Fuzzy processes

Kandel, Abraham; Byatt, W.J.

doi:10.1016/0165-0114(80)90032-9

Cited by 74 publications

(15 citation statements)

References 8 publications

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“…iii If f t ·f t < 0 and f t ·f t > 0, then F t is (2,1) we have the following i for 0 < t < 1: φ t · φ t < 0 and φ t · φ t < 0 then by iv , F t is 2-2 -differentiable and its second derivative,…”

Section: Theorem 315 Let F : I → R Be Twice Differentiable On I γ mentioning

confidence: 99%

New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability

Khastan

Bahrami

Ivaz

2009

Boundary Value Problems

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We firstly present a generalized concept of higher-order differentiability for fuzzy functions. Then we interpret Nth-order fuzzy differential equations using this concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for which both the new solutions and the former ones to the fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation with initial conditions having four different solutions.

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Section: Theorem 315 Let F : I → R Be Twice Differentiable On I γ mentioning

confidence: 99%

New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability

Khastan

Bahrami

Ivaz

2009

Boundary Value Problems

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“…Using, according to Ref. [26], d P ( x ) = f ( x ) d x , can be written as where f ( x ) is the distribution of probability density of S L .…”

Section: Fuzzy Failure Probabilitymentioning

confidence: 99%

A fuzzy approach for drawing S-N curves in the finite life region

Zonfrillo

Rossi

2011

Fatigue &Amp; Fracture of Engineering Materials &Amp; Structures

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A B S T R A C T In this work a fuzzy approach has been developed in order to estimate the probability of fatigue failure. In particular, with the proposed method the S-N curves of a material in the finite life region can be drawn. The experimental data are represented in terms of fuzzy sets and are fitted using a fuzzy linear regression. Data scattering and uncertainty in the empirical failure model are reflected in the definition of membership functions. Several examples are shown to illustrate the procedure. Failure probabilities and fatigue curves obtained by the fuzzy method are similar to those obtained by traditional statistical analysis, based on normal distributions of strength with standard deviation that remains constant with different load levels. In particular, the results obtained indicate that the possibilities offered by fuzzy systems are also applicable for estimating the Wöhler curve of a material under fatigue stresses. To evaluate its reliability, the proposed method is compared with the traditional one, with particular attention to the case in which a small amount of experimental data is available. The new fuzzy method is slightly less accurate than traditional statistical analysis to outline S-N curves in the finite life region. This is mainly due to the fact that the method is influenced by the nonuniformity of data dispersion at each level of stress. = Arithmetic mean between S Lmin and S Lmax N L = Arithmetic mean of the various N L at a given level of stress a m = Centre of triangular fuzzy number N = Cycle number N L = Decimal logarithm of N S L = Decimal logarithm of S D m1 = Distribution of the means D m2 = Distribution of the medians d er = Error distribution R = Fuzzy event 'strength is smaller than the stress amplitude' c L = Left width of triangular fuzzy number α = Material parameter in the S-N curve β = Material parameter in the S-N curve A 0 = Material parameter in the S-N curve A 1 = Material parameter in the S-N curve M = Mean of D m1 distribution M e = Mean of D m2 distribution μ() = Membership function of a fuzzy set Correspondence: G. Zonfrillo.

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“…In the past decades, many works have been appeared on the aspects of theories and applications on fuzzy differential equations; see [5][6][7][8][9][10][11][12]. The notation of fuzzy differential equation was initially introduced by Kandel and Byatt [13,14] and later they applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems [7,15]. A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [16,17], Wu and Song [18], Ouyang and Wu [19], Kim and Sakthivel [20], and M. D. Wu [21].…”

Section: Introductionmentioning

confidence: 99%