improved and more feasible numerical methods for riesz space fractional partial differential equations abstract in this article the numerical solutions of fractional order partial differential equations with riesz space fractional derivatives on a finite domain have been considered two kinds of equations have been considered the riesz fractional diffusion equation and the riesz fractional advection dispersion equation the riesz fractional diffusion equation is obtained from the standard diffusion equation by replacing the second order space derivative with the riesz fractional derivative of order where 1 2 the riesz fractional advection dispersion equation is obtained from the standard advection dispersion equation by replacing the first and second order space derivatives with the riesz fractional derivatives of order 1 1 and 1 2 respectively to obtain the solutions firstly the system of odes is obtained using the improved matrix transform method with respect to the space variable after this the 3 1 pade approximation can be used to construct the exponential matrix in the analytic solution of the ode and we obtain two difference schemes it has also been shown that the two difference schemes are unconditionally stable and feasible using the matrix analysis method finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

topological properties of eschenburg spaces and 3 sasakian manifolds we examine topological properties of the seven dimensional positively curved eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic a special subfamily of these manifolds also carries a 3 sasakian metric among these we construct a pair of 3 sasakian spaces which are diffeomorphic to each other thus giving rise to the first example of a manifold which carries two non isometric 3 sasakian metrics

some exact results for the trapping of subdiffusive particles in one dimension we study a generalization of the standard trapping problem of random walk theory in which particles move subdiffusively on a one dimensional lattice we consider the cases in which the lattice is filled with a one sided and a two sided random distribution of static absorbing traps with concentration c the survival probability t that the random walker is not trapped by time t is obtained exactly in both versions of the problem through a fractional diffusion approach comparison with simulation results is made

asymptotically cylindrical calabi yau manifolds let m be a complete ricci flat kahler manifold with one end and assume that this end converges at an exponential rate to 0 oo x x for some compact connected ricci flat manifold x we begin by proving general structure theorems for m in particular we show that there is no loss of generality in assuming that m is simply connected and irreducible with hol m su n where n is the complex dimension of m if n 2 we then show that there exists a projective orbifold barm and a divisor bard in k barm with torsion normal bundle such that m is biholomorphic to barm bard thereby settling a long standing question of yau in the asymptotically cylindrical setting we give examples where barm is not smooth the existence of such examples appears not to have been noticed previously conversely for any such pair barm bard we give a short and self contained proof of the existence and uniqueness of exponentially asymptotically cylindrical calabi yau metrics on barm bard

approximate nonlinear receding horizon control laws in closed form the receding horizon control problem for continuous time affine nonlinear systems is approximated by a multi step predictive control formulation under a controllability condition a closed form nonlinear feedback control law is obtained the output tracking problem is treated in a similar fashion and a closed form tracking control law is constructed the stability properties of these control laws are examined

molecular evolution modeled as a fractal statistical process abstract modeling the rate of nucleotide substitutions in dna as a dichotomous stochastic process with an inverse power law correlation function describes evolution by a fractal stochastic process fsp this fsp model agrees with recent findings on the relationship between the variance and mean number of synonymous and nonsynonymous substitutions in 49 different genes in mammals that being a power law increase in the ratio of the variance to the mean the index of dispersion with the number of substitutions in a protein the probability of a given number of substitutions occuring in a time t is determined by a fractional diffusion equation whose solution is a truncated levy distribution implying that evolution is a levy process in time and yields the same functional behavior for the variance in the number of substitutions as does the fsp model in addition to obtaining these relationships the fsp model implies lognormal statistics for the index of dispersion as a function of the mean number of substitutions in a protein which is confirmed in the regression of the fsp model to data lognormal statistics suggest that molecular evolution can be viewed as a multiplicative stochastic process rather than the linear additive processes of darwinian selection and drift

linear systems with levy fluctuations abstract we have examined the response of multivariable linear systems to levy fluctuations the characteristic function of the response is obtained explicitly as a function of time using the characteristic function the fractional moments of the process are determined the nature of the product rule obeyed by the characteristic function of a general linear dynamical system is elucidated

