Convergence, Consistency and Stability in Fuzzy Differential Equations

Document Type: Research Paper

Authors

1 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

2 Department of Mathematics, Islamic Azad University - South Tehran Branch, Tehran, Iran

3 Department of Mathematics, Institute for Advanced Studies in Basic Sciences(IASBS), P.O. BOX 45195-1159, Zanjan, Iran

Abstract

In this paper, we consider First-order fuzzy differential equations with initial value conditions. The convergence, consistency and stability of difference method for approximating the solution of fuzzy differential equations involving generalized H-differentiability, are studied. Then the local truncation error is defined and sufficient conditions for convergence, consistency and stability of difference method are provided and fuzzy stiff differential equation and one example are presented to illustrate the accuracy and capability of our proposed concepts.

Keywords


[1] S. Abbasbandy and T. Allahviranloo, Numerical solutions of fuzzy di erential equations by
Taylor method, Journal of Computational Methods in Applied Mathematics, 2 (2002), 113{
124.
[2] T. Allahviranloo, N. A. Kiani and M. Barkhordari, Toward the existence and uniquness
of solution of second- order fuzzy di erential equations, Information Sciences, 179 (2009),
1207{1215.
[3] T. Allahviranloo and M. Barkhordari, Fuzzy laplace transforms, Soft Computing, 14 (2010),
235-243.
[4] B. Bede and SG. Gal, Almost periodic fuzzy-number valued functions, Fuzzy Sets and Sys-
tems, 147 (2004), 385{403.
[5] B. Bede and SG. Gal, Generalizations of di erentiablity of fuzzy number valued function with
application to fuzzy di erential equations, Fuzzy Sets and Systems, 151 (2005), 581-599.
[6] B. Bede, Imre J. Rudas c and Attila L., First order linear fuzzy di erential equations under
generalized di erentiability, Information Sciences, 177 (2007), 3627-3635.
[7] Y. Chalco-Cano and H. Roman-Flores, On new solutions of fuzzy di erential equations,
Chaos, solitons and Fractals (2006), 1016-1043.
[8] S. L. Chang and L. A. Zadeh, On fuzzy mapping and control, IEEE Trans, Systems Man
Cybernet., 2 (1972), 30-34.
[9] D. Dubois and H. Prade, Towards fuzzy di erential calculus: Part 3, di erentiation, Fuzzy
Sets and Systems, 8 (1982), 225-233.
[10] S. G. Gal, Approximation theory in fuzzy setting, in: G.A. Anastassiou (Ed.), Handbook of
Analytic-Computational Methods in Applied Mathematics, Chapman Hall CRC Press, (2000),
617-666.
[11] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy sets and Systems, 18 (1986),
31-43.
[12] O. Kaleva, Fuzzy di erential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
[13] O. Kaleva, The Cuachy problem for fuzzy di erential equations, Fuzzy Sets and Systems, 35
(1990), 389-396.
[14] M. Ma, M. Friedman and A. Kandel, Numerical solutions of fuzzy di erential equations,
Fuzzy Sets and Systems, 105 (1999), 133-138.
[15] M. L. Puri and D. A. Ralescu, Di erentials of fuzzy functions, J. math. Analysis. Appl., 91
(1983), 552-558.
[16] S. Salahshour and T. Allahviranloo, Applications of fuzzy Laplace transforms, Soft comput-
ing, 17 (2013), 145-158.
[17] S. Salahshour and T. Allahviranloo, A new method for solving fuzzy rst order di erential
equations, IPMU, 2012.
[18] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.
[19] C. Wu and Z. Gong, On Henstock integral of fuzzy-number-valued functions I, Fuzzy Sets
and Systems, 120 (2001), 523-532.