Document Type : Research Paper
Authors
1 Department of Mathematics, Islamic Azad University, Qazvin Branch Qazvin Iran
2 Department of Mathematics, Islamic Azad University, Science and Research Branch, Tehran Iran
Abstract
In this paper, The Picard method is proposed to solve the system of first-order fuzzy differential equations $(FDEs)$ with fuzzy initial conditions under generalized $H$-differentiability. The
existence and uniqueness of the solution and convergence of the
proposed method are proved in details. Finally, the method is illustrated by solving some examples.
Keywords
[1] S. Abbasbandy and T. Allahviranloo, Numerical solutions of fuzzy di erential equations by
Taylor method, J. Comput. Meth. Appl. Math., 2 (2002), 113-124.
[2] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso and J. J. Nieto, Numerical methods for
fuzzy di erential inclusions, Comput. Math. Appl., 48 (2004), 1633-1641.
[3] S. Abbasbandy, J. J. Nieto and M. Alavi, Tuning of reachable set in one dimensional fuzzy
di erential inclusions, Chaos Soliton and Fractals., 26 (2005), 1337-1345.
[4] T. Allahviranloo, N. Ahmady and E. Ahmady, Numerical solution of fuzzy di erential equa-
tions by predictorcorrector method, Inform. Sci., 177 (2007), 1633-1647.
[5] B. Bede, Note on Numerical solutions of fuzzy di erential equations by predictorcorrector
method, Inform. Sci., 178 (2008), 1917-1922.
[6] B. Bede and S. G. Gal, Generalizations of the di erentiability of fuzzy number valued func-
tions with applications to fuzzy di erential equation, Fuzzy Set.Syst., 151 (2005), 581-599.
[7] B. Bede, J. Imre, C. Rudas and L. Attila, First order linear fuzzy di erential equations under
generalized di erentiability, Inform. Sci., 177 (2007), 3627-3635.
[8] J. J. Buckley and T. Feuring, Fuzzy di erential equations, Fuzzy Set. Syst., 110 (2000),
43-54.
[9] J.J. Buckley, T. Feuring and Y. Hayashi, Linear systems of rst order ordinary di erential
equations: fuzzy initial conditions, Soft Comput., 6 (2002), 415-421.
[10] J. J. Buckley and L. J. Jowers, Simulating Continuous Fuzzy Systems, Springer-Verlag, Berlin
Heidelberg, 2006.
[11] Y. Chalco-Cano and H. Romn-Flores,On new solutions of fuzzy di erential equations, Chaos
Soliton and Fractals., 45 (2006), 1016-1043.
[12] Y. Chalco-Cano, Romn-Flores, M. A. Rojas-Medar, O. Saavedra and M. Jimnez-Gamero, The
extension principle and a decomposition of fuzzy sets, Inform. Sci., 177 (2007), 5394-5403.
[13] C. K. Chen and S. H. Ho,Solving partial di erential equations by two-dimensional di erential
transform method, Appl. Math. Comput., 106 (1999), 171-179.
[14] Y. J. Cho and H. Y. Lan, The existence of solutions for the nonlinear rst order fuzzy
di erential equations with discontinuous conditions, Dyn. Contin.Discrete., 14 (2007) , 873-
884.
[15] W. Congxin and S. Shiji,Exitance theorem to the Cauchy problem of fuzzy di erential equa-
tions under compactance-type conditions, Inform. Sci., 108 (1993), 123-134.
[16] P. Diamond, Time-dependent di erential inclusions, cocycle attractors and fuzzy di erential
equations, IEEE Trans. Fuzzy Syst., 7 (1999), 734-740.
[17] P. Diamond, Brief note on the variation of constants formula for fuzzy di erential equations,
Fuzzy Set. Syst., 129 (2002), 65-71.
[18] Z. Ding, M. Ma and A. Kandel, Existence of solutions of fuzzy di erential equations with
parameters, Inform. Sci., 99 (1997), 205-217.
[19] D. Dubois and H. Prade, Towards fuzzy di erential calculus: Part 3, di erentiation, Fuzzy
Set. Syst., 8 (1982), 225-233.
[20] O. S. Fard, Z. Hadi, N. Ghal-Eh and A. H. Borzabadi,A note on iterative method for solving
fuzzy initial value problems, J. Adv. Res. Sci. Comput., 1 (2009), 22-33.
[21] O. S. Fard, A numerical scheme for fuzzy cauchy problems, J. Uncertain Syst., 3 (2009),
307-314.
Taylor method, J. Comput. Meth. Appl. Math., 2 (2002), 113-124.
