Adomian Method for Second-order Fuzzy Differential Equation
In this paper, we study the numerical method for solving second-order fuzzy differential equations using Adomian method under strongly generalized differentiability. And, we present an example with initial condition having four different solutions to illustrate the efficiency of the proposed method under strongly generalized differentiability.
[1] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential
equations by Taylor method, Journal of Computer and Mathematics with
Applications 2 (2002) 113-124
[2] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso, J.J. Nieto, Numerical
methods for fuzzy differential inclusions, Journal of Computer and
Mathematics with Applications 48 (2004) 1633-1641.
[3] G. Adomian, Nonlinear stochastic systems and application to physics,
Kluwer, Dordecht,1989.
[4] G. Adomian, A review of the decomposition method and some results for
nonlinear equations, Math. Compute Model 7 (1990) 17-43.
[5] G. Adomian, Solving Frontier Problems of Physics: The Decomposition
Method, Kluwer,Dordecht, 1994.
[6] T. Allahviranloo, N. Ahmadi, E. Ahmadi, Numerical solution of fuzzy differential
equations by predictor-corrector method, Information Sciences
177 (2007) 1633-1647.
[7] E. Babolian , H. Sadeghi b, Sh. Javadi. Numerically solution of fuzzy
differential equations by Adomian method, Applied Mathematics and
Computation 149 (2004) 547-557.
[8] B. Bede, S.G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy
Sets and Systems 147 (2004) 385-403.
[9] B. Bede, S.G. Gal, Generalizations of the differentiability of fuzzy number
value functions with applications to fuzzy differential equations, Fuzzy
Sets and Systems 151 (2005) 581-599.
[10] B. Bede, I.J. Rudas, A.L. Bencsik, First order linear fuzzy differential
equations under generalized differentiability, Information Sciences 177
(2007)1648-1662.
[11] J.J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets and
Systems 110 (2000) 43-54.
[12] Y. Chalco-Cano, H. Romn-Flores, Comparison between some approaches
to solve fuzzy differential equations, Fuzzy Sets and Systems
160 (2009)1517-1527.
[13] S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE
Transactions on Systems Man Cybernetics 2 (1972) 330-340.
[14] D. Dubois, H. Prade, Towards fuzzy differential calculus: Part 3,
Differentiation. Fuzzy Sets and Systems 8 (1982) 225-233.
[15] D. Dubois, H. Prade, Fuzzy numbers: an overview, in: J. Bezdek (Ed.),
Analysis of Fuzzy Information, CRC Press (1987) 112-148.
[16] M. Friedman, M. Ma, A. Kandel, Numerical solution of fuzzy differential
and integral equations, Fuzzy Sets and Systems 106 (1999) 35-48.
[17] 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.
[18] E. Hllermeier, An approach to modelling and simulation of uncertain
systems, International Journal of Uncertainty Fuzziness Knowledge-Based
Systems 5 (1997) 117-137.
[19] E. Hllermeier, Numerical methods for fuzzy initial value problems,
International Journal of Uncertainty Fuzziness Knowledge-Based Systems
7 (1999) 439-461.
[20] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24
(1987) 301-317.
[21] A. Kandel, W. J. Byatt, Fuzzy differential equations, in: Proceedings of
International Conference Cybernetics and Society, Tokyo: (1978) 1213-
1216.
[22] A. Kandel, W. J. Byatt, Fuzzy processes, Fuzzy Sets and Systems 4
(1980 )117-152.
[23] A. Khastan, F. Bahrami, K. Ivaz, New results on multiple solutions
for nth-order fuzzy differential equation under generalized differentiability.
Boundary Value Problem (Hindawi Publishing Corporation).
doi:10.1155/2009/395714 (2010).
[24] A. Khastan, K. Ivaz, Numerical solution of fuzzy differential equations
by Nystr?m method. Chaos Solitons Fractals 41 (2009) 859-868.
[25] J.J. Nieto, R. Rodrłguez-Lpez, D. Franco, Linear first-order fuzzy
differential equation, International Journal of Uncertainty Fuzziness
Knowledge-Based Systems 14 (2006) 687-709.
[26] M. Puri, D. Ralescu. Differentials of fuzzy functions.Journal of Mathematical
Analysis and Applications 91(1983) 552-558.
[27] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems
24 (1987) 319-330.
[28] S. J .Song, C. X. Wu, Existence and uniqueness of solutions to the
Cauchy problem of fuzzy differential equations, Fuzzy Sets and Systems
110 (2000) 55-67.
[29] J. P. Xu, Z. Liao, Z. Hu, A class of linear differential dynamical systems
with fuzzy initial condition. Fuzzy Sets and Systems 158 (2007) 2339-
2358.
