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Erschienen in: Soft Computing 11/2014

01.11.2014 | Methodologies and Application

Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method

verfasst von: T. Allahviranloo, S. Abbasbandy, Sh. S. Behzadi

Erschienen in: Soft Computing | Ausgabe 11/2014

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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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Literatur
Zurück zum Zitat Abbasbandy S, Allahviranloo T (2002) Numerical solutions of fuzzy differential equations by Taylor method. J Comput Methods Appl Math 2:113–124MathSciNet Abbasbandy S, Allahviranloo T (2002) Numerical solutions of fuzzy differential equations by Taylor method. J Comput Methods Appl Math 2:113–124MathSciNet
Zurück zum Zitat Abbasbandy S, Allahviranloo T, Lopez-Pouso O, Nieto JJ (2004) Numerical methods for fuzzy differential inclusions. Comput Math Appl 48:1633–1641MathSciNetCrossRefMATH Abbasbandy S, Allahviranloo T, Lopez-Pouso O, Nieto JJ (2004) Numerical methods for fuzzy differential inclusions. Comput Math Appl 48:1633–1641MathSciNetCrossRefMATH
Zurück zum Zitat Abbasbandy S, Nieto JJ, Alavi M (2005) Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos Solitons Fract 26:1337MathSciNetCrossRefMATH Abbasbandy S, Nieto JJ, Alavi M (2005) Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos Solitons Fract 26:1337MathSciNetCrossRefMATH
Zurück zum Zitat Abbod MF, Von Keyserlingk DG, Linkens DA, Mahfouf M (2001) Survey of utilisation of fuzzy technology in medicine and healthcare. Fuzzy Sets Syst 120:331–349CrossRef Abbod MF, Von Keyserlingk DG, Linkens DA, Mahfouf M (2001) Survey of utilisation of fuzzy technology in medicine and healthcare. Fuzzy Sets Syst 120:331–349CrossRef
Zurück zum Zitat Allahviramloo T (2005) The Adomian decomposition method for fuzzy system of linear equations. Appl Math Comput 163:553– 563 Allahviramloo T (2005) The Adomian decomposition method for fuzzy system of linear equations. Appl Math Comput 163:553– 563
Zurück zum Zitat Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equations by predictor–corrector method. Inf Sci 177:1633–1647MathSciNetCrossRefMATH Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equations by predictor–corrector method. Inf Sci 177:1633–1647MathSciNetCrossRefMATH
Zurück zum Zitat Bede B (2008) Note on numerical solutions of fuzzy differential equations by predictor–corrector method. Inf Sci 178:1917– 1922 Bede B (2008) Note on numerical solutions of fuzzy differential equations by predictor–corrector method. Inf Sci 178:1917– 1922
Zurück zum Zitat Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equation. Fuzzy Set Syst 151:581–599MathSciNetCrossRefMATH Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equation. Fuzzy Set Syst 151:581–599MathSciNetCrossRefMATH
Zurück zum Zitat Bede B, Imre J, Rudas C, Attila L (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:3627–3635CrossRef Bede B, Imre J, Rudas C, Attila L (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:3627–3635CrossRef
Zurück zum Zitat Buckley JJ, Jowers LJ (2006) Simulating continuous fuzzy systems. Springer, BerlinMATH Buckley JJ, Jowers LJ (2006) Simulating continuous fuzzy systems. Springer, BerlinMATH
Zurück zum Zitat Buckley JJ, Feuring T, Hayashi Y (2002) Linear systems of first order ordinary differential equations: fuzzy initial conditions. Soft Comput 6:415–421 Buckley JJ, Feuring T, Hayashi Y (2002) Linear systems of first order ordinary differential equations: fuzzy initial conditions. Soft Comput 6:415–421
Zurück zum Zitat 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 H (2006) On new solutions of fuzzy differential Buckley and Jowers equations. Chaos Solitons Fract: 1016–1043