[2] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso and J. J. Nieto, Numerical methods for
fuzzy di erential inclusions, Comput. Math. Appl., 48 (2004), 1633-1641.
[3] S. Abbasbandy, J. J. Nieto and M. Alavi, Tuning of reachable set in one dimensional fuzzy
di erential inclusions, Chaos Soliton and Fractals., 26 (2005), 1337-1345.
[4] T. Allahviranloo, N. Ahmady and E. Ahmady, Numerical solution of fuzzy di erential equa-
tions by predictorcorrector method, Inform. Sci., 177 (2007), 1633-1647.
[5] B. Bede, Note on Numerical solutions of fuzzy di erential equations by predictorcorrector
method, Inform. Sci., 178 (2008), 1917-1922.
[6] B. Bede and S. G. Gal, Generalizations of the di erentiability of fuzzy number valued func-
tions with applications to fuzzy di erential equation, Fuzzy Set.Syst., 151 (2005), 581-599.
[7] B. Bede, J. Imre, C. Rudas and L. Attila, First order linear fuzzy di erential equations under
generalized di erentiability, Inform. Sci., 177 (2007), 3627-3635.
[8] J. J. Buckley and T. Feuring, Fuzzy di erential equations, Fuzzy Set. Syst., 110 (2000),
43-54.
[9] J.J. Buckley, T. Feuring and Y. Hayashi, Linear systems of rst order ordinary di erential
equations: fuzzy initial conditions, Soft Comput., 6 (2002), 415-421.
[10] J. J. Buckley and L. J. Jowers, Simulating Continuous Fuzzy Systems, Springer-Verlag, Berlin
Heidelberg, 2006.
[11] Y. Chalco-Cano and H. Romn-Flores,On new solutions of fuzzy di erential equations, Chaos
Soliton and Fractals., 45 (2006), 1016-1043.
[12] Y. Chalco-Cano, Romn-Flores, M. A. Rojas-Medar, O. Saavedra and M. Jimnez-Gamero, The
extension principle and a decomposition of fuzzy sets, Inform. Sci., 177 (2007), 5394-5403.
[13] C. K. Chen and S. H. Ho,Solving partial di erential equations by two-dimensional di erential
transform method, Appl. Math. Comput., 106 (1999), 171-179.
[14] Y. J. Cho and H. Y. Lan, The existence of solutions for the nonlinear rst order fuzzy
di erential equations with discontinuous conditions, Dyn. Contin.Discrete., 14 (2007) , 873-
884.
[15] W. Congxin and S. Shiji,Exitance theorem to the Cauchy problem of fuzzy di erential equa-
tions under compactance-type conditions, Inform. Sci., 108 (1993), 123-134.
[16] P. Diamond, Time-dependent di erential inclusions, cocycle attractors and fuzzy di erential
equations, IEEE Trans. Fuzzy Syst., 7 (1999), 734-740.
[17] P. Diamond, Brief note on the variation of constants formula for fuzzy di erential equations,
Fuzzy Set. Syst., 129 (2002), 65-71.
[18] Z. Ding, M. Ma and A. Kandel, Existence of solutions of fuzzy di erential equations with
parameters, Inform. Sci., 99 (1997), 205-217.
[19] D. Dubois and H. Prade, Towards fuzzy di erential calculus: Part 3, di erentiation, Fuzzy
Set. Syst., 8 (1982), 225-233.
[20] O. S. Fard, Z. Hadi, N. Ghal-Eh and A. H. Borzabadi,A note on iterative method for solving
fuzzy initial value problems, J. Adv. Res. Sci. Comput., 1 (2009), 22-33.
[21] O. S. Fard, A numerical scheme for fuzzy cauchy problems, J. Uncertain Syst., 3 (2009),
307-314.
[22] O. S. Fard, An iterative scheme for the solution of generalized system of linear fuzzy di er-
ential equations, World Appl. Sci. J., 7 (2009), 1597-1604.
[23] O. S. Fard and A. V. Kamyad, Modi ed k-step method for solving fuzzy initial value problems,
Iranian Journal of Fuzzy Systems, 8(1) (2011), 49-63.
[24] O. S. Fard, T. A. Bidgoli and A. H. Borzabadi, Approximate-analytical approach to nonlinear
FDEs under generalized di erentiability, J. Adv. Res. Dyn. Control Syst., 2 (2010), 56-74.
[25] W. Fei,Existence and uniqueness of solution for fuzzy random di erential equations with
non-Lipschitz coecients, Inform. Sci., 177 (2007), 329-4337.
