efficiency analysis generating an efficient extreme point for an molp and some comparisons this paper starts with analyzing efficiency of an alternative in a multi objective problem by combination of concepts in data envelopment analysis dea and multiple objective linear programming molp we search a composite alternative to compare with a given alternative and assess its efficiency in an molp problem in addition to some results this leads us to produce some procedures for generating an initial efficient extreme point to an molp problem we compare our procedures with some other existing methods and exhibit some advantages of our methods in addition we consider some models of dea as molp problems and use them to illustrate some contents of the paper which also appears some analogies between these areas

cauchy problems for fractional differential equations with riemann liouville fractional derivatives abstract in this paper we are concerned with cauchy problems of fractional differential equations with riemann liouville fractional derivatives in infinite dimensional banach spaces we introduce the notion of fractional resolvent obtain some its properties and present a generation theorem for exponentially bounded fractional resolvents moreover we prove that a homogeneous order cauchy problem is well posed if and only if its coefficient operator is the generator of an order fractional resolvent and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous order cauchy problem

random flights connecting porous medium and euler poisson darboux equations in this paper we consider the porous medium equation and establish a relationship between its kompanets zel dovich barenblatt solution u xd t xd rd t 0 and random flights the time rescaled version of u xd t is the fundamental solution of the euler poisson darboux equation which governs the distribution of random flights performed by a particle whose displacements have a dirichlet probability distribution and choosing directions uniformly on a d dimensional sphere we consider the space fractional version of the euler poisson darboux equation and present the solution of the related cauchy problem in terms of the probability distributions of random flights governed by the classical euler poisson darboux equation furthermore this research is also aimed at studying the relationship between the solutions of a fractional porous medium equation and the fractional euler poisson darboux equation a considerable part of this paper is devoted to the analysis of the probabilistic tools of the solutions of the fractional equations the extension to the higher order euler poisson darboux equation is considered and the solutions are interpreted as compositions of laws of pseudoprocesses

fractional kinetic equations and universal decoupling of a memory function in mesoscale region it is proved that kinetic equations containing non integer integrals and derivatives appear in the result of reduction of a set of micromotions to some averaged collective motion in the mesoscale region in other words it means that after a proper statistical average the microscopic dynamics is converted into a collective motion in the mesoscopic regime a fractal medium containing weakly and strongly correlated relaxation units has been considered it is shown that in most cases the original of the memory function recovers the riemann liouville integral for a strongly correlated fractal medium a generalization of the riemann liouville integral is obtained for the fractal branched processes one can derive the stretched exponential law of relaxation that is widely used for description of relaxation phenomena in disordered media it is shown that the generalized stretched exponential function describes the averaged collective motion in the fractal branched complex systems it is confirmed that the generalized stretched exponential law can be definitely applicable for description of a set of spontaneous evoked postsynaptic signals in the living matter the application of these fractional kinetic equations for the description of the dielectric relaxation phenomena is also discussed these kinetic equations and their generalizations can be applicable for the description of different relaxation or diffusion processes in the intermediate mesoscale region

an asymmetry similarity measure based neural fuzzy inference system in this paper a new asymmetry similarity measure based neural fuzzy inference system asm nfis is proposed a pseudo gaussian membership function can provide a neural fuzzy inference system which has a higher flexibility and can approach the optimized result more accurately an on line self constructing learning algorithm is proposed to automatically construct the asm nfis it consists of structure learning and parameter learning that would create adaptive fuzzy logic rules the structure learning is based on the similarity measure of asymmetric gaussian membership functions and the parameter learning is based on a supervised gradient descent method computer simulations were conducted to illustrate the performance and applicability of the proposed model

