Artificial neural network approach for solving fuzzy differential equations

Effati, Sohrab; Pakdaman, Morteza

doi:10.1016/j.ins.2009.12.016

Cited by 113 publications

(41 citation statements)

References 25 publications

(12 reference statements)

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“…Definition (7), [18]: Let I be a real interval. The r-level set of the fuzzy function y ∶ I → E 1 can be denoted by :…”

Section: Definition ( ) [19]: Fuzzy Functionmentioning

confidence: 99%

“…Definition (8), [18]: let u, v ∈ E 1 . If there exist w∈ E 1 such that u= v + w, then w is called the H-difference (Hukuhara-difference) of u, v and it is denoted by w= u ⊝ v.…”

Section: Definition ( ) [19]: Fuzzy Functionmentioning

confidence: 99%

“…In (2010) Effati and pakdaman [18] used ANN for solving FDE, they used for the first time the ANN to approximate fuzzy initial value problems. In (2012) Mosleh, Otadi [19] used ANN for solving fuzzy Fredholm integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Artificial Neural Network for Solving Fuzzy Differential Equations under Generalized H – Derivation

Suhhiem¹

2017

IJPDEA

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The aim of this work is to present a novel approach based on the artificial neural network for finding the numerical solution of first order fuzzy differential equations under generalized H-derivation. The differentiability concept used in this paper is the generalized differentiability since a fuzzy differential equation under this differentiability can have two solutions. The fuzzy trial solution of fuzzy initial value problem is written as a sum of two parts. The first part satisfies the fuzzy condition, it contains no adjustable parameters. The second part involves feed-forward neural networks containing adjustable parameters. Under some conditions the proposed method provides numerical solutions with high accuracy.

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“…Definition (7), [18]: Let I be a real interval. The r-level set of the fuzzy function y ∶ I → E 1 can be denoted by :…”

Section: Definition ( ) [19]: Fuzzy Functionmentioning

confidence: 99%

“…Definition (8), [18]: let u, v ∈ E 1 . If there exist w∈ E 1 such that u= v + w, then w is called the H-difference (Hukuhara-difference) of u, v and it is denoted by w= u ⊝ v.…”

Section: Definition ( ) [19]: Fuzzy Functionmentioning

confidence: 99%

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Artificial Neural Network for Solving Fuzzy Differential Equations under Generalized H – Derivation

Suhhiem¹

2017

IJPDEA

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“…There are few works on FDE. [31] suggests a static neural network to solve FDE. Since the structure of the neural network is not suitable for FDE, the approximation accuracy is poor.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

Jafari¹,

Yu²,

Li³

et al. 2017

IJCIS

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The uncertain nonlinear systems can be modeled with fuzzy equations or fuzzy differential equations (FDEs) by incorporating the fuzzy set theory. The solutions of them are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs.In this paper, the solutions of FDEs are approximated by two types of Bernstein neural networks. Here, the uncertainties are in the sense of Z-numbers. Initially the FDE is transformed into four ordinary differential equations (ODEs) with Hukuhara differentiability. Then neural models are constructed with the structure of ODEs. With modified back propagation method for Z-number variables, the neural networks are trained. The theory analysis and simulation results show that these new models, Bernstein neural networks, are effective to estimate the solutions of FDEs based on Z-numbers.

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“…One of the important topics in fuzzy mathematics and its applications is to solve fuzzy differential equations (FDEs) based on the definition of fuzzy derivative [7], [14], [15], [26], [27], [30]. One of the fundamental difference schemes to solve FDEs is Runge-Kutta methods which play the important role in numerical methods of solving ordinary differential equations [8], [16], [19], [21], [23], [36], [38].…”

Section: Introductionmentioning

confidence: 99%

Dynamical control of accuracy in the fuzzy Runge-Kutta methods to estimate the solution of a fuzzy differential equation

Araghi¹,

Kelishami²

2016

Journal of Fuzzy Set Valued Analysis

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In this paper, a reliable scheme is proposed to solve fuzzy differential equations by fuzzy Runge-Kutta method of order m. For this purpose, the stochastic arithmetic and CESTAC method are applied to validate the results. In order to implement the C++ codes, the CADNA library is used. In this case, the optimal step size is found. The examples illustrate the efficiency and importance of using the stochastic arithmetic in place of the floating-point arithmetic.

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Artificial neural network approach for solving fuzzy differential equations

Cited by 113 publications

References 25 publications

Artificial Neural Network for Solving Fuzzy Differential Equations under Generalized H – Derivation

Artificial Neural Network for Solving Fuzzy Differential Equations under Generalized H – Derivation

Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

Dynamical control of accuracy in the fuzzy Runge-Kutta methods to estimate the solution of a fuzzy differential equation

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