Abstract
In this paper, the fuzzy variational iteration method is proposed to solve the nonlinear fuzzy differential equation (NFDE). The convergence and the maximum absolute truncation error of the proposed method are proved in details. Some examples are investigated to verify convergence results and to illustrate the efficiently of the method.
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Abbasbandy S, Allahviranloo T (2002) Numerical solutions of fuzzy differential equations by Taylor method. J Comput Methods Appl Math 2:113–124
Abbasbandy S, Allahviranloo T, Lopez-Pouso O, Nieto JJ (2004) Numerical methods for fuzzy differential inclusions. Comput Math Appl 48:1633–1641
Abbasbandy S, Nieto JJ, Alavi M (2005) Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos Solitons Fract 26:1337
Abbod MF, Von Keyserlingk DG, Linkens DA, Mahfouf M (2001) Survey of utilisation of fuzzy technology in medicine and healthcare. Fuzzy Sets Syst 120:331–349
Allahviramloo T (2005) The Adomian decomposition method for fuzzy system of linear equations. Appl Math Comput 163:553– 563
Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equations by predictor–corrector method. Inf Sci 177:1633–1647
Barro S, Marin R (2002) Fuzzy logic in medicine. Physica-Verlag, Heidelberg
Bede B (2008) Note on numerical solutions of fuzzy differential equations by predictor–corrector method. Inf Sci 178:1917– 1922
Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equation. Fuzzy Set Syst 151:581–599
Bede B, Imre J, Rudas C, Attila L (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:3627–3635
Buckley JJ, Feuring T (2000) Fuzzy differential equations. Fuzzy Set Syst 110:43–54
Buckley JJ, Jowers LJ (2006) Simulating continuous fuzzy systems. Springer, Berlin
Buckley JJ, Feuring T, Hayashi Y (2002) Linear systems of first order ordinary differential equations: fuzzy initial conditions. Soft Comput 6:415–421
Chalco-Cano Y, Romn-Flores H (2006) On new solutions of fuzzy differential Buckley and Jowers equations. Chaos Solitons Fract: 1016–1043
Chalco-Cano Y, Romn-Flores, Rojas-Medar MA, Saavedra O, Jimnez-Gamero M (2007) The extension principle and a decomposition of fuzzy sets. Inf Sci 177:5394–5403
Chalco-Cano Y, Roman-Flores H, Jimnez-Gamero MD (2011) Generalized derivative and \(\pi \)-derivative for set-valued functions. Inf Sci 181:2177–2188
Chen CK, Ho SH (1999) Solving partial differential equations by two-dimensional differential transform method. Appl Math Comput 106:171–179
Cho YJ, Lan HY (2007) The existence of solutions for the nonlinear first order fuzzy differential equations with discontinuous conditions. Dyn Contin Discrete 14:873–884
Congxin W, Shiji S (1993) Exitance theorem to the Cauchy problem of fuzzy differential equations under compactance-type conditions. Inf Sci 108:123–134
Datta DP (2003) The golden mean, scale free extension of real number system, fuzzy sets and \(1/f\) spectrum in physics and biology. Chaos Solitons Fract 17:781–788
Diamond P (1999) Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations. IEEE Trans Fuzzy Syst 7:734–740
Diamond P (2002) Brief note on the variation of constants formula for fuzzy differential equations. Fuzzy Set Syst 129:65–71
Ding Z, Ma M, Kandel A (1997) Existence of solutions of fuzzy differential equations with parameters. Inf Sci 99:205–217
Dubois D, Prade H (1980) Theory and application, fuzzy sets and systems. Academic Press, New York
Dubois D, Prade H (1982) Towards fuzzy differential calculus: Part 3, differentiation. Fuzzy Set Syst 8:225–233
El Naschie MS (2005) From experimental quantum optics to quantum gavity via a fuzzy Kahler manifold. Chaos Solitons Fract 25: 969–977
Fard OS (2009a) A numerical scheme for fuzzy cauchy problems. J Uncertain Syst 3:307–314
Fard OS (2009b) An iterative scheme for the solution of generalized system of linear fuzzy differential equations. World Appl Sci J 7:1597–1604
Fard OS, Kamyad AV (2011) Modified k-step method for solving fuzzy initial value problems. Iran J Fuzzy Syst 8(3):49–63
Fard OS, Hadi Z, Ghal-Eh N, Borzabadi AH (2009) A note on iterative method for solving fuzzy initial value problems. J Adv Res Sci Comput 1:22–33
Fard OS, Bidgoli TA, Borzabadi AH (2010) Approximate-analytical approach to nonlinear FDEs under generalized differentiability. J Adv Res Dyn Control Syst 2:56–74
Fei W (2007) Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients. Inf Sci 177:329–4337
Feng G, Chen G (2005) Adaptative control of discrete-time chaotic system: a fuzzy control approach. Chaos Solitons Fract 23:459–467
Goetschel R, Voxman W (1986) Elementary calculus. Fuzzy Sets Syst 18:31–43
Guo M, Xue X, Li R (2003) Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets Syst 138:601–615
Jang MJ, Chen CL, Liy YC (2000) On solving the initial-value problems using the differential transformation method. Appl Math Comput 115:145–160
Jiang W, Guo-Dong Q, Bin D (2005) \(H_{\infty }\) variable universe adaptive fuzzy control for chaotic systems. Chaos Solitons Fract 24:1075–1086
Jowers LJ, Buckley JJ, Reilly KD (2007) Simulating continuous fuzzy systems. Inf Sci 177:436–448
Kaleva O (1987) Fuzzy differential equations. Fuzzy Set Syst 24:301–317
Kaleva O (1990) The Cauchy problem for fuzzy differential equations. Fuzzy Set Syst 35:389–396
Kaleva O (2006) A note on fuzzy differential equations. Nonlinear Anal 64:895–900
Kauffman A, Gupta MM (1991) Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York
Lopez RR (2008) Comparison results for fuzzy differential equations. Inf Sci 178:1756–1779
Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy differential equations. Fuzzy Set Syst 105:133–138
Mizukoshi MT, Barros LC, Chalco-Cano Y, Romn-Flores H, Bassanezi RC (2007) Fuzzy differential equations and the extension principle. Inf Sci 177:3627–3635
Nguyen HT (1978) A note on the extension principle for fuzzy sets. J Math Anal Appl 64:369–380
Oberguggenberger M, Pittschmann S (1999) Differential equations with fuzzy parameters. Math Modul Syst 5:181–202
Papaschinopoulos G, Stefanidou G, Efraimidi P (2007) Existence uniqueness and asymptotic behavior of the solutions of a fuzzy differential equation with piecewise constant argument. Inf Sci 177:3855–3870
Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552–558
Puri ML, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114:409–422
Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Set Syst 24:319–330
Song S, Guo L, Feng C (2000) Global existence of solutions to fuzzy differential equations. Fuzzy Set Syst 115:371–376
Solaymani Fard O, Ghal-Eh N (2011) Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients. Inf Sci 181:4765–4779
Zadeh LA (1965) Fuzzy sets. Inf. Control 8:338–353
Zimmermann HJ (1991) Fuzzy sets theory and its applications. Kluwer Academic Press, Dordrecht
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Communicated by G. Acampora.
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Allahviranloo, T., Abbasbandy, S. & Behzadi, S.S. Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method. Soft Comput 18, 2191–2200 (2014). https://doi.org/10.1007/s00500-013-1193-5
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DOI: https://doi.org/10.1007/s00500-013-1193-5