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Research Papers

An Efficient Numerical Simulation for Solving Dynamical Systems With Uncertainty

[+] Author and Article Information
Ali Ahmadian

Department of Mathematics,
Faculty of Science,
University Putra Malaysia,
Serdang 43400 UPM, Selangor, Malaysia
e-mail: ahmadian.hosseini@gmail.com

Soheil Salahshour

Young Researchers and Elite Club,
Mobarakeh Branch,
Islamic Azad University,
Mobarakeh 19166, Iran
e-mail: soheilsalahshour@yahoo.com

Chee Seng Chan

Centre of Image and Signal Processing,
Faculty of Computer Science and
Information Technology,
University of Malaya,
Kuala Lumpur 50603, Malaysia
e-mail: cs.chan@um.edu.my

Dumitur Baleanu

Department of Mathematics and
Computer Science,
Cankaya University,
Balgat 06530, Ankara, Turkey;
Institute of Space Sciences,
Magurele-Bucharest R 76911, Romania
e-mail: dumitru@cankaya.edu.tr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 9, 2016; final manuscript received March 27, 2017; published online May 4, 2017. Assoc. Editor: Corina Sandu.

J. Comput. Nonlinear Dynam 12(5), 051008 (May 04, 2017) (13 pages) Paper No: CND-16-1186; doi: 10.1115/1.4036419 History: Received April 09, 2016; Revised March 27, 2017

In a wide range of real-world physical and dynamical systems, precise defining of the uncertain parameters in their mathematical models is a crucial issue. It is well known that the usage of fuzzy differential equations (FDEs) is a way to exhibit these possibilistic uncertainties. In this research, a fast and accurate type of Runge–Kutta (RK) methods is generalized that are for solving first-order fuzzy dynamical systems. An interesting feature of the structure of this technique is that the data from previous steps are exploited that reduce substantially the computational costs. The major novelty of this research is that we provide the conditions of the stability and convergence of the method in the fuzzy area, which significantly completes the previous findings in the literature. The experimental results demonstrate the robustness of our technique by solving linear and nonlinear uncertain dynamical systems.

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Figures

Grahic Jump Location
Fig. 1

Fuzzy exact and approximate solutions of Example 4.1: (−O) Exact,(×) ARK4,(*) ARK4−4,(+) RK4,(□) ERK4

Grahic Jump Location
Fig. 2

Fuzzy approximate solution using the ARK4-4 method, Example 4.1

Grahic Jump Location
Fig. 3

Fuzzy exact and approximate solutions of Example 4.2: (−O) Exact,(×) ARK4,(*) ARK4−4,(+) RK4,(◻) ERK4

Grahic Jump Location
Fig. 4

Fuzzy approximate solution using the ARK4-4 method, Example 4.2

Grahic Jump Location
Fig. 5

Fuzzy exact and approximate solutions of Example 4.3: (×) ARK4,(−O) ARK4−4,(+) RK4

Grahic Jump Location
Fig. 6

Fuzzy approximate solution using the ARK4-4 method, Example 4.3

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