A Hybrid Modified Approach for Solving Fuzzy Differential Equations
DOI:
https://doi.org/10.24203/ajfam.v5i2.4478Keywords:
Fuzzy differential equation, Laplace transformation, Variational iteration methods, Lagrange multiplierAbstract
In this paper, a hybrid method is presented by combining Laplace transformation and variational iteration method for solving fuzzy differential equations with fuzzy initial and boundaries values. A defuzzification technique had been implemented to convert the fuzzy parameters into crisp values by building an extended ranking method. Then the method is implemented on the new problem in which two approaches has been built according to the formula of Lagrange multiplier obtaining the lower, upper and center solutions.
References
L. A. Zadeh, “Fuzzy sets,†Inf. Control, vol. 8, no. 3, pp. 338–353, Jun. 1965.
O. Kaleva, “Fuzzy differential equations,†Fuzzy Sets Syst., vol. 24, no. 3, pp. 301–317, Dec. 1987.
A. Khastan and J. J. Nieto, “A boundary value problem for second order fuzzy differential equations,†Nonlinear Anal. Theory Methods Appl., vol. 72, no. 9–10, pp. 3583–3593, May 2010.
L. Ahmad, M. Farooq, S. Ullah, and S. Abdullah, “Solving fuzzy two-point boundary value problem using fuzzy Laplace transform,†ArXiv Prepr. ArXiv14030571, 2014.
L. Ahmad, M. Farooq, and S. Abdullah, “Solving nth order fuzzy differential equation by fuzzy Laplace transform,†ArXiv14030242 Math, Mar. 2014.
M. Ghanbari, “Numerical solution of fuzzy initial value problems under generalized di erentiability by HPM,†Int. J. Ind. Math., vol. 1, no. 1, pp. 19–39, 2009.
E. Babolian, H. Sadeghi, and S. Javadi, “Numerically solution of fuzzy differential equations by Adomian method,†Appl. Math. Comput., vol. 149, no. 2, pp. 547–557, Feb. 2004.
G. Adomian, “A review of the decomposition method in applied mathematics,†J. Math. Anal. Appl., vol. 135, no. 2, pp. 501–544, Nov. 1988.
T. Allahviranloo, S. Khezerloo, and M. Mohammadzaki, “Numerical solution for differential inclusion by adomian decomposition method,†J. Appl. Math., vol. 5, no. 17, pp. 51–62, 2008.
J.-H. He, “Variational iteration method – a kind of non-linear analytical technique: some examples,†Int. J. Non-Linear Mech., vol. 34, no. 4, pp. 699–708, Jul. 1999.
J.-H. He, “Some asymptotic methods for strongly nonlinear equations,†Int. J. Mod. Phys. B, vol. 20, no. 10, pp. 1141–1199, Apr. 2006.
H. Jafari, M. Saeidy, and D. Baleanu, “The variational iteration method for solving n-th order fuzzy differential equations,†Cent. Eur. J. Phys., vol. 10, no. 1, pp. 76–85, Feb. 2012.
D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications. Academic Press, 1980.
P. Diamond and P. Kloeden, “Metric Topology of Fuzzy Numbers and Fuzzy Analysis,†in Fundamentals of Fuzzy Sets, D. Dubois and H. Prade, Eds. Springer US, 2000, pp. 583–641.
O. S. Fard, “An iterative scheme for the solution of generalized system of linear fuzzy differential equations,†World Appl. Sci. J., vol. 7, no. 12, pp. 1597–1604, 2009.
N. A. Alaulden and M. Y. Sanar, “Solving Fuzzy Network Problems by Defuzzification Techniques,†Int. J. Innov. Res. Sci. Eng. Technol., 2014.
R. R. Yager, “A procedure for ordering fuzzy subsets of the unit interval,†Inf. Sci., vol. 24, no. 2, pp. 143–161, Jul. 1981.
H. Jafari and A. Alipoor, “A new method for calculating general lagrange multiplier in the variational iteration method,†Numer. Methods Partial Differ. Equ., vol. 27, no. 4, pp. 996–1001, Jul. 2011.
G.-C. Wu and D. Baleanu, “Variational iteration method for fractional calculus - a universal approach by Laplace transform,†Adv. Differ. Equ., vol. 2013, no. 1, p. 18, Dec. 2013.
G.-C. Wu, “Challenge in the variational iteration method – A new approach to identification of the Lagrange multipliers,†J. King Saud Univ. - Sci., vol. 25, no. 2, pp. 175–178, Apr. 2013.
H. Jafari, M. Saeidy, and D. Baleanu, “The variational iteration method for solving n-th order fuzzy differential equations,†Open Phys., vol. 10, no. 1, pp. 76–85, 2011.
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