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

using an efficient immune symbiotic evolution learning for compensatory neuro fuzzy controller this paper presents an efficient immune symbiotic evolution learning isel algorithm for the compensatory neuro fuzzy controller cnfc the proposed isel method includes three major components initial population subgroup symbiotic evolution and immune system algorithm first the self clustering algorithm that determines proper input space partitioning and finds the mean and variance of the gaussian membership functions and number of rules is applied to the initial population second the subgroup symbiotic evolution method that uses each subantibody represents a single fuzzy rule and the evolution of the rule itself third the immune system algorithm uses the clonal selection principle such that antibodies between others of high similar degree are canceled and these antibodies after processing will have higher quality accelerating the search and increasing the global search capacity finally the proposed cnfc with isel cnfc isel method is adopted to solve several nonlinear control problems the simulation results have shown that the proposed cnfc isel can outperform other methods

time fractional wave diffusion equation in an inhomogeneous half space we consider the fundamental solution of the time fractional wave diffusion equation in a three dimensional half space medium which contains an inhomogeneity in form of a plane parallel layer the corresponding green s function which is derived by means of the fourier and laplace transforms can be accurately and efficiently evaluated without recourse to the mittag leffler or the fox h function moreover it is shown that in the one dimensional case the fundamental solution in an inhomogeneous half space is no longer a probability density function in addition we consider the advection equation for the fractional laplacian and the caputo time fractional derivative of orders on a bounded domain simple algorithms for accurate evaluation of the m wright function and the mittag leffler function are enclosed at the end of this article

mittag leffler analysis ii application to the fractional heat equation abstract mittag leffler analysis is an infinite dimensional analysis with respect to non gaussian measures of mittag leffler type which generalizes the powerful theory of gaussian analysis and in particular white noise analysis in this paper we further develop the mittag leffler analysis by characterizing the convergent sequences in the distribution space moreover we provide an approximation of donsker s delta by square integrable functions then we apply the structures and techniques from mittag leffler analysis in order to show that a green s function to the time fractional heat equation can be constructed using generalized grey brownian motion ggbm by extending the fractional feynman kac formula from schneider moreover we analyse ggbm show its differentiability in a distributional sense and the existence of corresponding local times

selecting fuzzy if then rules for classification problems using genetic algorithms this paper proposes a genetic algorithm based method for selecting a small number of significant fuzzy if then rules to construct a compact fuzzy classification system with high classification power the rule selection problem is formulated as a combinatorial optimization problem with two objectives to maximize the number of correctly classified patterns and to minimize the number of fuzzy if then rules genetic algorithms are applied to this problem a set of fuzzy if then rules is coded into a string and treated as an individual in genetic algorithms the fitness of each individual is specified by the two objectives in the combinatorial optimization problem the performance of the proposed method for training data and test data is examined by computer simulations on the iris data of fisher

fractional non linear linear and sublinear death processes this paper is devoted to the study of a fractional version of non linear open image in new window t 0 linear m t t 0 and sublinear mathfrak m nu t t 0 death processes fractionality is introduced by replacing the usual integer order derivative in the difference differential equations governing the state probabilities with the fractional derivative understood in the sense of dzhrbashyan caputo we derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values a useful subordination relation is also proved allowing us to express the death processes as compositions of their classical counterparts with the random time process t2 t t 0 this random time has one dimensional distribution which is the folded solution to a cauchy problem of the fractional diffusion equation

a value efficiency approach to incorporating preference information in data envelopment analysis we develop a procedure and the requisite theory for incorporating preference information in a novel way in the efficiency analysis of decision making units the efficiency of deci sion making units is defined in the spirit of data envelopment analysis dea complemented with decision maker s preference information concerning the desirable structure of inputs and outputs our procedure begins by aiding the decision maker in searching for the most preferred combination of inputs and outputs of decision making units for short most preferred solution which are efficient in dea then assuming that the decision maker s most preferred solution maximizes his her underlying unknown value function we approximate the indifference contour of the value function at this point with its possible tangent hyperplanes value efficiency scores are then calculated for each decision making unit comparing the inefficient units to units having the same value as the most preferred solution the resulting value efficiency scores are optimistic approximations of the true scores the procedure and the resulting efficiency scores are immediately applicable to solving practical problems

