Fuzzy differential equation is an important tool for studying dynamic system in fuzzy environment. Therefore, finding the solutions to these equations is essential. In order to make more fuzzy differential equations easier to solve, variables substitution is performed in nonlinear fuzzy differential equation to judge whether the equation is reducible. In this process, we try to find out the method to reduce a nonlinear fuzzy differential equation to a linear fuzzy differential equation. In the last, the conditions of two types of fuzzy differential equations being reducible are given. Therefore, it becomes easy to distinguish reducible fuzzy differential equations and solve them. In addition, some examples are given to verify the effectiveness of the results.
JI Liye
,
YOU Cuilian
. Reducibility conditions for fuzzy differential equation[J]. Journal of Hebei University(Natural Science Edition), 2020
, 40(4)
: 337
-343
.
DOI: 10.3969/j.issn.1000-1565.2020.04.001
[1] ZADEH L A. Fuzzy set [J]. Information and Control,1965, 8(3):338-353. DOI:10.1016/S0019-9958(65)9024 1-X.
[2] ZADEH L A. Fuzzy sets as a basis for a theory of possibility [J]. Fuzzy Sets and Systems,1978,1(1): 3-28.DOI:10.1016/0165-0114(78)90029-5.
[3] LIU B D, LIU Y K. Expected value of fuzzy variable and fuzzy expected value models [J]. IEEE Transactions on Fuzzy Systems,2002,10(4):445-450. DOI:10.1109/TFUZZ.2002.800692.
[4] LIU B D. Uncertainty Theory[M]. Berlin:Springer-Verlag, 2004.
[5] LI X, LIU B D. A sufficient and necessary condition for credibility measures[J]. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems,2006,14(5):527-535.DOI:10.1142/S0218488506004175.
[6] LIU B D. Uncertainty Theory[M]. 2nd ed, Berlin: Springer-Verlag,2007.
[7] LIU B D. Fuzzy process, hybrid process and uncertain process[J]. Journal of Uncertain Systems,2008, 2(1):3-16.
[8] QIN Z F, WEN M L. On analytic functions of complex Liu process[J]. Journal of Intelligent and Fuzzy Systems,2015,28(4):1627-1633. DOI:10.3233/IFS-141448.
[9] YOU C L, WANG W Q. Some properties of complex fuzzy integral[J]. Mathematical Problems in Engineering,2015(Article ID 290539):7 pages. DOI:10.1155/2015/290539.
[10] YOU C L, HUO H E, WANG W Q. Multi-dimensional Liu process, differential and integral[J]. East Asian Mathematical Journal,2013,29(1):13-22. DOI:10.7858/eamj.2013.002.
[11] YOU C L, MA H Y, HUO H E. A new kind of generalized fuzzy integrals[J]. Journal of Nonlinear Science and Applications,2016, 3(9):1396-1402. DOI:10.22436/jnsa.009.03.63.
[12] KALEVA O. Fuzzy differential equations[J]. Fuzzy Sets and Systems,1987, 24(3):301-317. DOI:10.1016/016-50114(87)90029-7.
[13] ABBASBANDY S, ALLAH V T. Numerical solutions of fuzzy differential equations by Taylor method[J]. Computational Methods in Applied Mathematics, 2002,2(2):113-114. DOI:10.2478/cmam-2002-0006.
[14] DURAISAMY C, USHA B. Another approach to solution of fuzzy differential equations by modified Eulers method[C] //International Conference on Communication and Computational Intelligence, 2010.
[15] AHMAD M Z, HASAN M K. A new fuzzy version of Eulers method for solving differential equations with fuzzy initial values[J]. Sains Malaysiana, 2011,40(6):651-657.
[16] HOOSHANGIAN L. Analytic solution of fuzzy second order differential equations under H-derivation[J].Turkish Journal of Fuzzy Systems, 2016,7(1):55-67.
[17] ALTAIE S A, JAMEEL A F, SAABAN A. Homotopy perturbation method approximate analytical solution of fuzzy partial differential equation[J]. IAENG International Journal of Applied Mathematics,2019, 49(1):22-28.
[18] YOU C L, WANG W Q, HUO H E. Existence and uniqueness theorems for fuzzy differential equations[J]. Journal of Uncertain Systems, 2013, 7(4):303-315.
[19] YOU C L, HAO Y Y. Fuzzy Euler approximation and its local convergence[J]. Journal of Computational and Applied Mathematics, 2018,343:55-61.DOI:10.1016/j.cam.2018.04.031.
[20] 尤翠莲,郝杨阳. 基于泰勒展开式的模糊微分方程数值解法[J]. 河北大学学报(自然科学版),2018,38(2):113-118. DOI:10.3969/j.issn.1000-1565.2018.02.001.
[21] CHENG Y, YOU C L. Convergence of numerical methods for fuzzy differential equations[J]. Journal of Intelligent and Fuzzy Systems, 2020,38(4):5257-5266.DOI:10.3233/JIFS-191856.