compact riemannian 7 manifolds with holonomy g sb 2 ii this is the second of two papers about metrics of holonomy g2 on compact 7manifolds in our first paper 15 we established the existence of a family of metrics of holonomy g2 on a single compact simply connected 7 manifold m using three general results theorems a b and c our purpose in this paper is to explore the theory of compact riemannian 7 manifolds with holonomy g2 in greater detail by relying on theorems a c we will be able to avoid the emphasis on analysis that characterized 15 so that this sequel will have a more topological flavour the paper has four chapters the first chapter consists of introductory material section 1 1 gives some elementary geometric and topological material on compact 7 manifolds with torsion free g2structures then 1 2 describes the holonomy groups su 2 and su 3 and 1 3 explains the concept of asymptotically locally euclidean riemannian manifolds shortened to ale spaces with special holonomy recall that in 15 a compact 7 manifold m was defined by desingularizing a quotient t of the 7 torus by a finite group of isometries z2 the subject of chapters 2 and 3 is a generalization of this idea chapter 2 defines a general construction for compact 7 manifolds with torsion free g2structures which works by desingularizing quotients t for finite groups the ale spaces with holonomy su 2 and su 3 discussed in 1 3 are an essential ingredient in performing this desingularization the central result of chapter 2 is theorem 2 2 3 which states that given a suitable finite group and certain other data one may construct a compact 7manifold m from t that admits torsion free g2structures this result is proved using theorems a c of 15 chapter 3 is devoted entirely to examples of this construction we give many examples of compact 7 manifolds with holonomy g2 and determine their basic topological invariants the betti numbers and fundamental group finally in chapter 4 we discuss some areas of interest and give a number of open problems this paper is not written to be read independently of 15 the language and results of 15 will be used freely in particular the introductory material in 15 1 1 for reference we reproduce here the model 3and 4forms defining the flat g2structure on r as given in 15 1 1

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

fractional diffusion waiting time distributions and cattaneo type equations we discuss a generalized diffusion equation resulting from an additive two state process in combination with an asymptotically fractal asymptotic power law waiting time distribution the obtained equation is an extension to previously discussed fractional diffusion equations our description leads to a mean squared displacement which describes enhanced subballistic transport for long times the short time behavior however is of a ballistic nature this separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting time distribution this is also mirrored by the observation that for small times our generalized diffusion equation reduces to the standard cattaneo equation the asymptotic probability density is of compressed gaussian type and thus differs from the l evy tail generally found for these kinds of processes

a continuous variant for grunwald letnikov fractional derivatives abstract the names of grunwald and letnikov are associated with discrete convolutions of mesh h multiplied by h when h tends to zero the result tends to a marchaud s derivative of the order of of the function to which the convolution is applied the weights w k of such discrete convolutions form well defined sequences proportional to k 1 near infinity and all moments of integer order r are equal to zero provided is not an integer we present a continuous variant of grunwald letnikov formulas with integrals instead of series it involves a convolution kernel which mimics the above mentioned features of grunwald letnikov weights a first application consists in computing the flux of particles spreading according to random walks with heavy tailed jump distributions possibly involving boundary conditions

fractional diffusion probability distributions and random walk models abstract we present a variety of models of random walk discrete in space and time suitable for simulating random variables whose probability density obeys a space time fractional diffusion equation

multidimensional solutions of space fractional diffusion equations green s functions and propagators for the multidimensional anisotropic spacetime fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework

boundary problems for diffusion in a fluctuating potential abstract using a newly developed formalism adapted for diffusion in the presence of partially absorbing boundaries we treat the overdamped motion of a particle in a fluctuating potential and in a restricted space the potential is assumed to be linear with a slope switching between two possible values the switching process is taken as the standard markov alternating process we first construct the free dynamics propagator no boundaries which controls the evolution of the joint probability for both the particle s position and the state of the potential secondly at a given spatial point we affix a two channel boundary a given channel absorbs the trajectories arriving at the boundary with the potential in a given state we demonstrate the asymmetry of the filling process for the individual channels and we discuss their asymptotic occupation depending on the mean switching frequency finally we study the motion in a domain restricted by two boundaries of the above two channel type and the corresponding first passage time the simple problem setting allows for an explicit analysis of the correlation between the particle s motion and the potential switching process

