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Symmetry 2018, 10(2), 42; doi:10.3390/sym10020042

Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives

1
Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia
2
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia
3
Department of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn Malaysia (Pagoh Campus), Muar 84000, Johor, Malaysia
*
Author to whom correspondence should be addressed.
Received: 31 October 2017 / Revised: 22 December 2017 / Accepted: 8 January 2018 / Published: 8 February 2018
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Abstract

In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper the method will be used to solve the fuzzy initial value problems where the initial value is a symmetric triangular fuzzy interval. The triangular fuzzy number is not necessarily symmetric, however by imposing symmetry the definition of a triangular fuzzy number can be simplified. The symmetric triangular fuzzy interval is a triangular fuzzy interval that has same left and right width of membership function from the center. Due to this, the parametric form of symmetric triangular fuzzy number is simple and the performing arithmetic operations become easier. In order to interpret the fuzzy problems, Seikkala’s derivative approach is implemented. Characterization theorem is then used to translate the problems into a system of ordinary differential equations. The convergence of the introduced method is also proved. Numerical examples are given to investigate the performance of the proposed method. It is clearly shown in the results that the proposed method is comparable and reliable in solving fuzzy differential equations. View Full-Text
Keywords: block method; fuzzy differential equations; predictor-corrector; Seikkala’s derivatives block method; fuzzy differential equations; predictor-corrector; Seikkala’s derivatives
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Isa, S.; Abdul Majid, Z.; Ismail, F.; Rabiei, F. Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives. Symmetry 2018, 10, 42.

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