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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 3, 2011

The variational iteration method for solving n-th order fuzzy differential equations

Hossein Jafari, Mohammad Saeidy and Dumitru Baleanu
From the journal Open Physics

Abstract

The variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

[1] A. Kandel, In: P. P. Wang, S. K. Chang (Eds.), Fuzzy sets theory and application to policy analysis and information systems (Plenum Press, New York, 1980) 93 Search in Google Scholar

[2] A. Kandel, W. J. Byatt, Proceedings of the international conference on cybernetics and society, Nov. 1978, Tokyo, 1213 Search in Google Scholar

[3] M. A. Abdou, A. A. Soliman, J. Comput. Appl. Math. 181, 245 (2005) http://dx.doi.org/10.1016/j.cam.2004.11.03210.1016/j.cam.2004.11.032Search in Google Scholar

[4] M. A. Abdou, A. A. Soliman, Physica D 211, 1 (2005) http://dx.doi.org/10.1016/j.physd.2005.08.00210.1016/j.physd.2005.08.002Search in Google Scholar

[5] S. Momani, S. Abuasad, Chaos, Soliton. Fract. 27, 119 (2006) http://dx.doi.org/10.1016/j.chaos.2005.04.11310.1016/j.chaos.2005.04.113Search in Google Scholar

[6] H. T. Nguyen, J. Math. Anal. Appl. 64, 369 (1978) http://dx.doi.org/10.1016/0022-247X(78)90045-810.1016/0022-247X(78)90045-8Search in Google Scholar

[7] J. H. He, J. Comput. Appl. Math. 207, 3 (2007) http://dx.doi.org/10.1016/j.cam.2006.07.00910.1016/j.cam.2006.07.009Search in Google Scholar

[8] J. H. He, Chaos Soliton. Fract. 19, 847 (2004) http://dx.doi.org/10.1016/S0960-0779(03)00265-010.1016/S0960-0779(03)00265-0Search in Google Scholar

[9] J. H. He, Appl. Math. Comput. 114, 115 (2000) http://dx.doi.org/10.1016/S0096-3003(99)00104-610.1016/S0096-3003(99)00104-6Search in Google Scholar

[10] J. H. He, Int. J. Mod. Phys. B 20, 1141 (2006) http://dx.doi.org/10.1142/S021797920603379610.1142/S0217979206033796Search in Google Scholar

[11] S. Q. Wang, J. H. He, Phys. Lett. A 367, 188 (2007) http://dx.doi.org/10.1016/j.physleta.2007.02.04910.1016/j.physleta.2007.02.049Search in Google Scholar

[12] M. Inokuti et al., In: S. Nemat-Nasser (Ed.), Variational method in the mechanics of solids (Pergamon Press, Oxford, 1978) 156 Search in Google Scholar

[13] A. M. Wazwaz, Comput. Math. Appl. 54, 926 (2007) http://dx.doi.org/10.1016/j.camwa.2006.12.03810.1016/j.camwa.2006.12.038Search in Google Scholar

[14] A. M. Wazwaz, J. Comput. Appl. Math. 207, 18 (2007) http://dx.doi.org/10.1016/j.cam.2006.07.01010.1016/j.cam.2006.07.010Search in Google Scholar

[15] B. Bede, I. J. Rudas, A. L. Bencsik, Inform. Sciences 177, 1648 (2007) http://dx.doi.org/10.1016/j.ins.2006.08.02110.1016/j.ins.2006.08.021Search in Google Scholar

[16] D. Dubois, H. Prade, Fuzzy Set. Syst. 8, 225 (1982) http://dx.doi.org/10.1016/S0165-0114(82)80001-810.1016/S0165-0114(82)80001-8Search in Google Scholar

[17] J. Y. Park, Y. C. Kwan, J. V. Jeong, Fuzzy Set. Syst. 72, 373 (1995) http://dx.doi.org/10.1016/0165-0114(94)00296-J10.1016/0165-0114(94)00296-JSearch in Google Scholar

[18] J. J. Buckley, T. Feuring, Fuzzy Set. Syst. 121, 247 (2001) http://dx.doi.org/10.1016/S0165-0114(00)00028-210.1016/S0165-0114(00)00028-2Search in Google Scholar

[19] J. J. Buckley, Simulating continuous fuzzy systems (Springer, Heildelberg, 2006) Search in Google Scholar