surgery and duality surgery as developed by browder kervaire milnor novikov sullivan wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension 5 in this paper a modiflcation of this theory is presented where instead of flxing a homotopy type one considers a weaker information roughly speaking one compares n dimensional compact manifolds with topological spaces whose k skeletons are flxed where k is at least n 2 a particularly attractive example which illustrates the concept is given by complete intersections by the lefschetz hyperplane theorem a complete intersection of complex dimension n has the same n skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known the theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group this was under certain restrictions carried out in tr the restrictions are if d d1 dr is the total degree of a complete intersection x n1 dr of complex dimension n then the assumption is that for all primes p with p pi1 n 1 the total degree d is divisible by p 2n 1 2pi1 1 theorem a two complete intersections x n

hierarchical spatio temporal coupling in fractional wanderings i continuous time weierstrass flights abstract the one dimensional continuous time weierstrass flights ctwf model is considered in the framework of the nonseparable continuous time random walks formalism ctrw a novel spatio temporal coupling is introduced by assuming that in each scale the probability density for the flight and for waiting are joined hence we treat the spatio temporal relations in terms of the self similar structure of the weierstrass process this stochastic structure is characterized by the spatial fractional dimension 1 representing the flights and the temporal one 1 representing the waiting time was assumed here as the only independent truncation range in this work we study the asymptotic properties of the ctwf model for example by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form p x t t pt 2 f where the scalingfunction f pt exp const pt while the scaling variable x t pt 2 is large the principal result of our analysis is that the exponents and depend on more fundamental ones and what leads to a novel scaling as a result of competition between exponents and an enhanced dispersive or normal diffusion was recognized in distinction from the prediction of the separable ctrw model where the enhanced diffusion is lost and the dispersive one is strongly limited it should be noted that we compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes having the propogator we calculated for example the particle mean square displacement and found its novel asymptotic scaling with time for enhanced diffusion given by t1 2 1 in distinction from its diverging for

visibility to discern local from nonlocal dynamic processes we compare using visibility the usual kardar parisi zhang kpz universality class and a fractional edward wilkinson ewf equation with correlated noise which share the same kinetic roughening exponents the kpz universality class is described by an equation in terms of the usual derivatives uncorrelated noise and therefore is intrinsically local the second model includes fractional powers of the laplace operator and correlated noise both of which are nonlocal from their scaling properties one could be tempted to conclude that both dynamics belong to the same universality class specifically to the kpz universality class however this is a wrong conclusion that calls the attention against the indiscriminate application of this approach in real systems without taking into consideration basic physical assumptions e g locality these examples reveal the necessity of finding new algorithms for detecting characteristics that remain unnoticed to classical scaling analysis where only the two first moments of the interface distribution mean and variance are used to classify the dynamics we show that visibility and in particular the kinetic roughening exponents of the visibility interface are able to distinguish between these two dynamics which are confused by standard techniques

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

closed form control laws for linear time varying systems closed form control laws are developed for continuous linear time varying ltv systems based on approximate solutions to a receding horizon control problem these control laws can be derived in the first or higher order closed forms once obtained the control laws need no explicit gain scheduling or online integrations to implement the notion of practical stability is used and practical or uniform asymptotic stability of the closed loop system depending on conditions imposed on the system is established