elicitation and fine tuning of fuzzy control rules using symbiotic evolution this paper exploits the ability of symbiotic evolution se as a generic methodology to elicit a fuzzy rule base of the mamdani type almost all fuzzy rule base generation algorithms produce rule bases with redundant and overlapped membership functions that limit their interpretability elegance in their application we address this problem by applying an algorithm to merge any similar membership functions it is shown that our proposed algorithm leads generally to a more transparent and more interpretable rule base with a minimum number of membership functions and a reduced number of rules in addition a new post processing approach is proposed for recovering any probable performance lost after membership functions merging the proposed methodology has been applied successfully for the design of an active control suspension system using a non linear bond graphs bg based half car model with parameters that relate to a ford fiesta mk2

the possibilistic c means algorithm insights and recommendations recently the possibilistic c means algorithm pcm was proposed to address the drawbacks associated with the constrained memberships used in algorithms such as the fuzzy c means fcm in this issue barni et al 1996 report a difficulty they faced while applying the pcm and note that it exhibits an undesirable tendency to converge to coincidental clusters the purpose of this paper is not just to address the issues raised by barni et al but to go further and analytically examines the underlying principles of the pcm and the possibilistic approach in general we analyze the data sets used by barni et al and interpret the results reported by them in the light of our findings

fractional diffusions with time varying coefficients this paper is concerned with the fractionalized diffusion equations governing the law of the fractional brownian motion bh t we obtain solutions of these equations which are probability laws extending that of bh t our analysis is based on mcbride fractional operators generalizing the hyper bessel operators l and converting their fractional power l into erdelyi kober fractional integrals we study also probabilistic properties of the random variables whose distributions satisfy space time fractional equations involving caputo and riesz fractional derivatives some results emerging from the analysis of fractional equations with time varying coefficients have the form of distributions of time changed random variables

brownian time processes the pde connection ii and the corresponding feynman kac formula we delve deeper into our study of the connection of brownian time processes btps to fourth order parabolic pdes which we introduced in a recent joint article with w zheng probabilistically btps and their cousins btps with excursions form a unifying class of interesting stochastic processes that includes the celebrated ibm of burdzy and other new intriguing processes and is also connected to the markov snake of le gall btps also offer a new connection of probability to pdes that is fundamentally different from the markovian one they solve fourth order pdes in which the initial function plays an important role in the pde itself not only as initial data we connect two such types of interesting and new pdes to btps the first is obtained by running the btp and then integrating along its path and the second type of pdes is related to what we call the feynman kac formula for btps a special case of the second type is a step towards a probabilistic solution to linearized cahn hilliard and kuramoto sivashinsky type pdes which we tackle in an upcoming paper

brownian time processes the pde connection and the half derivative generator we introduce a class of interesting stochastic processes based on brownian time processes these are obtained by taking markov processes and replacing the time parameter with the modulus of brownian motion they generalize the iterated brownian motion ibm of burdzy and the markov snake of le gall and they introduce new interesting examples after defining brownian time processes we relate them to fourth order parabolic partial differential equations pde s we then study their exit problem as they exit nice domains in r d and connect it to elliptic pde s we show that these processes have the peculiar property that they solve fourth order parabolic pde s but their exit distribution at least in the standard brownian time process case solves the usual second order dirichlet problem we recover fourth order pde s in the elliptic setting by encoding the iterative nature of the brownian time process through its exit time in a standard brownian motion we also show that it is possible to assign a formal generator to these non markovian processes by giving such a generator in the half derivative sense

vibrations and fractional vibrations of rods plates and fresnel pseudo processes different initial and boundary value problems for the equation of vibrations of rods also called fresnel equation are solved by exploiting the connection with brownian motion and the heat equation the equation of vibrations of plates is considered and the case of circular vibrating disks cr is investigated by applying the methods of planar orthogonally reflecting brownian motion within cr the analysis of the fractional version of order of the fresnel equation is also performed and in detail some specific cases like 1 2 1 3 2 3 are analyzed by means of the fundamental solution of the fresnel equation a pseudo process f t t 0 with real sign varying density is constructed and some of its properties examined the composition of f with reflecting brownian motion b yields the law of biquadratic heat equation while the composition of f with the first passage time tt of b produces a genuine probability law strictly connected with the cauchy process