[26] M. J. Jang and C. L. Chen, Y.C. Liy,On solving the initial-value problems using the di er-
ential transformation method, Appl. Math. Comput., 115 (2000), 145- 160.
[27] L. J. Jowers, J. J. Buckley and K. D. Reilly, Simulating continuous fuzzy systems, Inform.
Sci., 177 (2007), 436-448.
[28] O. Kaleva, Fuzzy di erential equations, Fuzzy Set. Syst., 24 (1987), 301-317.
[29] O. Kaleva, The Cauchy problem for fuzzy di erential equations, Fuzzy Set. Syst., 35 (1990),
389-396.
[30] O. Kaleva, A note on fuzzy di erential equations, Nonlinear Anal.,64 (2006) 895-900.
[31] R. R. Lopez, Comparison results for fuzzy di erential equations, Inform. Sci., 178 (2008),
1756-1779.
[32] M. Ma, M. Friedman and A. Kandel, Numerical solutions of fuzzy di erential equations,
Fuzzy Set. Syst., 105 (1999), 133-138.
[33] M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Romn-Flores and R. C. Bassanezi, Fuzzy
di erential equations and the extension principle, Inform. Sci., 177(2007) , 3627-3635.
[34] M. Oberguggenberger and S. Pittschmann,Di erential equations with fuzzy parameters,
Math. Mod. Syst., 5 (1999), 181-202.
[35] G. Papaschinopoulos, G. Stefanidou and P. Efraimidi, Existence uniqueness and asymptotic
behavior of the solutions of a fuzzy di erential equation with piecewise constant argument,
Inform. Sci., 177 (2007), 3855-3870.
[36] M. L. Puri and D. A. Ralescu, Di erentials of fuzzy functions, J. Math. Anal. Appl., 91
(1983), 552-558.
[37] M. L. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986),
409{422.
[38] H. J. Zimmermann, Fuzzy sets theory and its applications, Kluwer Academic Press, Dor-
drecht, 1991.
ential equations, World Appl. Sci. J., 7 (2009), 1597-1604.
[23] O. S. Fard and A. V. Kamyad, Modi ed k-step method for solving fuzzy initial value problems,
Iranian Journal of Fuzzy Systems, 8(1) (2011), 49-63.
[24] O. S. Fard, T. A. Bidgoli and A. H. Borzabadi, Approximate-analytical approach to nonlinear
FDEs under generalized di erentiability, J. Adv. Res. Dyn. Control Syst., 2 (2010), 56-74.
[25] W. Fei,Existence and uniqueness of solution for fuzzy random di erential equations with
non-Lipschitz coecients, Inform. Sci., 177 (2007), 329-4337.
[26] M. J. Jang and C. L. Chen, Y.C. Liy,On solving the initial-value problems using the di er-
ential transformation method, Appl. Math. Comput., 115 (2000), 145- 160.
[27] L. J. Jowers, J. J. Buckley and K. D. Reilly, Simulating continuous fuzzy systems, Inform.
Sci., 177 (2007), 436-448.
[28] O. Kaleva, Fuzzy di erential equations, Fuzzy Set. Syst., 24 (1987), 301-317.
[29] O. Kaleva, The Cauchy problem for fuzzy di erential equations, Fuzzy Set. Syst., 35 (1990),
389-396.
[30] O. Kaleva, A note on fuzzy di erential equations, Nonlinear Anal.,64 (2006) 895-900.
[31] R. R. Lopez, Comparison results for fuzzy di erential equations, Inform. Sci., 178 (2008),
1756-1779.
[32] M. Ma, M. Friedman and A. Kandel, Numerical solutions of fuzzy di erential equations,
Fuzzy Set. Syst., 105 (1999), 133-138.
[33] M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Romn-Flores and R. C. Bassanezi, Fuzzy
di erential equations and the extension principle, Inform. Sci., 177(2007) , 3627-3635.
[34] M. Oberguggenberger and S. Pittschmann,Di erential equations with fuzzy parameters,
Math. Mod. Syst., 5 (1999), 181-202.
[35] G. Papaschinopoulos, G. Stefanidou and P. Efraimidi, Existence uniqueness and asymptotic
behavior of the solutions of a fuzzy di erential equation with piecewise constant argument,
Inform. Sci., 177 (2007), 3855-3870.
[36] M. L. Puri and D. A. Ralescu, Di erentials of fuzzy functions, J. Math. Anal. Appl., 91
(1983), 552-558.
[37] M. L. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986),
409{422.
[38] H. J. Zimmermann, Fuzzy sets theory and its applications, Kluwer Academic Press, Dor-
drecht, 1991.
)