On Existence and Uniqueness of Solution of Fuzzy Fractional Differential Equations

Document Type : Research Paper

Author

Comsats Institute of information Technology, Lahore, Pakistan and Ab- dus Salam School of Mathematical Sciences GC University, Lahore, Pakistan

Abstract

The purpose of this paper is to study the fuzzy fractional differential
equations. We prove that fuzzy fractional differential equation is
equivalent to the fuzzy integral equation and then using this equivalence
existence and uniqueness result is establish. Fuzzy derivative is consider
in the Goetschel-Voxman sense and fractional derivative is consider in the
Riemann Liouville sense. At the end, we give the applications of the main
result.

Keywords


\bibitem{fuzzyfac 1}R. P. Agarwal, S. Arshad, D. O'Regan and V. Lupulescu, {\it Fuzzy fractional integral equations
under compactness type condition}, Fract. Calc. Appl. Anal., {\bf15(4)} (2012), 572-590.

\bibitem{fuzzyfac} R. P. Agarwal, V. Lakshmikantham and J. J. Nieto, \textit{ On the concept of solution for fractional differential equations with
uncertainty}, Nonlinear Analysis, {\bf72} (2010), 2859-2862.

\bibitem{TSS} T. Allahviranloo, S. Salahshour and S. Abbasbandy, \textit{ Explicit solutions of fractional differential equations with uncertainty},
Soft Comput., {\bf16} (2012), 297-302.

\bibitem{LS} S. Arshad and V. Lupulescu,\textit{ On the fractional
differential equations with uncertainty}, Nonlinear Analysis, {\bf74} (2011),
3685-3693.

\bibitem{SL2} S. Arshad and V. Lupulescu,\textit{ Fractional differential
equation with fuzzy initial condition}, Electronic Journal of Differential
Equations, {\textbf{2011}(34)} (2011), 1-8.

\bibitem{bsg} B. Bede and S. G. Gal,\textit{ Generalizations of the
differentiability of fuzzy-number-valued functions with applications to
fuzzy differential equations}, Fuzzy Sets and Systems, {\bf151} (2005), 581-599.

\bibitem{jbf} J. J. Buckley and T. Feuring,\textit{ Fuzzy differential
equations}, Fuzzy Sets and Systems, {\bf110} (2000), 43-54.

\bibitem{MCXG} M. Chen, C. Wu, X. Xue and G. Liu,\textit{ On fuzzy boundary
value problems}, Information Sciences, {\bf178} (2008), 1877-1892.

\bibitem{AF} K. Diethelm,\textit{ The analysis of fractional differential
equations}, Springer, 2004.\vspace{0.05cm}

\bibitem{rg} R. Goetsch Jr. and W. Voxman,\textit{ Elementary fuzzy calculus}%
, Fuzzy Sets and Systems, {\bf18} (1986), 31-43.

\bibitem{hul} E. Hullermeier,\textit{ An approach to modeling and simulation
of uncertain dynamical systems}, International Journal of Uncertainty,
Fuzziness Knowledge-Bases System, {\bf5} (1997), 117-137.

\bibitem{jjj} D. Junsheng, A. Jianye and Xu Mingyu,\textit{ Solution of
system of fractional differential equations by adomian decomposition method},
Appl. Math. J. Chinese Univ. Ser. B, {\bf22} (2007), 7-12.

\bibitem{kh} A. Khastan and J.J. Nieto,\textit{ A boundary value problem for
second order fuzzy differential equations}, Nonlinear Analysis, {\bf72} (2010),
3583-3593.

\bibitem{KST} A. A. Kilbas, H. M. Srivastava and J. J. Trujillo,\textit{ Theory and applications of fractional differential equations},  North-Holland Mathematics Studies, Elsevier, New York, NY, USA, {\bf204} (2006).

\bibitem{lak5} V. Lakshmikantham and R. N. Mohapatra,\textit{ Theory of fuzzy
differential equations and inclusions}, Taylor \& Francis, London, 2003.

\bibitem{16} M. Mazandarani and A. V. Kamyad, {\it Modified fractional Euler method for solving fuzzy frac-
tional initial value problem}, Commun Nonlinear Sci Numer Simulat, {\bf18} (2013), 1221.

\bibitem{mr} K. S. Miller and B. Ross,\textit{ An introduction to fractional
calculus and fractional differential equations}, Wiley, New York, 1993.

\bibitem{miz} M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Rom\'{a}%
n-Flores and R. C. Bassanezi,\textit{ Fuzzy differential equations and the
extension principle}, Information Sciences, {\bf177} (2007), 3627-3635.

\bibitem{os} K. B. Oldham and J. Spanier,\textit{ The frational calculus:
theory and application of differentiation and integration to an arbitrary
order}, Academic, New York, London, 1974.

\bibitem{sts} S. Salahshour, T. Allahviranloo and S. Abbasbandy,\textit{ Solving fuzzy fractional differential equations by fuzzy Laplace transforms}, Commun Nonlinear Sci Numer Simulat, {\bf17} (2012), 1372-1381.

\bibitem{seik} S. Seikkala,\textit{ On the fuzzy initial value problem},
Fuzzy Sets and Systems, {\bf24} (1987), 319-330.

\bibitem{ta} A. J. Turski, B. Atamaniuk and E. Turska,\textit{ On the
appearance of the fractional derivative in the behavior of real materials},
J. Appl. Mechanics, {\bf51} (1984), 294-298.

\bibitem{dvss} D. Vorobiev and S. Seikkala,\textit{ Towards the theory of
fuzzy differential equations}, Fuzzy Sets and Systems, {\bf125} (2002), 231-237.

\bibitem{xu} J. Xu, Z. Liao and J. J. Nieto,\textit{ A class of linear
differential dynamical systems with fuzzy matrices}, Journal of Mathematical
Analysis and Applications, {\bf368} (2010), 54-68.