solution of boundary value problems for integro differential equations by using differential transform method in this study we extended the differential transform method dtm to solve the integro differential equations new theorems for the transformation of integrals are introduced and prooved the commonly encountered linear and nonlinear integro differential equations that appear in literature are solved as an illustration for the efficiency of the method some numerical results are also given to demonstrate the superiority of the method to other common techniques the results we obtained by using this technique is much more accurate compared to the existing ones

a theory of independent fuzzy probability for system reliability fuzzy fault trees provide a powerful and computationally efficient technique for developing fuzzy probabilities based on independent inputs the probability of any event that can be described in terms of a sequence of independent unions intersections and complements may be calculated by a fuzzy fault tree unfortunately fuzzy fault trees do not provide a complete theory many events of substantial practical interest cannot be described only by independent operations thus the standard fuzzy extension based on fuzzy fault trees is not complete since not all events are assigned a fuzzy probability other complete extensions have been proposed but these extensions are not consistent with the calculations from fuzzy fault trees we propose a new extension of crisp probability theory our model is based on n independent inputs each with a fuzzy probability the elements of our sample space describe exactly which of the n input events did and did not occur our extension is complete since a fuzzy probability is assigned to every subset of the sample space our extension is also consistent with all calculations that can be arranged as a fault tree our approach allows the reliability analyst to develop complete and consistent fuzzy reliability models from existing crisp reliability models this allows a comprehensive analysis of the system computational algorithms are provided both to extend existing models and develop new models the technique is demonstrated on a reliability model of a three stage industrial process

a fuzzy reliability assessment of basic events of fault trees through qualitative data processing abstract probabilistic approaches are common in the analysis of reliability of complex engineering systems however they require quantitative historical failure data for determining reliability characteristics in many real world areas such as e g nuclear engineering quantitative historical failure data are unavailable or become inadequate and only qualitative data such as expert opinions which are described in linguistic terms can be collected and then used to assess system reliability moreover experts are more comfortable justifying event failure likelihood using linguistic terms which capture uncertainties rather than by expressing judgments in a quantitative manner new techniques are therefore needed that will help construct models of reliability of complex engineering system without being confined to quantitative historical failure data the objective of this study is to develop a fuzzy reliability algorithm to effectively generate basic event failure probabilities without reliance on quantitative historical failure data through qualitative data processing the originality of this study comes with an introduction of linguistic values articulated in terms of component failure possibilities in order to qualitatively assess basic event failure possibilities treated as inputs of the proposed model and generate basic event failure probabilities as its outputs to demonstrate the feasibility and effectiveness of the proposed algorithm actual basic event failure probabilities collected from nuclear power plant operating experiences are compared with the failure probabilities generated by the algorithm the results demonstrate that the proposed fuzzy reliability algorithm arises as a suitable alternative for the probabilistic reliability approach when quantitative historical failure data are unavailable

fuzzy system reliability analysis based on level λ 1 interval valued fuzzy numbers this study uses level 1 interval valued fuzzy numbers to examine the fuzzy reliability of a serial system and a parallel system and obtain the estimated reliability of both systems in the fuzzy sense

a fuzzy universal generating function based method for the reliability evaluation of series systems with performance sharing between adjacent units under parametric uncertainty abstract this paper considers fuzzy multi state systems msss with performance sharing between adjacent units in which each fuzzy multi state unit has a random performance and a random demand if the performance of a unit exceeds its individual demand performance demand its surplus performance can be transmitted to its adjacent unit that experiences performance deficiency performance

a note on the conditional probability of fuzzy subsets of a continuous domain abstract the notion of a mass assignment corresponding to a discrete fuzzy set is generalised to include all fuzzy sets the definition is motivated by considering a possible mechanism by which an intelligent agent could extend set theoretic operations to fuzzy sets this same mechanism also suggests an alternative definition for the conditional probability of fuzzy sets it is noted that this definition is probability possibility consistent and it is shown that in the case where the probability distribution on the domain has a density function then the conditional probability of f given g can be expressed as the probability of f calculated relative to a conditional density function dependent on g this result is used to develop an analytical method for calculating conditional probabilities of fuzzy subsets of the reals where the membership functions of f and g and the density function are continuous piecewise linear functions such a method has applications to the development of a general semantic unification in fril

use of fuzzy set theory for level i studies in probabilistic risk assessment at a post chernobyl conference on probabilistic safety assessment and risk management psa 87 held in september 1987 the inadequacies of the current probabilistic risk assessment psa came into sharp focus highlighting the importance of improvements in the current psa professor rasmussen a pioneer in the psa methodology made a strong plea for reduction of rather large uncertainties in the results of current psa obviously amongst the ways to achieve this one of the approaches would be to encourage collection and use of accurate failure data however as the state of affairs is a better alternative would be to develop a methodology which could handle imprecision of data and uncertainties of models which are existence in the current psa fuzzy set theory fst provides a natural framework to achieve the above goal and therefore the present paper is intended to emphasize and explore the possibility of the use of such a theory to develop a methodology which is computationally simple and easy to use in the quantitative assessment of failure probability of catastrophic events in psa particularly in level i studies some illustrative examples are provided to demonstrate the feasibility and effectiveness of such an approach it is demonstrated that a general purpose computer code can be developed without much difficulty to recast the pra studies to incorporate fuzzy probabilities

on the integrable perturbations of the camassa holm equation we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure

redundancy optimization problems with uncertainty of combining randomness and fuzziness by using random fuzzy lifetimes as basic parameters three types of system performance expected system lifetime system lifetime and system reliability are presented some random fuzzy simulations are employed to estimate the system performance furthermore a spectrum of random fuzzy models are constructed for redundancy optimization problems in order to solve these models a hybrid intelligent algorithm is used finally some numerical examples are provided for the sake of illustration

convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh we study here the discretisation of the nonlinear hyperbolic equationu t div vf u 0 in 3 with given initial conditionu 0 u 0 in 2 wherev is a function from 2 to 2 such that divv 0 andf is a given nondecreasing function from to an explicit euler scheme is used for the time discretisation of the equation and a triangular mesh for the spatial discretisation under a usual stability condition we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation

the method of generalized generating sequences abstract the method of generalized generating sequences ggs is based on some modifications of the standard generating function this method is convenient for a computerized calculation of enumeration problems that arise in discrete optimization this method will be described below in the context of optimal redundancy problems

fuzzy system reliability analysis by interval of confidence abstract the use of fuzzy number over interval of confidence instead of probability and possibilitic considerations for evaluating the range value and interval of the fuzzy reliability is proposed this approach of evaluating fuzzy system reliability by interval arithmetic and cuts has a considerable advantage we can calculate reliability problems easily by the use of derive and mathcad packages through theoretical analysis an illustrative example and computational results we show that the proposed approach is more general and simplified

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

bayesian reliability analysis for fuzzy lifetime data lifetime data are important in reliability analysis classical reliability estimation is based on precise lifetime data it is usually assumed that observed lifetime data are precise real numbers however some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers thus it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers bayesian methods have proved to be very useful when the sample size is small there is little study on bayesian reliability estimation based on fuzzy lifetime data most of the reported works in this area is limited to single parameter lifetime distributions in this paper we propose a new method to determine the membership function of the estimates of the parameters and the reliability function of multi parameter lifetime distributions an artificial neural network is used to approximate the calculation process of parameter estimation and reliability prediction the genetic algorithm is used to find the boundary values of the membership function of the estimate of interest at any cut level this method can be used to determine the membership functions of the bayesian estimates of multi parameter distributions the effectiveness of the proposed method is illustrated with normal and weibull distributions

on solutions and distribution problems of the linear programming with fuzzy random variable coefficients abstract this paper studies the solution method and distribution problems for two kinds of linear programming with fuzzy random variable coefficients some concepts of solutions for example the fuzzy pseudo random optimization solution are introduced for these new programming problems we prove some equivalent theorems that transform the fuzzy random programming problems into a series of random programming problems by using the simplex method of linear programming with random variable coefficients we give the solution method of linear programming with fuzzy random variable coefficients the formulas of probability distribution function projective distribution function and expectation on these new programmings are presented