[1] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential
equations by Taylor method, Journal of Computer and Mathematics with
Applications 2 (2002) 113-124
[2] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso, J.J. Nieto, Numerical
methods for fuzzy differential inclusions, Journal of Computer and
Mathematics with Applications 48 (2004) 1633-1641.
[3] G. Adomian, Nonlinear stochastic systems and application to physics,
Kluwer, Dordecht,1989.
[4] G. Adomian, A review of the decomposition method and some results for
nonlinear equations, Math. Compute Model 7 (1990) 17-43.
[5] G. Adomian, Solving Frontier Problems of Physics: The Decomposition
Method, Kluwer,Dordecht, 1994.
[6] T. Allahviranloo, N. Ahmadi, E. Ahmadi, Numerical solution of fuzzy differential
equations by predictor-corrector method, Information Sciences
177 (2007) 1633-1647.
[7] E. Babolian , H. Sadeghi b, Sh. Javadi. Numerically solution of fuzzy
differential equations by Adomian method, Applied Mathematics and
Computation 149 (2004) 547-557.
[8] B. Bede, S.G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy
Sets and Systems 147 (2004) 385-403.
[9] B. Bede, S.G. Gal, Generalizations of the differentiability of fuzzy number
value functions with applications to fuzzy differential equations, Fuzzy
Sets and Systems 151 (2005) 581-599.
[10] B. Bede, I.J. Rudas, A.L. Bencsik, First order linear fuzzy differential
equations under generalized differentiability, Information Sciences 177
(2007)1648-1662.
[11] J.J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets and
Systems 110 (2000) 43-54.
[12] Y. Chalco-Cano, H. Romn-Flores, Comparison between some approaches
to solve fuzzy differential equations, Fuzzy Sets and Systems
160 (2009)1517-1527.
[13] S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE
Transactions on Systems Man Cybernetics 2 (1972) 330-340.
[14] D. Dubois, H. Prade, Towards fuzzy differential calculus: Part 3,
Differentiation. Fuzzy Sets and Systems 8 (1982) 225-233.
[15] D. Dubois, H. Prade, Fuzzy numbers: an overview, in: J. Bezdek (Ed.),
Analysis of Fuzzy Information, CRC Press (1987) 112-148.
[16] M. Friedman, M. Ma, A. Kandel, Numerical solution of fuzzy differential
and integral equations, Fuzzy Sets and Systems 106 (1999) 35-48.
[17] 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.
[18] E. Hllermeier, An approach to modelling and simulation of uncertain
systems, International Journal of Uncertainty Fuzziness Knowledge-Based
Systems 5 (1997) 117-137.
[19] E. Hllermeier, Numerical methods for fuzzy initial value problems,
International Journal of Uncertainty Fuzziness Knowledge-Based Systems
7 (1999) 439-461.
[20] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24
(1987) 301-317.
[21] A. Kandel, W. J. Byatt, Fuzzy differential equations, in: Proceedings of
International Conference Cybernetics and Society, Tokyo: (1978) 1213-
1216.
[22] A. Kandel, W. J. Byatt, Fuzzy processes, Fuzzy Sets and Systems 4
(1980 )117-152.
[23] A. Khastan, F. Bahrami, K. Ivaz, New results on multiple solutions
for nth-order fuzzy differential equation under generalized differentiability.
Boundary Value Problem (Hindawi Publishing Corporation).
doi:10.1155/2009/395714 (2010).
[24] A. Khastan, K. Ivaz, Numerical solution of fuzzy differential equations
by Nystr?m method. Chaos Solitons Fractals 41 (2009) 859-868.
[25] J.J. Nieto, R. Rodrłguez-Lpez, D. Franco, Linear first-order fuzzy
differential equation, International Journal of Uncertainty Fuzziness
Knowledge-Based Systems 14 (2006) 687-709.
[26] M. Puri, D. Ralescu. Differentials of fuzzy functions.Journal of Mathematical
Analysis and Applications 91(1983) 552-558.
[27] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems
24 (1987) 319-330.
[28] S. J .Song, C. X. Wu, Existence and uniqueness of solutions to the
Cauchy problem of fuzzy differential equations, Fuzzy Sets and Systems
110 (2000) 55-67.
[29] J. P. Xu, Z. Liao, Z. Hu, A class of linear differential dynamical systems
with fuzzy initial condition. Fuzzy Sets and Systems 158 (2007) 2339-
2358.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53415", author = "Lei Wang and Sizong Guo", title = "Adomian Method for Second-order Fuzzy Differential Equation", abstract = "In this paper, we study the numerical method for solving second-order fuzzy differential equations using Adomian method under strongly generalized differentiability. And, we present an example with initial condition having four different solutions to illustrate the efficiency of the proposed method under strongly generalized differentiability.
", keywords = "Fuzzy-valued function, fuzzy initial value problem, strongly generalized differentiability, adomian decomposition method.", volume = "5", number = "4", pages = "568-4", }