gensofnn a generic self organizing fuzzy neural network existing neural fuzzy neuro fuzzy networks proposed in the literature can be broadly classified into two groups the first group is essentially fuzzy systems with self tuning capabilities and requires an initial rule base to be specified prior to training the second group of neural fuzzy networks on the other hand is able to automatically formulate the fuzzy rules from the numerical training data no initial rule base needs to be specified prior to training a cluster analysis is first performed on the training data and the fuzzy rules are subsequently derived through the proper connections of these computed clusters however most existing neural fuzzy systems whether they belong to the first or second group encountered one or more of the following major problems they are 1 inconsistent rule base 2 heuristically defined node operations 3 susceptibility to noisy training data and the stability plasticity dilemma and 4 needs for prior knowledge such as the number of clusters to be computed hence a novel neural fuzzy system that is immune to the above mentioned deficiencies is proposed in this paper this new neural fuzzy system is named the generic self organizing fuzzy neural network gensofnn the gensofnn network has strong noise tolerance capability by employing a new clustering technique known as discrete incremental clustering dic the fuzzy rule base of the gensofnn network is consistent and compact as gensofnn has built in mechanisms to identify and prune redundant and or obsolete rules extensive simulations were conducted using the proposed gensofnn network and its performance is encouraging when benchmarked against other neural and neural fuzzy systems

a hybrid genetic algorithm for multiobjective problems with activity analysis based local search the objective of this research was the development of a method that integrated an activity analysis model of profits from production with a biophysical model and included the capacity for optimization over multiple objectives we specified a hybrid genetic algorithm using activity analysis as a local search method and nsga ii for calculation of the multiple objective pareto optimal set we describe a parallel computing approach to computation of the genetic algorithm and apply the algorithm to evaluation of an input tax to regulate pollution from agricultural production

structural comparison of data envelopment analysis and multiple objective linear programming the concept of efficiency as it applies to decision making units dmus solutions alternatives plays an important role both in data envelopment analysis dea and multiple objective linear programming molp despite this and other apparent similarities dea and molp research has developed separately we show that structurally the dea formulation to identify efficient units is quite similar to the molp model based on the reference point or the reference direction approach to generate efficient solutions dea and molp should not be seen as substitutes but rather as complements we show that they cross fertilize each other molp provides interesting extensions to dea and dea provides new areas of application to molp

asymptotic properties of some space time fractional stochastic equations consider non linear time fractional stochastic heat type equations of the following type begin aligned partial beta tu t x nu delta alpha 2 u t x i 1 beta t lambda sigma u mathop f limits cdot t x end aligned in d 1 dimensions where nu 0 beta in 0 1 alpha in 0 2 the operator partial beta t is the caputo fractional derivative while delta alpha 2 is the generator of an isotropic stable process and i 1 beta t is the riesz fractional integral operator the forcing noise denoted by mathop f limits cdot t x is a gaussian noise and the multiplicative non linearity sigma mathbb r rightarrow mathbb r is assumed to be globally lipschitz continuous mijena and nane stochastic process appl 125 9 3301 3326 2015 have introduced these time fractional spdes these types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory under suitable conditions on the initial function we study the asymptotic behaviour of the solution with respect to time and the parameter lambda in particular our results are significant extensions of those in ann probab to appear foondun and khoshnevisan electron j probab 14 21 548 568 2009 mijena and nane 2015 and mijena and nane potential anal 44 295 312 2016 along the way we prove a number of interesting properties about the deterministic counterpart of the equation

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

fractional diffusion equations and processes with randomly varying time in this paper the solutions u u x t to fractional diffusion equations of order 0 v 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes for the fractional equations of order v 1 2 n n 1 we show that the solutions u 1 2 n correspond to the distribution of the n times iterated brownian motion for these processes the distributions of the maximum and of the sojourn time are explicitly given the case of fractional equations of order v 2 3 n n 1 is also investigated and related to brownian motion and processes with densities expressed in terms of airy functions in the general case we show that u coincides with the distribution of brownian motion with random time or of different processes with a brownian time the interplay between the solutions u and stable distributions is also explored interesting cases involving the bilateral exponential distribution are obtained in the limit

a generalized model for data envelopment analysis abstract data envelopment analysis dea is a method to estimate a relative efficiency of decision making units dmus performing similar tasks in a production system that consumes multiple inputs to produce multiple outputs so far a number of dea models have been developed the ccr model the bcc model and the fdh model are well known as basic dea models these models based on the domination structure in primal form are characterized by how to determine the production possibility set from a viewpoint of dual form the convex cone the convex hull and the free disposable hull for the observed data respectively in this study we suggest a model called generalized dea gdea model which can treat the above stated basic dea models in a unified way in addition by establishing the theoretical properties on relationships among the gdea model and those dea models we prove that the gdea model makes it possible to calculate the efficiency of dmu incorporating various preference structures of decision makers furthermore we propose a dual approach to gdea gdea d and also show that gdea d can reveal domination relations among all dmus

levy mixing related to distributed order calculus subordinators and slow diffusions abstract the study of distributed order calculus usually concerns fractional derivatives of the form 0 1 u m d for some measure m eventually a probability measure in this paper an approach based on levy mixing is proposed non decreasing levy processes associated with levy triplets of the form a y b y d s y are considered and the parameter y is randomized by means of a probability measure the related subordinators are studied from different points of view some distributional properties are obtained and the interplay with inverse local times of markov processes is explored distributed order integro differential operators are introduced and adopted in order to write explicitly the governing equations of such processes an application to slow diffusions delayed brownian motion is discussed

nonlinear control via approximate input output linearization the ball and beam example approximate input output linearization of nonlinear systems which fail to have a well defined relative degree is studied for such systems a method for constructing approximate systems that are input output linearizable is provided the analysis presented is motivated through its application to a common undergraduate control laboratory experiment the ball and beam where it is shown to be more effective for trajectory tracking than the standard jacobian linearization

l kuramoto sivashinsky spdes in one to three dimensions l ks kernel sharp holder regularity and swift hohenberg law equivalence generalizing the l kuramoto sivashinsky l ks kernel from our earlier work we give a novel explicit kernel formulation useful for a large class of fourth order deterministic stochastic linear and nonlinear pdes in multispatial dimensions these include pattern formation equations like the swift hohenberg and many other prominent and new pdes we first establish existence uniqueness and sharp dimension dependent spatio temporal holder regularity for the canonical zero drift l ks spde driven by white noise on r rd d 13 the spatio temporal holder exponents are exactly the same as the striking ones we proved for our recently introduced brownian time brownian motion btbm stochastic integral equation associated with time fractional pdes the challenge here is that unlike the positive btbm density the l ks kernel is the gaussian average of a modified highly oscillatory and complex schrodinger propagator we use a combination of harmonic and delicate analysis to get the necessary estimates second attaching order parameters e1 to the l ks spatial operator and e2 to the noise term we show that the dimension dependent critical ratio e2 e1d 8 controls the limiting behavior of the l ks spde as e1 e2 0 and we compare this behavior to that of the less regular second order heat spdes finally we give a change of measure equivalence between the canonical l ks spde and nonlinear l ks spdes in particular we prove uniqueness in law for the swift hohenberg and the law equivalence and hence the same holder regularity of the swift hohenberg spde and the canonical l ks spde on compacts in one to three dimensions

a ga paradigm for learning fuzzy rules abstract in this paper we describe a paradigm for learning fuzzy rules using genetic algorithms ga we formulate our problem of learning as follows given a set of linguistic values that characterize the input and output state variables of the system in consideration derive an n rule fuzzy control algorithm the value n represents a specified constraint of the ga in searching for a functional ruleset the ga learning paradigm is powerful since it requires no prior knowledge about the system s behavior in order to formulate a set of functional control rules through adaptive learning we present our simulation results using the classical inverted pendulum control problem to demonstrate the effectiveness of the ga learning scheme results have shown that the approach has great potential as a tool for the learning of fuzzy control rules particularly in situations where the knowledge from a human expert is not easily accessible

implementation of evolutionary fuzzy systems evolutionary fuzzy systems are discussed in which the membership function shapes and types and the fuzzy rule set including the number of rules inside it are evolved using a genetic evolutionary algorithm in addition the genetic parameters operators of the evolutionary algorithm are adapted via a fuzzy system benefits of the methodology are illustrated in the process of classifying the iris data set possible extensions of the methods are summarized

random walks with strongly inhomogeneous rates and singular diffusions convergence localization and aging in one dimension let r i i z denote i i d positive random variables with common distribution f and conditional on r let x x t t 0 x 0 0 be a continuous time simple symmetric random walk on z with inhomogeneous rates 1 i i z when f is in the domain of attraction of a stable law of exponent a 0 which is independent of s 0 because of scaling self similarity properties of z p the scaling properties of z p are also closely related to the aging of x our main technical result is a general convergence criterion for localization and aging functionals of diffusions walks y e with nonrandom speed measures e in a sufficiently strong sense

strang type preconditioners for solving fractional diffusion equations by boundary value methods the finite difference scheme with the shifted grunwald formula is employed to semi discrete the fractional diffusion equations this spatial discretization can reduce to the large system of ordinary differential equations odes with initial values recently the boundary value method bvm was developed as a popular algorithm for solving the large systems of odes this method requires the solutions of one or more nonsymmetric and large scale linear systems in this paper the gmres method with the block circulant preconditioner is proposed to solve relevant linear systems some conclusions about the convergence analysis and spectrum of the preconditioned matrices are also drawn if the diffusion coefficients are constant finally extensive numerical experiments are reported to show the performance of our method for solving the fractional diffusion equations

simultaneous design of membership functions and rule sets for fuzzy controllers using genetic algorithms this paper examines the applicability of genetic algorithms ga s in the simultaneous design of membership functions and rule sets for fuzzy logic controllers previous work using genetic algorithms has focused on the development of rule sets or high performance membership functions however the interdependence between these two components suggests a simultaneous design procedure would be a more appropriate methodology when ga s have been used to develop both it has been done serially e g design the membership functions and then use them in the design of the rule set this however means that the membership functions were optimized for the initial rule set and not the rule set designed subsequently ga s are fully capable of creating complete fuzzy controllers given the equations of motion of the system eliminating the need for human input in the design loop this new method has been applied to two problems a cart controller and a truck controller beyond the development of these controllers we also examine the design of a robust controller for the cart problem and its ability to overcome faulty rules

analytic solutions of fractional differential equations by operational methods abstract we describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions as an example we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics fractionality is obtained by substituting the ordinary integer order derivative with the caputo fractional derivative furthermore operational relations between ordinary and fractional differentiation are shown and discussed in detail finally a last example concerns the application of the method to the study of a fractional poisson process

a multiple criteria approach to data envelopment analysis in this paper we present a multiple criteria data envelopment analysis mcdea model which can be used to improve discriminating power of dea methods and also effectively yield more reasonable input and output weights without a priori information about the weights in the proposed model several different efficiency measures including classical dea efficiency are defined under the same constraints each measure serves as a criterion to be optimized efficiencies are then evaluated under the framework of multiple objective linear programming molp the method is illustrated through three examples in which data sets are taken from previous research on dea s discriminating power and weight restriction

brownian subordinators and fractional cauchy problems a brownian time process is a markov process subordinated to the absolute value of an independent one dimensional brownian motion its transition densities solve an initial value problem involving the square of the generator of the original markov process an apparently unrelated class of processes emerging as the scaling limits of continuous time random walks involves subordination to the inverse or hitting time process of a classical stable subordinator the resulting densities solve fractional cauchy problems an extension that involves fractional derivatives in time in this paper we will show a close and unexpected connection between these two classes of processes and consequently an equivalence between these two families of partial differential equations

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems