2374.
12. J. Tenreiro Machado, V. Kiryakova, (2011), F. Mainardi, Recent history of fractional cal- culus, Communications in Nonlinear Sci. and Numerical Simulations, 16, No 3, 1140-1153; doi:10.1016/j.cnsns.2010.05.027.
13. Zadeh, L. A.,Fuzzy Sets, (1965), Information and Control, 8, 338-353.