hierarchical spatio temporal coupling in fractional wanderings ii diffusion phase diagram for weierstrass walks abstract the one dimensional weierstrass walks ww model is developed in the framework of the extended nonseparable continuous time random walk ctrw formalism 1 2 3 4 5 6 7 8 9 10 11 12 the ww model is a lacunary foundation of levy walks 6 7 8 9 10 11 12 generalized to a nonconstant velocity this nonconstant velocity is introduced by hierarchical coherent spatio temporal coupling adopted from the continuous time weierstrass flights ctwf model developed in the previous paper 13 hence for the probability density to pass by a walker in a single step a random displacement with finite velocity is constructed as a geometric series of the corresponding probability densities defined within a sequence of spatio temporal scales we calculated analytically and by monte carlo simulations the asymptotic in time mean square displacement msd of the walker obtaining very good agreement between both approaches also comparison with corresponding results of the ctwf model is made considering different types of the diffusion exponents we constructed a diffusion phase diagram on the plane defined by the spatial and temporal fractional dimensions which characterize our coupling we obtained a diffusion exponent as a function of these fractional dimensions covering all types of nonbiased diffusion known up to now from the dispersive one over the normal enhanced ballistic and hyperdiffusion up to the richardson law of diffusion which defines here a part of the borderline of the region where the msd diverges we observed that all kinds of anomalous diffusion are characterized by three types of diffusion exponents only for example we found an asymptotic scaling of msd to occur with time for enhanced diffusion which was discovered by us within the ctwf model but is valid for a much more extended range of spatio and temporal fractional dimensions

green function of the double fractional fokker planck equation path integral and stochastic differential equations the statistics of rare events the so called black swan events is governed by non gaussian distributions with heavy power like tails we calculate the green functions of the associated fokker planck equations and solve the related stochastic differential equations we also discuss the subject in the framework of path integration

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

dimensional influence on blow up in a superdiffusive medium thermal blow up in a superdiffusive medium with a localized energy source is examined for a spatial domain of infinite extent in one two and three dimensions an analysis of a nonlinear model of this problem reveals that the occurrence of a blow up depends upon the spatial dimension of the superdiffusive medium in one dimension a blow up always occurs in two and three dimensions a blow up can be avoided if the superdiffusive properties are sufficiently enhanced the asymptotic growth of the temperature near blow up is determined for energy sources whose output increases in either an algebraic or an exponential manner the results of the analysis have potential application to porous materials in which a pore filling agent is introduced to enhance the thermal transport properties

new numerical methods for the riesz space fractional partial differential equations in this paper we consider the numerical solution of the riesz space fractional diffusion equation and advection dispersion equation first a system of ordinary differential equations is obtained from the above equations with respect to the space variable by using the improved matrix transform method furthermore we use the 2 2 pade approximation to compute the exponential matrix in the analytic solution of the ordinary differential equation and get two difference schemes second using the matrix analysis method we prove that the two difference schemes are unconditionally stable finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

analytical solutions to time fractional partial differential equations in a two dimensional multilayer annulus abstract in this paper analytical solutions to time fractional partial differential equations in a multi layer annulus are presented the final solutions are obtained in terms of mittag leffler function by using the finite integral transform technique and laplace transform technique in addition the classical diffusion equation 1 the helmholtz equation 0 and the wave equation 2 are discussed as special cases finally an illustrative example problem for the three layer semi circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

time fractional schrodinger equation fox s h functions and the effective potential after introducing the formalism of the general space and time fractional schrodinger equation we concentrate on the time fractional schrodinger equation and present new results via the elegant language of fox s h functions we show that the general time dependent part of the wave function for the separable solutions of the time fractional schrodinger equation is the mittag leffler function with an imaginary argument by two different methods after separating the mittag leffler function into its real and imaginary parts in contrast to existing works we show that the total probability is 1 and decays with time introducing the effective potential approach we also write the mittag leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts in the light of these we reconsider the simple box problem

fractional diffusion exact representations of spectral functions for all the relevant transformed spaces i e fourier laplace and fourier laplace we present exact solutions of a fractional diffusion equation describing random transport on fractals the potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes in contrast to the well known asymptotic results the exact representations provide a much broader basis for comparison with data

diffusion equations for a markovian jumping process we consider a markovian jumping process which is defined in terms of the jump size distribution and the waiting time distribution with a position dependent frequency in the diffusion limit we assume the power law form for the frequency for small steps we derive the fokker planck equation and show the presence of the normal diffusion subdiffusion and superdiffusion for the l evy distribution of the step size we construct a fractional equation which possesses a variable coefficient and solve it in the diffusion limit then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance we also solve the master equation numerically and demonstrate that there are deviations from the l evy stable distribution for large wave numbers