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

a prototype neural network to perform early warning in nuclear power plant the paper presents some results of research work in the field of artificial neural networks ann applied to nuclear safety it shows how a priori knowledge in the form of qualitative physical reasoning can provide a powerful basis for designing a set of ann based detection subsystems in particular it explains how each ann is in charge of modelling a physical relationship between a set of state variables thermal balance mass balance etc by trying to predict one particular variable from other ones then the residual signal defined by the difference between the predicted value and the real one is used to decide whether abnormalities are present as far as the decision logic is concerned the paper describes how robustness can be improved by adequate filters on the residuals the proposed approach is then validated on data coming from a fullscope simulator of one of the belgian nuclear power units the neural based detection system is trained on normal scenarios and is able after learning to detect reliably and rapidly most of the incidental situations chosen as tests

a multi step differential transform method and application to non chaotic or chaotic systems the differential transform method dtm is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form in this paper we propose a reliable new algorithm of dtm namely multi step dtm which will increase the interval of convergence for the series solution the multi step dtm is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations this new algorithm is applied to lotka volterra chen and lorenz systems then a comparative study between the new algorithm multi step dtm classical dtm and the classical runge kutta method is presented the results demonstrate reliability and efficiency of the algorithm developed

a finite volume scheme for a noncoercive elliptic equation with measure data we show here the convergence of the finite volume approximate solutions of a convection diffusion equation to a weak solution without the usual coercitivity assumption on the elliptic operator and with weak regularity assumptions on the data numerical experiments are performed to obtain some rates of convergence in two and three space dimensions

reliability estimation and design with insufficient data based on possibility theory early in the engineering design cycle it is difficult to quantify product reliability or compliance to performance targets because of insufficient data or information for modeling the uncertainties design decisions are therefore based on fuzzy information that is vague imprecise qualitative linguistic or incomplete the uncertain information is usually available as intervals with lower and upper limits in this work the possibility theory is used to assess design reliability with incomplete information the possibility theory can be viewed as a variant of fuzzy set theory the formal theories to handle uncertainty are first introduced using the theoretical fundamentals of fuzzy measures a computationally efficient and accurate hybrid global local optimization approach is subsequently described for calculating the confidence level of fuzzy response the method combines the advantages of the commonly used vertex and discretization methods subsequently the possibility theory is used in design a possibility based design optimization method is proposed where all design constraints are expressed in a possibilistic way it is shown that the method gives a conservative solution compared with all conventional reliability based designs obtained with different probability distributions finally a general possibility based design optimization method which handles a combination of random and possibilistic design variables is presented several numerical examples demonstrate the application of possibility theory in design

fuzzy reliability assessment of systems with multiple dependent competing degradation processes components are often subject to multiple competing degradation processes for multicomponent systems the degradation dependence within one component or and among components need to be considered physics based models and multistate models are often used for component degradation processes particularly when statistical data are limited in this paper we treat dependence between degradation processes within a piecewise deterministic markov process pdmp modeling framework epistemic subjective uncertainty can arise due to the incomplete or imprecise knowledge about the degradation processes and the governing parameters to take this into account we describe the parameters of the pdmp model as fuzzy numbers then we extend the finite volume method to quantify the fuzzy reliability of the system the proposed method is tested on one subsystem of the residual heat removal system of a nuclear power plant and a comparison is offered with a monte carlo simulation solution the results show that our method can be most efficient

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

fuzzy random reliability of structures based on fuzzy random variables abstract using fuzzy random variables as basic variables the relationship between fuzzy random variables and fuzzy random events is established in this paper formulas of measures of a fuzzy random event and its cut set are presented the fuzzy random reliability of structures is defined on the basis of fuzzy measure and fuzzy integral further by using fuzzy random variables the fuzzy random failure function of structures is defined and the fuzziness between structural reliable state and failure state is also analyzed moreover the fuzzy random limit state equation of structures is built the probability calculation method of structural reliability under the reliable level is given finally the calculation procedures of structural fuzzy random reliability is proposed

strong form meshless implementation of taylor series method a new meshless approach is investigated by using taylor series expansion and technique of differential transform method which is called meshless implementation of taylor series method mitsm in particular strong form meshless implementation of taylor series method smitsm is studied in this paper then the basis functions are used to solve a 1d second order ordinary differential equation and 2d laplace equation by using the smitsm comparisons are made with the analytical solutions and results of symmetric smoothed particle hydrodynamics ssph method we also compared the effectiveness of the smitsm and ssph method by considering various particle distributions nonhomogeneous terms and number of terms in the basis functions it is observed that the mitsm has the conventional convergence properties and at the expense of cpu time yields smaller l 2 error norms than the ssph method especially in the existence of nonsmooth nonhomogeneous problems