Zurück zum Zitat 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, 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
Zurück zum Zitat Chalco-Cano Y, Roman-Flores H, Jimnez-Gamero MD (2011) Generalized derivative and \(\pi \)-derivative for set-valued functions. Inf Sci 181:2177–2188CrossRefMATH Chalco-Cano Y, Roman-Flores H, Jimnez-Gamero MD (2011) Generalized derivative and \(\pi \)-derivative for set-valued functions. Inf Sci 181:2177–2188CrossRefMATH
Zurück zum Zitat Chen CK, Ho SH (1999) Solving partial differential equations by two-dimensional differential transform method. Appl Math Comput 106:171–179MathSciNetCrossRefMATH Chen CK, Ho SH (1999) Solving partial differential equations by two-dimensional differential transform method. Appl Math Comput 106:171–179MathSciNetCrossRefMATH
Zurück zum Zitat Cho YJ, Lan HY (2007) The existence of solutions for the nonlinear first order fuzzy differential equations with discontinuous conditions. Dyn Contin Discrete 14:873–884MathSciNetMATH Cho YJ, Lan HY (2007) The existence of solutions for the nonlinear first order fuzzy differential equations with discontinuous conditions. Dyn Contin Discrete 14:873–884MathSciNetMATH
Zurück zum Zitat Congxin W, Shiji S (1993) Exitance theorem to the Cauchy problem of fuzzy differential equations under compactance-type conditions. Inf Sci 108:123–134CrossRef Congxin W, Shiji S (1993) Exitance theorem to the Cauchy problem of fuzzy differential equations under compactance-type conditions. Inf Sci 108:123–134CrossRef
Zurück zum Zitat 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 17:781–788CrossRefMATH 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 17:781–788CrossRefMATH
Zurück zum Zitat Diamond P (1999) Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations. IEEE Trans Fuzzy Syst 7:734–740MathSciNetCrossRef Diamond P (1999) Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations. IEEE Trans Fuzzy Syst 7:734–740MathSciNetCrossRef
Zurück zum Zitat Diamond P (2002) Brief note on the variation of constants formula for fuzzy differential equations. Fuzzy Set Syst 129:65–71MathSciNetCrossRefMATH Diamond P (2002) Brief note on the variation of constants formula for fuzzy differential equations. Fuzzy Set Syst 129:65–71MathSciNetCrossRefMATH
Zurück zum Zitat Dubois D, Prade H (1980) Theory and application, fuzzy sets and systems. Academic Press, New York Dubois D, Prade H (1980) Theory and application, fuzzy sets and systems. Academic Press, New York
Zurück zum Zitat El Naschie MS (2005) From experimental quantum optics to quantum gavity via a fuzzy Kahler manifold. Chaos Solitons Fract 25: 969–977CrossRefMATH El Naschie MS (2005) From experimental quantum optics to quantum gavity via a fuzzy Kahler manifold. Chaos Solitons Fract 25: 969–977CrossRefMATH
Zurück zum Zitat Fard OS (2009a) A numerical scheme for fuzzy cauchy problems. J Uncertain Syst 3:307–314 Fard OS (2009a) A numerical scheme for fuzzy cauchy problems. J Uncertain Syst 3:307–314
Zurück zum Zitat 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 (2009b) An iterative scheme for the solution of generalized system of linear fuzzy differential equations. World Appl Sci J 7:1597–1604
Zurück zum Zitat Fard OS, Kamyad AV (2011) Modified k-step method for solving fuzzy initial value problems. Iran J Fuzzy Syst 8(3):49–63MathSciNetMATH Fard OS, Kamyad AV (2011) Modified k-step method for solving fuzzy initial value problems. Iran J Fuzzy Syst 8(3):49–63MathSciNetMATH
Zurück zum Zitat 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 1:22–33MathSciNet 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 1:22–33MathSciNet