[20] J. J. Buckley, T. Feuring, Fuzzy Set. Syst. 105, 241 (1999) http://dx.doi.org/10.1016/S0165-0114(98)00323-610.1016/S0165-0114(98)00323-6Search in Google Scholar

[21] J. J. Buckley, T. Feuring, Fuzzy Set. Syst. 110, 43 (2000) http://dx.doi.org/10.1016/S0165-0114(98)00141-910.1016/S0165-0114(98)00141-9Search in Google Scholar

[22] L. J. Jowers, J. J. Buckley, K. D. Reilly, Inform. Sciences 177, 436 (2007) http://dx.doi.org/10.1016/j.ins.2006.03.00510.1016/j.ins.2006.03.005Search in Google Scholar

[23] O. Kaleva, Fuzzy Set. Syst. 35, 389 (1990) http://dx.doi.org/10.1016/0165-0114(90)90010-410.1016/0165-0114(90)90010-4Search in Google Scholar

[24] O. Kaleva, Fuzzy Set. Syst. 56, 297 (1993) http://dx.doi.org/10.1016/0165-0114(93)90209-Z10.1016/0165-0114(93)90209-ZSearch in Google Scholar

[25] Z. Ding, M. Ma, A. Kandel, Inform. Sciences 99, 205 (1997) http://dx.doi.org/10.1016/S0020-0255(96)00279-410.1016/S0020-0255(96)00279-4Search in Google Scholar

[26] O. He, W. Yi, Fuzzy Set. Syst. 24, 321 (1989). http://dx.doi.org/10.1016/0165-0114(89)90264-910.1016/0165-0114(89)90264-9Search in Google Scholar

[27] O. Kaleva, Fuzzy Set. Syst. 24, 301 (1987) http://dx.doi.org/10.1016/0165-0114(87)90029-710.1016/0165-0114(87)90029-7Search in Google Scholar

[28] P. Kloeden, Fuzzy Set. Syst. 44, 161 (1991) http://dx.doi.org/10.1016/0165-0114(91)90041-N10.1016/0165-0114(91)90041-NSearch in Google Scholar

[29] R. Goetschel, W. Voxman, Fuzzy Set. Syst. 18, 31 (1986) http://dx.doi.org/10.1016/0165-0114(86)90026-610.1016/0165-0114(86)90026-6Search in Google Scholar

[30] S. Abbasbandy, T. Allahviranloo, Comput. Meth. Appl. Math. 2, 113 (2002) 10.2478/cmam-2002-0006Search in Google Scholar

[31] S. Seikkala, Fuzzy Set. Syst. 24, 319 (1987) http://dx.doi.org/10.1016/0165-0114(87)90030-310.1016/0165-0114(87)90030-3Search in Google Scholar

[32] S. L. Chang, L. A. Zadeh? IEEE T. Syst. Man Cy. 2, 330 (1972) 10.1109/TSMC.1972.5408553Search in Google Scholar

[33] T. Allahviranloo, E. Ahmady, N. Ahmady, Inform. Sciences 178, 1309 (2008) http://dx.doi.org/10.1016/j.ins.2007.10.01310.1016/j.ins.2007.10.013Search in Google Scholar

[34] T. Allahviranloo, N. Ahmady, E. Ahmady, Inform. Sciences 177, 1633 (2007) http://dx.doi.org/10.1016/j.ins.2006.09.01510.1016/j.ins.2006.09.015Search in Google Scholar

[35] W. Congxin, S. Shiji, Inform. Sciences 108, 123 (1998) http://dx.doi.org/10.1016/S0020-0255(97)10064-010.1016/S0020-0255(97)10064-0Search in Google Scholar

[36] W. Menda, J. Fuzzy Syst. Math. 2, 51 (1988) (in Chinese) Search in Google Scholar

[37] H. Jafari, A. Alipoor, Numer. Meth. Part. D. E. 27, 996 (2011) http://dx.doi.org/10.1002/num.2056710.1002/num.20567Search in Google Scholar

[38] A. M. Wazwaz, Int. J. Comput. Math. 87, 1131 (2010) http://dx.doi.org/10.1080/0020716090312496710.1080/00207160903124967Search in Google Scholar

[39] S. Abbasbandy, T. Allahviranloo, P. Darabi, O. Sedaghatfar, Math. Comput. Appl. 16, 819 (2011) 10.3390/mca16040819Search in Google Scholar

Published Online: 2011-12-3
Published in Print: 2012-2-1

© 2011 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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