robust nonlinear generalised predictive control for a class of uncertain nonlinear systems via an integral sliding mode approach abstractin this paper a robust nonlinear generalised predictive control gpc method is proposed by combining an integral sliding mode approach the composite controller can guarantee zero steady state error for a class of uncertain nonlinear systems in the presence of both matched and unmatched disturbances indeed it is well known that the traditional gpc based on taylor series expansion cannot completely reject unknown disturbance and achieve offset free tracking performance to deal with this problem the existing approaches are enhanced by avoiding the use of the disturbance observer and modifying the gain function of the nonlinear integral sliding surface this modified strategy appears to be more capable of achieving both the disturbance rejection and the nominal prescribed specifications for matched disturbance simulation results demonstrate the effectiveness of the proposed approach

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

the time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems abstract in this paper a time fractional fourier law is obtained from fractional calculus according to the fractional fourier law a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built the fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases in addition we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized h function using integral transformation methods the fractional heat conduction equation in the case 0 1 interpolates the standard heat conduction equation 1 and the localized heat conduction equation 0 finally numerical results are presented graphically for various values of order of fractional derivative

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

levy processes on a generalized fractal comb comb geometry constituted of a backbone and fingers is one of the most simple paradigm of a two dimensional structure where anomalous diffusion can be realized in the framework of markov processes however the intrinsic properties of the structure can destroy this markovian transport these effects can be described by the memory and spatial kernels in particular the fractal structure of the fingers which is controlled by the spatial kernel in both the real and the fourier spaces leads to the levy processes levy flights and superdiffusion this generalization of the fractional diffusion is described by the riesz space fractional derivative in the framework of this generalized fractal comb model levy processes are considered and exact solutions for the probability distribution functions are obtained in terms of the fox h function for a variety of the memory kernels and the rate of the superdiffusive spreading is studied by calculating the fractional moments for a special form of the memory kernels we also observed a competition between long rests and long jumps finally we considered the fractal structure of the fingers controlled by a weierstrass function which leads to the power law kernel in the fourier space this is a special case when the second moment exists for superdiffusion in this competition between long rests and long jumps

solutions of the space time fractional cattaneo diffusion equation abstract the object of this paper is to present the exact solution of the fractional cattaneo equation for describing anomalous diffusion the classical cattaneo model has been generalised to the space time fractional cattaneo model the method of the joint laplace and fourier transform is used in deriving the solution the solutions of the fractional cattaneo equation are obtained under integral and series forms in terms of the h functions finally the fractional order moments are also investigated

time and space fractional partial differential equations the fundamental solution for time and space fractional partial differential operator dt a2 2 0 is given in terms of the fox s h function here the time fractional derivative in the sense of generalized functions distributions dt is defined by the convolution dt f t t f t where t t 1 and f t 0 as t 0 and the fractional n dimensional laplace operator 2 is defined by its fourier transform with respect to spatial variable f 2g x f g x the solutions for initial value problems for time and space fractional partial differential equation in the sense of caputo and riemann liouville time fractional derivatives respectively are obtained by the fundamental solution

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

superdiffusion of a random walk driven by an ergodic markov process with switching we propose a markov model with an ergodic two component switching mechanism that dynamically generates anomalous diffusion the first component plays the role of a hidden parameter the second one is the switching component generating the superdiffusion of a random walker and is itself non markovian the model is studied numerically using the monte carlo technique

discrete random walk models for symmetric levy feller diffusion processes we propose a variety of models of random walk discrete in space and time suitable for simulating stable random variables of arbitrary index 0 2 in the symmetric case we show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous markovian stochastic processes which we refer to as levy feller diffusion processes

first passage time moments of asymmetric levy flights we investigate the first passage dynamics of symmetric and asymmetric l evy flights in a semi infinite and bounded intervals by solving the space fractional diffusion equation we analyse the fractional order moments of the first passage time probability density function for different values of the index of stability and the skewness parameter a comparison with results using the langevin approach to l evy flights is presented for the semi infinite domain in certain special cases analytic results are derived explicitly and in bounded intervals a general analytical expression for the mean first passage time of l evy flights with arbitrary skewness is presented these results are complemented with extensive numerical analyses

anomalous diffusion modeling by fractal and fractional derivatives this paper makes an attempt to develop a fractal derivative model of anomalous diffusion we also derive the fundamental solution of the fractal derivative equation for anomalous diffusion which characterizes a clear power law this new model is compared with the corresponding fractional derivative model in terms of computational efficiency diffusion velocity and heavy tail property the merits and distinctions of these two models of anomalous diffusion are then summarized

the fundamental solutions of the space space time riesz fractional partial differential equations with periodic conditions in this paper the space time riesz fractional partial differential equations with periodic conditions are considered the equations are obtained from the integral partial differential equation by replacing the time derivative with a caputo fractional derivative and the space derivative with riesz potential the fundamental solutions of the space riesz fractional partial differential equation srfpde and the space time riesz fractional partial differential equation strfpde are discussed respectively using methods of fourier series expansion and laplace transform we derive the explicit expressions of the fundamental solutions for the srfpde and the strfpde respectively

non fickian flux for advection dispersion with immobile periods the fractal mobile immobile model mim is intermediate between advection dispersion ade and fractal fokker planck ffkpe equations it involves two time derivatives whose orders are 1 and between 0 and 1 on the left hand side whereas all mentioned equations have identical right hand sides the fractal mim model accounts for non fickian effects that occur when tracers spread in media because of through flow and can get trapped by immobile sites the solid matrix of a porous material may contain such sites so that non fickian spread is actually observed within the context of the fractal mim model we present a mapping that allows the computation of fluxes on the basis of the density of spreading particles the mapping behaves as fickian flux at early times and tends to a fractional derivative at late times by means of this mapping we recast the fractal mim model into conservative form which is suitable to deal with sources and bounded domains mathematical proofs are illustrated by comparing the discretized fractal p d e with monte carlo simulations

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

macroscopic current in fractional anomalous diffusion abstract fick s law which relates the macroscopic current with the density in ordinary diffusion is generalized to fractional diffusion where either the waiting time distribution or the displacement probability density exhibit power law decays these long tailed distributions give rise to respectively temporal or spatial nonlocality in connection between current and density in the case of power law displacement distributions the current is shown to also display such decays in the spatial coordinate an illustrative example is explicitly worked out in a one dimensional system

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

existence and regularity of a linear nonlocal fokker planck equation with growing drift abstract the nonlocal fokker planck equations for a class of stochastic differential equations with non gaussian stable levy motion in euclidean space are studied the existence and uniqueness of weak solution are obtained with rough drift the solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded moreover the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity

an asymptotic analytical solution to the problem of two moving boundaries with fractional diffusion in one dimensional drug release devices we set up a one dimensional mathematical model with a caputo fractional operator of a drug released from a polymeric matrix that can be dissolved into a solvent a two moving boundaries problem in fractional anomalous diffusion in time with order 0 1 under the assumption that the dissolving boundary can be dissolved slowly is presented in this paper the two parameter regular perturbation technique and fourier and laplace transform methods are used a dimensionless asymptotic analytical solution is given in terms of the wright function

second order accuracy finite difference methods for space fractional partial differential equations abstract space fractional partial differential equations are used for simulations of for example diffusion of radioactive materials and financial and other models which are characterized by heavy tailed distributions a number of first order accuracy finite difference methods have been proposed in the present paper we introduce second order accuracy finite difference methods with dirichlet boundary conditions these methods have a parameter in these schemes and the parameter stabilizes the schemes this means that there exist various schemes with second order accuracy but the stability of each scheme is different in the present paper we introduce the most stable scheme for any fractional calculus order by choosing the optimal parameter in addition we describe a phenomenon whereby the expected accuracy cannot be obtained if the analytical solution can be expanded to a series having less than second order around boundaries this also happens in both existing methods and the proposed methods in the present paper we develop the stability conditions for the proposed schemes and numerical examples of second order accuracy and accuracy decay are shown

solutions for a fractional nonlinear diffusion equation spatial time dependent diffusion coefficient and external forces we analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation we start our investigation by considering the linear case and the nonlinear case afterward the linear case is discussed taking fractional time and spatial derivatives into account in a unified approach we also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by d x t d t x for the nonlinear case we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation

approximate solutions of nonlinear fractional equations fractional exponential that are invariant under fractional derivatives elementary and special fractional functions are introduced approximate solutions to fractional burgers equation by using the homotopy perturbation method are obtained furthermore real integral representations for some h functions are found that may be very helpful in numerical computations

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

persistent random walk of cells involving anomalous effects and random death the purpose of this paper is to implement a random death process into a persistent random walk model which produces sub ballistic superdiffusion levy walk we develop a stochastic two velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction it is assumed that the switching rate is a decreasing function of residence running time this assumption leads to the power law for the velocity switching time distribution this describes the anomalous persistence of cell motility the longer the cell moves in one direction the smaller the switching probability to another direction becomes we derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives we show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process in the long time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a levy jump distribution we find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death rate dependent diffusion coefficient monte carlo simulations confirm these bounds

on the solution of fractional evolution equations this paper is devoted to the solution of the bi fractional differential equation for real 0 0 and 0 with the initial conditions here cd tu t x is the partial derivative coinciding with the caputo fractional derivative for 0 0 the laplace and fourier transforms are applied to solve the above problem in closed form the fundamental solution of these problems is established and its moments are calculated the special case 1 2 and 1 is presented and its application is given to obtain the dirac type decomposition for the ordinary diffusion equation

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

analytical solutions to the moving boundary problems with space time fractional derivatives in drug release devices a mathematical model of a solute release from a planar polymer matrix is presented and the analytical solution in terms of the fox h function is given the equation of space time fractional diffusion and a generalized fick s law are used in the paper three particular cases the standard diffusion the time fractional and the space fractional diffusions are discussed in detail the model and the solution are the generalization of the previous works and include them as special cases

g2 manifolds and associative submanifolds via semi fano 3 folds we construct many new topological types of compact g 2 manifolds i e riemannian 7 manifolds with holonomy group g 2 to achieve this we extend the twisted connected sum construction first developed by kovalev and apply it to the large class of asymptotically cylindrical calabi yau 3 folds built from semi fano 3 folds constructed previously by the authors in many cases we determine the diffeomorphism type of the underlying smooth 7 manifolds completely we find that many 2 connected 7 manifolds can be realised as twisted connected sums in a variety of ways raising questions about the global structure of the moduli space of g 2 metrics many of the g 2 manifolds we construct contain compact rigid associative 3 folds which play an important role in the the higher dimensional enumerative geometry gauge theory calibrated submanifolds approach to defining deformation invariants of g 2 metrics by varying the semi fanos used to build different g 2 metrics on the same 7 manifold we can change the number of rigid associative 3 folds we produce

effective medium equations for fractional fick s law in porous media abstract this paper studies reaction diffusion phenomena in disordered porous media with non fickian diffusion effects the aim is to obtain an effective medium equation of the concentration dynamics having a fractional fick s law description for the particles flux since the methodology is based on a volume averaging approach a fractional spatial averaging theorem is developed to interchange averaging integration and fractional differentiation model structure simplifications are made on the basis of an order of magnitude analysis from physical insights the closure problem associated with the effective diffusivity definition is also developed showing that the macroscale diffusion parameter is affected by i the scaling from mesoscales to macroscales and ii by the disordered structure of the porous medium

weak fano threefolds obtained by blowing up a space curve and construction of sarkisov links we characterise smooth curves in p3 whose blow up produces a threefold with anticanonical divisor big and nef these are curves c of degree d and genus g lying on a smooth quartic such that i 4d