Zurück zum Zitat Fard OS, Bidgoli TA, Borzabadi AH (2010) Approximate-analytical approach to nonlinear FDEs under generalized differentiability. J Adv Res Dyn Control Syst 2:56–74MathSciNet Fard OS, Bidgoli TA, Borzabadi AH (2010) Approximate-analytical approach to nonlinear FDEs under generalized differentiability. J Adv Res Dyn Control Syst 2:56–74MathSciNet
Zurück zum Zitat Fei W (2007) Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients. Inf Sci 177:329–4337CrossRef Fei W (2007) Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients. Inf Sci 177:329–4337CrossRef
Zurück zum Zitat Feng G, Chen G (2005) Adaptative control of discrete-time chaotic system: a fuzzy control approach. Chaos Solitons Fract 23:459–467MathSciNetCrossRefMATH Feng G, Chen G (2005) Adaptative control of discrete-time chaotic system: a fuzzy control approach. Chaos Solitons Fract 23:459–467MathSciNetCrossRefMATH
Zurück zum Zitat Guo M, Xue X, Li R (2003) Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets Syst 138:601–615MathSciNetCrossRefMATH Guo M, Xue X, Li R (2003) Impulsive functional differential inclusions and fuzzy population models. Fuzzy Sets Syst 138:601–615MathSciNetCrossRefMATH
Zurück zum Zitat Jang MJ, Chen CL, Liy YC (2000) On solving the initial-value problems using the differential transformation method. Appl Math Comput 115:145–160MathSciNetCrossRefMATH Jang MJ, Chen CL, Liy YC (2000) On solving the initial-value problems using the differential transformation method. Appl Math Comput 115:145–160MathSciNetCrossRefMATH
Zurück zum Zitat Jiang W, Guo-Dong Q, Bin D (2005) \(H_{\infty }\) variable universe adaptive fuzzy control for chaotic systems. Chaos Solitons Fract 24:1075–1086MathSciNetCrossRefMATH Jiang W, Guo-Dong Q, Bin D (2005) \(H_{\infty }\) variable universe adaptive fuzzy control for chaotic systems. Chaos Solitons Fract 24:1075–1086MathSciNetCrossRefMATH
Zurück zum Zitat Kauffman A, Gupta MM (1991) Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York Kauffman A, Gupta MM (1991) Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York
Zurück zum Zitat Lopez RR (2008) Comparison results for fuzzy differential equations. Inf Sci 178:1756–1779 Lopez RR (2008) Comparison results for fuzzy differential equations. Inf Sci 178:1756–1779
Zurück zum Zitat Mizukoshi MT, Barros LC, Chalco-Cano Y, Romn-Flores H, Bassanezi RC (2007) Fuzzy differential equations and the extension principle. Inf Sci 177:3627–3635CrossRefMATH Mizukoshi MT, Barros LC, Chalco-Cano Y, Romn-Flores H, Bassanezi RC (2007) Fuzzy differential equations and the extension principle. Inf Sci 177:3627–3635CrossRefMATH
Zurück zum Zitat Oberguggenberger M, Pittschmann S (1999) Differential equations with fuzzy parameters. Math Modul Syst 5:181–202MATH Oberguggenberger M, Pittschmann S (1999) Differential equations with fuzzy parameters. Math Modul Syst 5:181–202MATH
Zurück zum Zitat 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 177:3855–3870CrossRefMATH 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 177:3855–3870CrossRefMATH
Zurück zum Zitat Solaymani Fard O, Ghal-Eh N (2011) Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients. Inf Sci 181:4765–4779CrossRefMATH Solaymani Fard O, Ghal-Eh N (2011) Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients. Inf Sci 181:4765–4779CrossRefMATH
Zurück zum Zitat Zimmermann HJ (1991) Fuzzy sets theory and its applications. Kluwer Academic Press, DordrechtCrossRef Zimmermann HJ (1991) Fuzzy sets theory and its applications. Kluwer Academic Press, DordrechtCrossRef
Metadaten
Titel
Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method
verfasst von
T. Allahviranloo
S. Abbasbandy
Sh. S. Behzadi
Publikationsdatum
01.11.2014
Verlag
Springer Berlin Heidelberg
Erschienen in
Soft Computing / Ausgabe 11/2014
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-013-1193-5

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