fuzzy system reliability analysis using fuzzy number arithmetic operations abstract this paper presents a new method to analyze fuzzy system reliability using fuzzy number arithmetic operations where the reliability of each system component is represented by a triangular fuzzy number because the proposed method uses simplified fuzzy arithmetic operations of fuzzy numbers rather than the complicated interval arithmetic operations of fuzzy numbers mentioned by cheng and mon in fuzzy sets and systems 56 1993 29 35 or the complicated extended algebraic operations of fuzzy numbers mentioned by singer in fuzzy sets and systems 34 1990 145 155 its execution is faster than the ones presented by cheng and mon and singer

an implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes we are interested here in the numerical approximation of a family of probability measures solution of the chapman kolmogorov equation associated to some non diffusion markov process with uncountable state space such an equation contains a transport term and another term which implies redistribution of the probability mass on the whole space an implicit finite volume scheme is proposed which is intermediate between an upstream weighting scheme and a modified lax friedrichs one due to the seemingly unusual probability framework a new weak bounded variation inequality had to be developed in order to prove the convergence of the discretised transport term such an inequality may be used in other contexts such as for the study of finite volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in l 1 also due to the redistribution term the tightness of the family of approximate probability measures had to be proven numerical examples are provided showing the efficiency of the implicit finite volume scheme and its potentiality to be helpful in an industrial reliability context

taylor polynomial solutions of general linear differential difference equations with variable coefficients abstract in this paper a taylor method is developed to find an approximate solution of high order linear differential difference equations with variable coefficients under the mixed conditions the solution is obtained in terms of taylor polynomials examples are presented which illustrate the pertinent features of the method

structure functions with fuzzy states systems are modeled in reliability theory by means of structure functions which show how the system state of performance depends on the state of all its components in this paper we consider that the space of states allowed for both the system and its components is an appropriate family of fuzzy numbers basic concepts in classical crisp binary and multistate models in reliability theory are in this way generalized some particular consequences relative to component importance are also analyzed

a theorem of renewal process for fuzzy random variables and its application the fuzzy set was introduced by zadeh 1965 and the concept of fuzzy random variables was provided by kwakernaak 1981 sequences of independent and identical distributed fuzzy random variables were considered by kruse 1982 he also showed the strong law of large numbers for fuzzy random variables in this paper we consider the stochastic process for fuzzy random variables and prove a theorem for fuzzy rate of a fuzzy renewal process

fuzzy system reliability analysis by the use of t ω the weakest t norm on fuzzy number arithmetic operations abstract in general the sup min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability where the reliability of each system component is represented by fuzzy numbers it is well known that t based addition preserves the shape of l r type fuzzy numbers in this paper we show t based multiplication also preserves the shape of l r type fuzzy numbers we then apply t based arithmetic operations to fuzzy system reliability analysis in fact we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for l r type fuzzy system reliability while others singer fuzzy sets syst 34 1990 145 cheng and mon fuzzy sets syst 56 1993 29 chen fuzzy sets syst 64 1994 31 have got the approximate solutions using sup min convolution for evaluating fuzzy system reliability

solution of differential difference equations by using differential transform method abstract in this work we successfully extended differential transform method dtm by presenting and proving new theorems to the solution of differential difference equations ddes theorems are presented in the most general form to cover a wide range of ddes being linear or nonlinear and constant or variable coefficient in order to show the power and the robustness of the method and to illustrate the pertinent features of related theorems examples are presented

legendre spectral collocation method for fredholm integro differential difference equation with variable coefficients and mixed conditions in this article the legendre spectral collocation method has been applied to solve fredholm integro differential difference equations with variable coefficients the proposed method is based on the gauss legendre points with the basis functions of lagrange polynomials usually this type of integral equations are very difficult to solve analytically as well as numerically the presented method applied to the integral equation reduces to solve the system of algebraic equations also the numerical results obtained by legendre spectral collocation method have been compared with the results obtained by existing methods illustrative examples have been discussed to demonstrate the validity and applicability of the presented method

linearizability of the perturbed burgers equation inst de fis teorica universidade estadual paulista rua pamplona 145 01405 900 sao paulo

mixed function method for obtaining exact solution of nonlinear differential difference equations and coupled nddes in this paper we construct a new mixed function method for the first time by using this new method we study the two nonlinear differential difference equations named the generalized hybrid lattice and two component volterra lattice equations some new exact solutions of mixed function type such as discrete solitary wave solutions discrete kink and anti kink wave solutions and discrete breather solutions with kink and anti kink character are obtained and their dynamic properties are also discussed by using software mathematica we show their profiles

a fuzzy set approach to fault tree and reliability analysis abstract conventional fault tree and reliability analysis does not give a picture concerning the tolerances of the probability values of hazards to overcome this shortcoming the relative frequencies of the basic envents are considered as fuzzy numbers using standard approximations for the membership functions n ary possibilistic and or and neg operators are derived allowing a straightforward evaluation of the frequencies of hazard events in complex systems simultaneously with their tolerances the paper shows the use of these operators for constructing possibilistic fault trees the use of the method for numerical calculation is demonstrated on a real like safety problem

on the well posedness problem and the scattering problem for the dullin gottwald holm equation in this paper we study the well posedness of the cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation the so called dgh equation which was derived by dullin gottwald and holm the issue of passing to the limit as the dispersive parameter tends to zero for the solution of the dgh equation is investigated and the convergence of solutions to the dgh equation as 2 0 is studied and the scattering data of the scattering problem for the equation can be explicitly expressed the new exact peaked solitary wave solutions are obtained in the dgh equation after giving the condition of existing peakon in the dgh equation it turns out to be nonlinearly stable for the peakon in the dgh equation

random fuzzy renewal process abstract this paper attempts to discuss a random fuzzy renewal process based on random fuzzy theory the interarrival times are characterized as nonnegative random fuzzy variables which is a more reasonable consideration in the real world under this setting the rate of the random fuzzy renewal process is discussed and a random fuzzy elementary renewal theorem is presented furthermore the expected value of renewals in an arbitrary interval is investigated and blackwell s theorem in random fuzzy sense is also established

numerical study of the solution of the burgers and coupled burgers equations by a differential transformation method in this paper the differential transformation method dtm is employed to obtain the numerical analytical solutions of the burgers and coupled burgers equations we begin by showing how the differential transformation method applies to the linear and nonlinear parts of any pde and give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations we also compare it against three famous methods namely the homotopy perturbation method the homotopy analysis method and the variational iteration method these results show that the technique introduced here is accurate and easy to apply

fuzzy random chance constrained programming by fuzzy random programming we mean the optimization theory dealing with fuzzy random decision problems this paper presents a new concept of chance of fuzzy random events and constructs a general framework of fuzzy random chance constrained programming we also design a spectrum of fuzzy random simulations for computing uncertain functions arising in the area of fuzzy random programming to speed up the process of handling uncertain functions we train a neural network to approximate uncertain functions based on the training data generated by fuzzy random simulation finally we integrate the fuzzy random simulation neural network and genetic algorithm to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples

the efficacy of fuzzy representations of uncertainty advocates of the theory of fuzzy sets as a system for representing uncertainty have based their case on five basic arguments these are 1 the reality hypothesis which holds that imprecision is an inherent property of the world external to an observer 2 the subjectivity hypothesis which holds that probability is an exclusively objective measure of uncertainty and that therefore subjective uncertainty can only be represented with fuzzy sets 3 the behaviorist hypothesis which claims that uncertainty systems should emulate rather than prescribe human behavior in the face of uncertainty 4 the probability as fiction hypothesis which claims that probability does not comprise a field of study in its own right and 5 the superset hypothesis which holds that fuzzy set theory includes probability as a special case and thus provides a richer uncertainty modeling environment we discuss and criticize all five we then criticize the argument that fuzziness represents a type of uncertainty distinct from probability and also the inordinate complexity of fuzzy methods we present a method for assessing the efficacy of fuzzy representations of uncertainty and apply this method in three examples

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

representing parametric probabilistic models tainted with imprecision numerical possibility theory belief functions have been suggested as useful tools to represent imprecise vague or incomplete information they are particularly appropriate in uncertainty analysis where information is typically tainted with imprecision or incompleteness based on their experience or their knowledge about a random phenomenon experts can sometimes provide a class of distributions without being able to precisely specify the parameters of a probability model frequentists use two dimensional monte carlo simulation to account for imprecision associated with the parameters of probability models they hence hope to discover how variability and imprecision interact this paper presents the limitations and disadvantages of this approach and propose a fuzzy random variable approach to treat this kind of knowledge

solution of difference equations by using differential transform method abstract this work presents semi numerical analytical solutions to both linear and nonlinear difference equations with variable coefficients by applying differential transform method dtm and extending it with the related theorems and their proofs to make clear and illustrate the features and capabilities of the presented method examples are carried out and numerical results with comparison to exact solutions and other methods are given

d equalities of fuzzy sets abstract a equality of two fuzzy sets means the two fuzzy sets are equal to each other to an extent of s in this paper we present basic properties of 6 equality and various results on 6 equality with respect to algebraic operations fuzzy relations extension principle and triangular norms we also present a number of reliability examples to show usability of equality in reality

nonlinear elliptic equations in rn without growth restrictions on the data abstract we show existence and regularity of solutions in rn to nonlinear elliptic equations of the form div a x du g x u when is just a locally integrable function under appropriate growth conditions on a and g but not on roughly speaking in the model case p u u s 1u with p 2 1 n existence of a nonnegative solution in rn is guaranteed for every nonnegative l1loc rn if and only if s p 1

uncertainty in fault tree analysis a fuzzy approach abstract in fault tree analysis the uncertainties in the failure probability and or failure rate of system components or basic events can be propagated to find the uncertainty in the overall system failure probability the conventional approach is monte carlo simulation by assuming a probability distribution for the failure probability in addition a new methodology based on fuzzy set theory is also being used in the fault tree analysis for quantifying the basic event uncertainty and for propagating it however identification of the components which contribute maximum to the system failure probability is also important in fault tree analysis similarly ranking the components based on their contribution of uncertainty to the uncertainty of the system failure probability is also very important this paper presents a comparative study of probabilistic and fuzzy methodologies for top event uncertainty evaluation further it explains a new approach to rank the system components or basic events depending on 1 their contribution to the top event failure probability and 2 their uncertainty contribution to the uncertainty of the top event based on fuzzy set theory

system failure engineering and fuzzy methodology an introductory overview abstract system failure engineering may encompass reliability safety security maintainability risk and quality control in this paper we pay a unified view i e a failure oriented view to system failure engineering the notion of failure can be represented in terms of fuzzy sets and widely interpreted we then discuss various application aspects of fuzzy methodology in system failure engineering which include fuzzy methodology in fault diagnosis in probist systems in structural reliability in software reliability in human reliability in safety engineering in security engineering in risk engineering and in quality control as well as profust posbist and posfust reliability theories engineering case studies and mathematical problems raised by applications of fuzzy methodology in system failure engineering are also addressed in this paper

posbist reliability behavior of typical systems with two types of failure abstract the posbist reliability theory is based on the possibility assumption and the binary state assumption in this paper we study the posbist reliability behavior in the presence of two failure modes for various classes of typical systems including series systems parallel systems k out of n systems series parallel systems and parallel series systems it is shown that for each typical system the posbist system reliability is bounded by the minimum and the maximum of the posbist component reliabilities

on advanced computing with words using the generalized extension principle for type 1 fuzzy sets in this paper we propose and demonstrate an effective methodology for implementing the generalized extension principle to solve advanced computing with words acww problems such problems involve implicit assignments of linguistic truth probability and possibility to begin we establish the vocabularies of the words involved in the problems and then collect data from subjects about the words after which fuzzy set models for the words are obtained by using the interval approach ia or the enhanced interval approach eia next the solutions of the acww problems which involve the fuzzy set models of the words are formulated using the generalized extension principle because the solutions to those problems involve complicated functional optimization problems that cannot be solved analytically we then develop a numerical method for their solution finally the resulting fuzzy set solutions are decoded into natural language words using jaccard s similarity measure we explain how acww problems can solve some potential prototype engineering problems and connect the methodology of this paper with perceptual computing

fuzzy universal generating functions for multi state system reliability assessment the paper considers a multi state system mss where performance rates or and corresponding state probabilities are presented as fuzzy values in many real world msss it is difficult to estimate precise values of performance rates and their probabilities because of two main reasons the first reason is inaccuracy and insufficiency of data the second one is based on the fact that msss are often used as the approximation for continuous state systems in order to simplify computational burden to overcome these deficiencies performance rates levels or and corresponding probabilities are considered as fuzzy values in this paper the fuzzy universal generating function fugf is developed to extend the universal generating function with crisp sets the composition operators are extended for fugfs and introduced in more general forms based on this a special technique for reliability assessment of such msss is developed illustrative examples are presented to demonstrate the technique

reliability and covariance estimation of weighted k out of n multi state systems abstract in the literature of reliability engineering reliability of the weighted k out of n system can be calculated using component reliability based on the structure function the calculation usually assumes that the true component reliability is completely known however this is not the case in practical applications instead component reliability has to be estimated using empirical sample data uncertainty arises during this estimation process and propagates to the system level this paper studies the propagation mechanism of estimation uncertainty through the universal generating function method equations of the complete solution including the unbiased system reliability estimator and the corresponding unbiased covariance estimator are derived this is a unified approach it can be applied to weighted k out of n systems with multi state components to weighted k out of n systems with binary components and to simple series and parallel systems it may also serve as building blocks to derive estimators of system reliability and uncertainty measures for more complicated systems

elicitation assessment and pooling of expert judgments using possibility theory the problem of modeling expert knowledge about numerical parameters in the field of reliability is reconsidered in the framework of possibility theory usually expert opinions about quantities such as failure rates are modeled assessed and pooled in the setting of probability theory this approach does not seem to always be natural since probabilistic information looks too rich to be currently supplied by individuals indeed information supplied by individuals is often incomplete imprecise rather than tainted with randomness moreover the probabilistic framework looks somewhat restrictive to express the variety of possible pooling modes in this paper the authors formulate a model of expert opinion by means of possibility distributions that are thought to better reflect the imprecision pervading expert judgments they are weak substitutes to unreachable subjective probabilities assessment evaluation is carried out in terms of calibration and level of precision respectively measured by membership grades and fuzzy cardinality indexes finally drawing from previous works on data fusion using possibility theory the authors present various pooling modes with their formal model under various assumptions concerning the experts a comparative experiment between two computerized systems for expert opinion analysis has been carried out and its results are presented in this paper

a numerical integration based simulation algorithm for expected values of strictly monotone functions of ordinary fuzzy variables fuzzy simulation is used to approximate the expected values of functions of fuzzy variables which plays an important role in the solution algorithms of fuzzy optimization problems the traditional discretization based simulation algorithms fail to return a satisfactory approximation within an acceptable computation time which hinders the applications of fuzzy optimization methods in large size or even middle size problems in this paper we first prove some equivalent formulas for the expected values of strictly monotone functions of ordinary fuzzy variables then we propose a new fuzzy simulation algorithm based on the numerical integration technique finally we present some numerical examples to make comparisons between the traditional approach and our approach the results show that our approach has higher performances on the reliability stability and computation time

solutions of integral and integro differential equation systems by using differential transform method in this study differential transform method dtm is applied to both integro differential and integral equation systems the method is further expanded with a formulation to treat fredholm integrals if the system considered has a solution in terms of the series expansion of known functions this powerful method catches the exact solution so as to show this capability and robustness some systems of integral and integro differential equations are solved as numerical examples

reliability and mean time to failure of unrepairable systems with fuzzy random lifetimes in practice the lifetimes of unrepairable systems contain both randomness and fuzziness so it is appropriate to assume the lifetimes of systems to be fuzzy random variables in this paper the definitions of reliability and mean time to failure mttf of unrepairable systems with fuzzy random lifetimes are given respectively then basic mathematical models of unrepairable systems with fuzzy random lifetimes are established furthermore the reliability and mttf of series systems parallel systems series parallel systems parallel series systems and cold standby systems are discussed respectively finally some numerical examples are presented to illustrate how to calculate the reliability and mttf of unrepairable systems with fuzzy random lifetimes

a taylor polynomial approach for solving differential difference equations the purpose of this study is to give a taylor polynomial approximation for the solution of mth order linear differential difference equations with variable coefficients under the mixed conditions about any point for this purpose taylor matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the differential difference equations and then substituting their matrix forms into the equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed also we have discussed the accuracy of the method we use the symbolic algebra program maple to prove our results

computer assisted reliability analysis an application of possibilistic reliability theory to a subsystem of a nuclear power plant abstract since the late 1970s multistate structure functions have been introduced to overcome the shortcomings of the classical and binary approach to the structural aspects of systems and their components the uncertainty about the state of a system and its components classically is described by probability theory recently several new models are proposed that are based upon fuzzy set theory and possibility theory the spread and success of these models however highly depend upon the development of computer tools that allow the reliability engineer to apply these new methods rather easily in this paper we present such tool cara computer assisted reliability analysis which is able to represent and study the multistate aspects of systems and their components when only incomplete information is available

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

the integral factor method for solving a class of generalized kdv equation this paper deals with a class of generalized kdv equation by making use of a mathematical technique based on using integral factors for solving differential equations and give rise to the solitary wave periodic cusp wave and periodic wave solutions the work confirms the power of the proposed method

fuzzy probabilistic design of water distribution networks 1 the primary aim of this paper is to present a fuzzy probabilistic approach for optimal design and rehabilitation of water distribution systems combining aleatoric and epistemic uncertainties in a unified framework the randomness and imprecision in future water consumption are characterized using fuzzy random variables whose realizations are not real but fuzzy numbers and the nodal head requirements are represented by fuzzy sets reflecting the imprecision in customers requirements the optimal design problem is formulated as a two objective optimization problem with minimization of total design cost and maximization of system performance as objectives the system performance is measured by the fuzzy random reliability defined as the probability that the fuzzy head requirements are satisfied across all network nodes the satisfactory degree is represented by necessity measure or belief measure in the sense of the dempster shafer theory of evidence an efficient algorithm is proposed within a monte carlo procedure to calculate the fuzzy random system reliability and is effectively combined with the nondominated sorting genetic algorithm ii nsgaii to derive the pareto optimal design solutions the newly proposed methodology is demonstrated with two case studies the new york tunnels network and hanoi network the results from both cases indicate that the new methodology can effectively accommodate and handle various aleatoric and epistemic uncertainty sources arising from the design process and can provide optimal design solutions that are not only cost effective but also have higher reliability to cope with severe future uncertainties

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

repairable consecutive k out of n f system with fuzzy states in this paper we study a repairable consecutive k out of n f system with fuzzy states the maintenance of the system is carried out by r repairmen or repair facilities the system is either circular or linear it is assumed that the working time and the repair time of each component in the system are both exponentially distributed and every component after repair is as same as new each component is classified as either a key component or an ordinary one according to its priority role to the system s repair when the system displays a gradual degradation of its performance its reliability is then analyzed in terms of fuzzy failure and success states the difference between fuzzy reliability theory and the conventional reliability theory is highlighted

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context abstract conventional reliability theory is based on the binary state assumption and the probability assumption namely 1 the systems are precisely defined as functioning or failed 2 the system behavior is fully characterized in the context of probability measures however the probability assumption is no longer reasonable in the case of small size of samples since the term probability stems from the sample generality in this way possibility measures should be used to characterize the sample particularity in this paper the binary state assumption is reserved and the possibility assumption is taken in place of the probability assumption thus a concept of fuzzy reliability i e posbist reliability is introduced and a theory of fuzzy reliability i e posbist reliability theory is established various conclusions which are contrary to statements in conventional reliability theory i e probist reliability theory are then drawn

a general formal approach for fuzzy reliability analysis in the possibility context abstract in this paper we focus on the fuzzy reliability analysis in the posbist reliability theory in accordance with this theory the system failure state is defined precisely and the system failure and repair behaviour is fully characterized in the context of possibility measures to formalize the reliability analysis in the posbist theory we propose a general approach on the basis of a system of functional equations the solution of the system allows us to obtain fuzzy reliability characteristics of the arbitrary repairable and unrepairable systems to illustrate the construction and application of the system of functional equations various examples are given

a fuzzy set approach for event tree analysis the traditional event tree analysis uses a single probability to represent each top event however it is unrealistic to evaluate the occurrence of each event by using a crisp value without considering the inherent uncertainty and imprecision a state has the fuzzy set theory is universally applied to deal with this kind of phenomena the main purpose of this study is to construct an easy method to evaluate human errors and integrate them into event tree analysis by using fuzzy concepts a systematic fuzzy event tree analysis algorithm is developed to evaluate the risk of a large scale system a practical example in a nuclear power plant is used to demonstrate this procedure

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem