a second order operator splitting method for allen cahn type equations with nonlinear source terms abstract allen cahn ac type equations with nonlinear source terms have been applied to a wide range of problems for example the vector valued ac equation for phase separation and the phase field equation for dendritic crystal growth in contrast to the well developed first and second order methods for the ac equation not many second order methods are suggested for the ac type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically in this paper we propose a simple and stable second order operator splitting method a core idea of the method is to decompose the original equation into three subequations with the free energy evolution term the heat evolution term and a nonlinear source term respectively it is important to combine these three subequations in proper order to achieve the second order accuracy and stability we propose a method with a half time free energy evolution solver a half time heat evolution solver a full time midpoint solver for the nonlinear source term and a half time heat evolution solver followed by a final half time free energy evolution solver we numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth

fourier spectral exponential time differencing methods for multi dimensional space fractional reaction diffusion equations abstract in this paper we propose two numerical methods for solving the space fractional reaction diffusion equations which are based on a fourier spectral approach in space and exponential time differencing schemes in time the advantages of the approaches are that they attain spectral convergence produce a full diagonal representation of the fractional operator and the extension to multiple spatial dimensions is the same as the one dimensional space that can overcome the constraints associated with many of the numerical schemes for these equations such as the computational efficiency caused by the non locality of the fractional operator which results in full dense matrices moreover the proposed schemes are second order convergent unconditionally stable and highly efficient due to the predictor corrector feature when comparing them with the existing method it is observed that the scheme based on using pade 1 1 approximations to the matrix exponential function introduces oscillations with non smooth initial data for some time steps due to high frequency components present in the solution which diminish as the fractional order decreases however the scheme based on pade 0 2 approximations to the matrix exponential function is oscillation free for any time step numerical experiments for well known models from the literature are performed to show the reliability and effectiveness of the proposed methods

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

obtaining full diversity space frequency codes from space time codes via mapping this paper considers the problem of space frequency code design for frequency selective multiple input multiple output mimo orthogonal frequency division multiplexing ofdm modulation we show that space time codes achieving full diversity in quasistatic flat fading environment can be used to construct space frequency codes that can achieve the maximum diversity available in frequency selective mimo fading channels since the codes are constructed via a simple mapping from space time codes to space frequency codes the abundant classes of existing space time block and trellis codes can be used for full diversity transmission in mimo ofdm systems the proposed mapping provides a tradeoff between the achieved diversity order and the symbol rate moreover we characterize the performance of the space frequency codes obtained via the mapping by finding lower and upper bounds on their coding advantages as functions of the coding advantages of the underlying space time codes this result will allow us to investigate the effects of the delay distribution and the power distribution of the channel impulse responses on the performance of the resulting space frequency codes extensive simulation results are also presented to illustrate and support the theory

bayesian analysis of switching regression models abstract the paper considers a class of switching regression models where the change of regime is represented by a min operator after reviewing a maximum likehood method a local identification criterion is proposed contrary to classical analysis it is shown that bayesian analysis requires here local identification everywhere on the parameter space the paper then proposes a monte carlo procedure to compute posterior moments with an importance function based on an approximate posterior density of the model feasibility of the method is shown in a numerical example and then in an economic example

subordinated advection dispersion equation for contaminant transport a mathematical method called subordination broadens the applicability of the classical advection dispersion equation for contaminant transport in this method the time variable is randomized to represent the operational time experienced by different particles in a highly heterogeneous aquifer the operational time captures the fractal properties of the medium this leads to a simple parsimonious model of contaminant transport that exhibits many of the features heavy tails skewness and non fickian growth rate typically seen in real aquifers we employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives applied to a one dimensional approximation of the made 2 data set the model shows excellent agreement

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

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

a unified construction of space time codes with optimal rate diversity tradeoff the problem of constructing space time st block codes over a fixed desired signal constellation is considered in this situation there is a tradeoff between the transmission rate as measured in constellation symbols per channel use and the transmit diversity gain achieved by the code the transmit diversity is a measure of the rate of polynomial decay of pairwise error probability of the code with increase in the signal to noise ratio snr in the setting of a quasi static channel model let n sub t denote the number of transmit antennas and t the block interval for any n sub t spl les t a unified construction of n sub t spl times t st codes is provided here for a class of signal constellations that includes the familiar pulse amplitude pam quadrature amplitude qam and 2 sup k ary phase shift keying psk modulations as special cases the construction is optimal as measured by the rate diversity tradeoff and can achieve any given integer point on the rate diversity tradeoff curve an estimate of the coding gain realized is given other results presented here include i an extension of the optimal unified construction to the multiple fading block case ii a version of the optimal unified construction in which the underlying binary block codes are replaced by trellis codes iii the providing of a linear dispersion form for the underlying binary block codes iv a gray mapped version of the unified construction and v a generalization of construction of the ary case corresponding to constellations of size sup k items ii and iii are aimed at simplifying the decoding of this class of st codes

a mass balance based numerical method for the fractional advection dispersion equation theory and application 1 the inherent heterogeneity of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement and one approach to model solute transport through such media is the continuous time random walk ctrw one special asymptotic case of the ctrw is the fractional advection dispersion equation fade which has proven to be a promising alternative to model anomalous dispersion and has been increasingly used in hydrology to model chemical transport in both surface and subsurface water most practical problems in hydrology have complicated initial and boundary conditions and need to be solved numerically but the numerical solution of the fade is not trivial in this paper we present a finite volume approach to solve the fade where the spatial derivative of the dispersion term is fractional we also give methods to solve different boundary conditions often encountered in practical applications the linear system resulting from the temporal spatial discretization is solved using a semi implicit scheme the numerical method is derived on the basis of mass balance and its accuracy is tested against analytical solutions the method is then applied to simulate tracer movement in a stream and a near saturated hillslope in a naturally structured upland podzol field in northeast scotland

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

high order time stepping fourier spectral method for multi dimensional space fractional reaction diffusion equations abstract this paper introduces a high order time stepping scheme that is based on using fourier spectral in space and a fourth order diagonal pade approximation to the matrix exponential function for solving multi dimensional space fractional reaction diffusion equations the resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non linear system at each time step while maintaining the stability of the scheme the non locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices this scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator it also attains spectral convergence for multiple spatial dimensions the stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions which provides an indication of the stability of the method the convergence analysis is performed empirically to show that the scheme is fourth order accurate in time as expected numerical experiments on reaction diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space fractional reaction diffusion equations and to validate the effectiveness of the scheme

a new conservative swift hohenberg equation and its mass conservative method abstract the swift hohenberg sh energy functional has been widely used to study pattern formation the l 2 and h 1 gradient flows for the sh energy functional are the sh and phase field crystal pfc equations respectively the sh equation is of lower order in space than the pfc equation but does not conserve the total mass furthermore when the sh energy functional does not have the cubic nonlinearity the sh equation unlike the pfc equation only shows striped patterns even for various parameter values in this study we introduce a new mass conservative sh equation by using a lagrange multiplier in order to solve the conservative sh equation that is an integro partial differential equation we propose operator splitting methods that are shown analytically to inherit the mass conservation numerical examples including standard tests in the pfc equation are presented to show the applicability of the proposed framework

econometric illustrations of novel numerical integration strategies for bayesian inference abstract recent progress with numerical techniques for implementing bayesian methods for irregular multi parameter likelihoods is reviewed and illustrated with two econometric examples the numerical integration strategies involve novel iterative adaptive uses of cartesian product and spherical rule quadrature formulae together with importance sampling techniques

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

diversity embedded space time codes rate and diversity impose a fundamental tradeoff in wireless communication high rate space time codes come at a cost of lower reliability diversity and high reliability diversity implies a lower rate however wireless networks need to support applications with very different quality of service qos requirements and it is natural to ask what characteristics should be built into the physical layer link in order to accommodate them in this paper we design high rate space time codes that have a high diversity code embedded within them this allows a form of communication where the high rate code opportunistically takes advantage of good channel realizations while the embedded high diversity code provides guarantees that at least part of the information is received reliably we provide constructions of linear and nonlinear codes for a fixed transmit alphabet constraint the nonlinear constructions are a natural generalization to wireless channels of multilevel codes developed for the additive white gaussian noise awgn channel that are matched to binary partitions of quadrature amplitude modulation qam and phase shift keying psk constellations the importance of set partitioning to code design for the wireless channel is that it provides a mechanism for translating constraints in the binary domain into lower bounds on diversity protection in the complex domain we investigate the systems implications of embedded diversity codes by examining value to unequal error protection rate opportunism and packet delay optimization these applications demonstrate that diversity embedded codes have the potential to outperform traditional single layer codes in moderate signal to noise snr regimes

adaptive eulerian lagrangian finite element method for advection dispersion a new adaptive finite element method is proposed for the advection dispersion equation using an eulerian lagrangian formulation the method is based on a decomposition of the concentration field into two parts one advective and one dispersive in a rigorous manner that does not leave room for ambiguity the advective component of steep concentration fronts is tracked forward with the aid of moving particles clustered around each front away from such fronts the advection problem is handled by an efficient modified method of characteristics called single step reverse particle tracking when a front dissipates with time its forward tracking stops automatically and the corresponding cloud of particles is eliminated the dispersion problem is solved by an unconventional lagrangian finite element formulation on a fixed grid which involves only symmetric and diagonal matrices preliminary tests against analytical solutions of one and two dimensional dispersion in a uniform steady state velocity field suggest that the proposed adaptive method can handle the entire range of peclet numbers from 0 to with courant numbers well in excess of 1

testing for unit roots in a bayesian framework abstract classical tests for unit roots have been criticized for their unusual asymptotic theory leading to disconnected confidence intervals and their lack of power in small samples such critiques were initiated by sims 1988 who promoted a bayesian approach to the question various papers then appeared which took up this idea the bayesian approach is however not exempt from criticism as shown by phillips 1991 who emphasizes the role of the prior as a bias toward stationarity in this paper i emphasize the role of the model and of the initial conditions i show that with a structural model and an adequate treatment of the first observation a bayesian unit root test based on the computation of pr p 1 y can produce results which are more or less in accordance with classical results the extended nelson plosser data set serves as an illustration

discontinuous decision processes and threshold autoregressive time series modelling where p is a smooth function suppose that we approximate 4u x by 0 a constant for all x in subsequent discussion we suppress the argument x whenever this may be done without obscuring the context clearly this approximation will incur some errors on the other hand as argued by tong lim 1980 we could approximate arbitrarily closely by a step function at first sight it seems that this could pose a horrendous computational problem indeed this is the case from a purely deterministic point of view however we usually approximate the true model with some purpose in mind e g forecasting control filtering thus we should really specify what we mean by approximating u arbitrarily closely bayesian decision theory seems a natural approach to this problem in particular the recent general results of smith et al 1981 provide the necessary framework

on the design of space time and space frequency codes for mimo frequency selective fading channels the authors introduced an algebraic design framework for space time coding in flat fading channels we extend this framework to design algebraic codes for multiple input multiple output mimo frequency selective fading channels the proposed codes strive to optimally exploit both the spatial and frequency diversity available in the channel we consider two design approaches the first uses space time coding and maximum likelihood decoding to exploit the multi path nature of the channel at the expense of increased receiver complexity within this time domain framework we also propose a serially concatenated coding construction which is shown to offer a performance gain with a reasonable complexity iterative receiver in some scenarios the second approach utilizes the orthogonal frequency division multiplexing technique to transform the mimo multipath channel into a mimo flat block fading channel the algebraic framework is then used to construct space frequency codes sfc that optimally exploit the diversity available in the resulting flat block fading channel finally the two approaches are compared in terms of decoder complexity maximum achievable diversity advantage and simulated frame error rate performance in certain representative scenarios

an efficient robust numerical method for singularly perturbed burgers equation abstract in this article we propose a parameter uniformly convergent numerical method for viscous burgers equation in order to find a numerical approximation to burgers equation we linearize the equation to obtain sequence of linear pdes the linear pdes are solved by a finite difference scheme which comprises of the backward difference scheme for the time derivative and upwind finite difference scheme for the spatial derivatives layer adapted nonuniform meshes are invoked at each time level to exhibit layer nature of the solution the nonuniform meshes are obtained by equidistribution of a positive integrable monitor function which involves the derivative of the solution it is shown that the methods converges uniformly with respect to the perturbation parameter numerical experiments are carried out to validate the e uniform error estimate of o n 1 t

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

novel numerical methods for solving the time space fractional diffusion equation in two dimensions in this paper a time space fractional diffusion equation in two dimensions tsfde 2d with homogeneous dirichlet boundary conditions is considered the tsfde 2d is obtained from the standard diffusion equation by replacing the first order time derivative with the caputo fractional derivative t d gamma gamma in 0 1 and the second order space derivatives with the fractional laplacian delta alpha 2 alpha in 1 2 using the matrix transfer technique proposed by ilic et al m ilic f liu i turner and v anh fract calc appl anal 9 2006 pp 333 349 the tsfde 2d is transformed into a time fractional differential system as t d gamma mathbf u k alpha mathbf a alpha 2 mathbf u where mathbf a is the approximate matrix representation of delta traditional approximation of mathbf a alpha 2 requires diagonalization of mathbf a which is very time consuming for large sparse matrices the novelty of our proposed numerical schemes is that using either the finite difference method or the laplace transform to handle the caputo time fractional derivative the solution of the tsfde 2d is written in terms of a matrix function vector product f mathbf a mathbf b at each time step where mathbf b is a suitably defined vector depending on the method used to generate the matrix mathbf a the product f mathbf a mathbf b can be approximated using either the preconditioned lanczos method when mathbf a is symmetric or the mathbf m lanzcos method when mathbf a is nonsymmetric which are powerful techniques for solving large linear systems we give error bounds for the new methods and illustrate their roles in solving the tsfde 2d we also derive the analytical solution of the tsfde 2d in terms of the mittag leffler function finally numerical results are presented to verify the proposed numerical solution strategies

an improved version of the quandt ramsey mgf estimator for mixtures of normal distributions and switching regressions quandt and ramsey have suggested an estimator for normal mixtures and switching regressions which minimizes a sum of squared differences between empirical and theoretical values of the moment generating function this paper demonstrates how their estimator can be improved by minimizing a generalized sum of squares rather than an ordinary sum of squares when this is done more points of evaluation moments are unambiguously better than less most of the results presented are also applicable to method of moments estimators in general

bent cable regression theory and applications we use the so called bent cable model to describe natural phenomena that exhibit a potentially sharp change in slope the model comprises two linear segments joined smoothly by a quadratic bend the class of bent cables includes as a limiting case the popular piecewise linear model with a sharp kink otherwise known as the broken stick associated with bent cable regression is the estimation of the bend width parameter through which the abruptness of the underlying transition may be assessed we present worked examples and simulations to demonstrate the regularity and irregularity of bent cable regression encountered in finite sample settings we also extend existing bent cable asymptotics that previously were limited to the basic model with known linear slopes of 0 and 1 practical conditions on the design are given to ensure regularity of the full bent cable estimation problem if the underlying bend segment has nonzero width under such conditions the least squares estimators are shown to be co

parameter estimation for fractional transport a particle tracking approach 1 space fractional advection dispersion models provide attractive alternatives to the classical advection dispersion equation for model applications that exhibit early arrivals and plume skewness this paper develops a flexible method for estimating the parameters of the fractional transport model on the basis of spatial plume snapshots or temporal breakthrough curve data a particle tracking approach provides error bars for the parameter estimates and a general method for model fitting and comparison via optimal weighted least squares a simple model of concentration variance based on the particle tracking approach identifies the optimal weights

shift an implementation for lattice boltzmann simulation in low porosity porous media the lattice boltzmann lb method has proven to be a promising method for simulating fluid dynamics in porous media when fluid flow in pores is the only concern a standard lb implementation which stores one or two sets of particle distribution functions pdfs for both pore and solid cells wastes a large amount of memory especially for low porosity media this paper proposes a lb implementation scheme that stores a single set of pdfs for pore cells only and therefore makes it possible to simulate flow through larger and more realistic porous models a unique feature of this scheme is that it decomposes all pdfs into a set of 1d arrays in such a way that each array corresponds to a set of pore cells that connect one another along a pair of opposite lb velocity directions this allows lb propagation and a standard bounce back rule to be realized together as one or two circular shifting operations on every array for this reason this scheme is called shift although pdfs are not stored in an efficient way for lb collision operation it is shown that the incurred overhead could be reduced by properly arranging pdf arrays according to the pore structures a d3q15 lb implementation of shift using the lattice bhatnagar gross krook model is applied to simulate the stokes flow through models of four natural and synthetic rock samples with porosities ranging from about 10 to 38 results show that shift requires 36 82 less memory than a comparable d3q15 lb does which stores a single set of pdf for both pore and solid cells shift achieves minimum performances of over 11 and 3 8 mega lattice updates per second mlups for the combined propagation and bounce back operation and the collision operation respectively and therefore a minimum of 2 8 mlups in total on a computer with one amd opteron 2218 the performance of the collision operation is significantly improved for all cases when a simple k mean clustering technique is employed to rearrange pdf arrays it is argued and shown that the number of pdf arrays per pore cell and the length frequency of pdf arrays are useful measurements on the geometry and topology of the pore structures and these characteristics are able to explain shift performance variations

bayesian bent cable growth mixture tobit models for longitudinal data with skewness and detection limit application to aids studies this paper presents a new bayesian methodology for identifying a transition period for the development of drug resistance to antiretroviral drug or therapy in hiv aids studies or other related fields estimation of such a transition period requires an availability of longitudinal data where growth trajectories of a response variable tend to exhibit a gradual change from a declining trend to an increasing trend rather than an abrupt change we assess this clinically important feature of the longitudinal hiv aids data using the bent cable framework within a growth mixture tobit model to account for heterogeneity of drug resistance among subjects the parameters of the bent cable growth mixture tobit model are also allowed to differ by subgroups subpopulations of patients classified into latent classes on the basis of trajectories of observed viral load data with skewness and left censoring the proposed methods are illustrated using real data from an aids clinical study copyright 2016 john wiley sons ltd

a banded preconditioner for the two sided nonlinear space fractional diffusion equation the method of lines is a standard method for advancing the solution of partial differential equations pdes in time in one sense the method applies equally well to space fractional pdes as it does to integer order pdes however there is a significant challenge when solving space fractional pdes in this way owing to the non local nature of the fractional derivatives each equation in the resulting semi discrete system involves contributions from every spatial node in the domain this has important consequences for the efficiency of the numerical solver especially when the system is large first the jacobian matrix of the system is dense and hence methods that avoid the need to form and factorise this matrix are preferred second since the cost of evaluating the discrete equations is high it is essential to minimise the number of evaluations required to advance the solution in time in this paper we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two sided nonlinear space fractional diffusion equation a key contribution is to show how to construct suitable banded approximations to the system jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration without requiring dense matrices to be formed the results of numerical experiments are presented that demonstrate the effectiveness of this approach

rate diversity tradeoff of space time codes with fixed alphabet and optimal constructions for psk modulation we show that for any q spl times m space time code s having a fixed finite signal constellation there is a tradeoff between the transmission rate r and the transmit diversity gain spl nu achieved by the code the tradeoff is characterized by r spl les q spl nu 1 where q is the number of transmit antennas when either binary phase shift keying bpsk or quaternary phase shift keying qpsk is used as the signal constellation a systematic construction is presented to achieve the maximum possible rate for every possible value of transmit diversity gain

stability and convergence of a finite volume method for the space fractional advection dispersion equation we consider the space fractional advection dispersion equation which is obtained from the classical advection diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order we derive a finite volume method that utilises fractionally shifted grunwald formulae for the discretisation of the fractional derivative to numerically solve the equation on a finite domain with homogeneous dirichlet boundary conditions we prove that the method is stable and convergent when coupled with an implicit timestepping strategy results of numerical experiments are presented that support the theoretical analysis

theory of anomalous chemical transport in random fracture networks we show that dominant aspects of chemical particle transport in fracture networks char22 non gaussian propagation char22 result from subtle features of the steady flow field distribution through the network this is an outcome of a theory based on a continuous time random walk formalism structured to retain the key space time correlations of particles as they are advected across each fracture segment the approach is designed to treat the complex geometries of a large variety of fracture networks and multiscale interactions monte carlo simulations of steady flow in these networks are used to determine the distribution of velocities in individual fractures as a function of their orientation the geometry and velocity distributions are used in conjunction with particle mixing rules to map the particle movement between fracture intersections onto a joint probability density ensuremath psi mathbf r t the chemical concentration plume and breakthrough curves can then be calculated analytically particle tracking simulations on these networks exhibit the same non gaussian profiles demonstrating quantitative agreement with the theory the analytic plume shapes display the same basic behavior as extensive field observations at the columbus air force base mississippi the quantitative correlation between the time dependence of the mean and standard deviation of the field plumes and their shape is predicted by the theory

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

a comparison study of adi and operator splitting methods on option pricing models in this paper we perform a comparison study of alternating direction implicit adi and operator splitting os methods on multi dimensional black scholes option pricing models the adi method is used extensively in mathematical finance for numerically solving multi factor option pricing problems however numerical results from the adi scheme show oscillatory solution behaviors with nonsmooth payoffs or discontinuous derivatives at the exercise price with large time steps in the adi scheme there are source terms which include y derivatives when we solve x derivative involving equations then due to the nonsmooth payoffs source terms contain abrupt changes which are not in the range of implicit discrete operators and this leads to difficulty in solving the problem on the other hand the os method does not contain the other variable s derivatives in the source terms we provide computational results showing the performance of the methods for two asset option pricing problems the results show that the os method is very efficient and gives better accuracy and robustness than the adi method with large time steps

a novel finite volume method for the riesz space distributed order diffusion equation abstract in recent years considerable attention has been devoted to distributed order differential equations mainly because they appear to be more effective for modelling complex processes which obey a mixture of power laws or flexible variations in space in this paper we propose a novel finite volume method fvm for a distributed order space fractional diffusion equation fde firstly we use the mid point quadrature rule to transform the space distributed order diffusion equation into a multi term fractional equation secondly the transformed multi term fractional equation is solved by discretising in space using the finite volume method and then in time using the crank nicolson scheme thirdly we prove that the crank nicolson scheme with fvm is unconditionally stable and convergent with second order accuracy in both time and space finally two numerical examples are presented to show the effectiveness of the numerical method these methods and techniques can also be used to solve other types of fractional partial differential equations

galerkin finite element method for two dimensional riesz space fractional diffusion equations abstract in this article a class of two dimensional riesz space fractional diffusion equations is considered some fractional spaces are established and some equivalences between fractional derivative spaces and fractional sobolev space are presented by the galerkin finite element method and backward difference method a fully discrete scheme is obtained according to lax milgram theorem the existence and uniqueness of the solution to the fully discrete scheme are investigated the stability and convergence of the scheme are also derived finally some numerical examples are given for verification of our theoretical analysis

fractional fick s law the direct way levy flights which are markovian continuous time random walks possibly accounting for extreme events serve frequently as small scale models for the spreading of matter in heterogeneous media among them brownian motion is a particular case where fick s law holds for a cloud of walkers the flux is proportional to the gradient of the probability density of finding a particle at some place levy flights resemble brownian motion except that jump lengths are distributed according to an stable levy law possibly showing heavy tails and skewness for between 1 and 2 a fractional form of fick s law is known to hold in infinite media that the flux is proportional to a combination of fractional derivatives or the order of 1 of the density of walkers was obtained as a consequence of a fractional dispersion equation we present a direct and natural proof of this result based upon a novel definition of usual fractional derivatives involving a convolution and a limiting process taking account of the thus obtained fractional fick s law yields fractional dispersion equation for smooth densities the method adapts to domains limited by boundaries possibly implying non trivial modifications to this equation

a new maximum likelihood algorithm for piecewise regression abstract this paper presents a piecewise regression method for continuous models containing max or min operators or both this method does not require knowledge of the zone in which a shift in regimes occurs moreover it allows the application of analytical derivatives to maximize the likelihood function which greatly simplifies the estimation of the model the method proposed exhibits fast convergence and can be used for an arbitrary number of regimes and variables

embedded rank distance codes for isi channels designs for transmit alphabet constrained space time codes naturally lead to questions about the design of rank distance codes recently diversity embedded multilevel space time codes for flat fading channels have been designed from sets of binary matrices with rank distance guarantees over the binary field by mapping them onto quadrature amplitude modulation qam and phase shift keying psk constellations in this paper we demonstrate that diversity embedded space time codes for fading intersymbol interference isi channels can be designed with provable rank distance guarantees as a corollary we obtain an asymptotic characterization of the fixed transmit alphabet rate diversity tradeoff for multiple antenna fading isi channels the key idea is to construct and analyze properties of binary matrices with a particular structure toeplitz structure induced by isi channels

testing and modeling threshold autoregressive processes abstract the threshold autoregressive model is one of the nonlinear time series models available in the literature it was first proposed by tong 1978 and discussed in detail by tong and lim 1980 and tong 1983 the major features of this class of models are limit cycles amplitude dependent frequencies and jump phenomena much of the original motivation of the model is concerned with limit cycles of a cyclical time series and indeed the model is capable of producing asymmetric limit cycles the threshold autoregressive model however has not received much attention in application this is due to a the lack of a suitable modeling procedure and b the inability to identify the threshold variable and estimate the threshold values the primary goal of this article therefore is to suggest a simple yet widely applicable model building procedure for threshold autoregressive models based on some predictive residuals a simple statistic is proposed to test for threshold nonlinearity and specify the

fast estimation of regression parameters in a broken stick model for longitudinal data abstractestimation of change point locations in the broken stick model has significant applications in modeling important biological phenomena in this article we present a computationally economical likelihood based approach for estimating change point s efficiently in both cross sectional and longitudinal settings our method based on local smoothing in a shrinking neighborhood of each change point is shown via simulations to be computationally more viable than existing methods that rely on search procedures with dramatic gains in the multiple change point case the proposed estimates are shown to have n consistency and asymptotic normality in particular they are asymptotically efficient in the cross sectional setting allowing us to provide meaningful statistical inference as our primary and motivating longitudinal application we study the michigan bone health and metabolism study cohort data to describe patterns of change in log estradiol levels before and after the final menstrual period

a bayesian analysis of some threshold switching models abstract this paper presents a bayesian analysis of various threshold switching regression models including simple time series models where the change of regime is governed by a known function of exogenous variables some special features arising from the choice of a two dimensional linear dichotomy function are then discussed and the concepts of concurrency and duality introduced within this framework we compare the maximum likelihood and bayesian methodologies for inference and prediction in particular we show that the bayesian approach solves the non uniqueness problem which affects maximum likelihood prediction in certain situations these results are illustrated with two numerical examples

numerical simulation of non fickian transport in geological formations with multiple scale heterogeneities 1 we develop a numerical method to model contaminant transport in heterogeneous geological formations the method is based on a unified framework that takes into account the different levels of uncertainty often associated with characterizing heterogeneities at different spatial scales it treats the unresolved small scale heterogeneities residues probabilistically using a continuous time random walk ctrw formalism and the large scale heterogeneity variations trends deterministically this formulation leads to a fokker planck equation with a memory term fpme and a generalized concentration flux term the former term captures the non fickian behavior arising from the residues and the latter term accounts for the trends which are included with explicit treatment at the heterogeneity interfaces the memory term allows a transition between non fickian and fickian transport while the coupling of these dynamics with the trends quantifies the unique nature of the transport over a broad range of temporal and spatial scales the advection dispersion equation ade is shown to be a special case of our unified framework the solution of the ade is used as a reference for the fpme solutions for the same formation structure numerical treatment of the equations involves solution for the laplace transformed concentration by means of classical finite element methods and subsequent inversion in the time domain we use the numerical method to quantify transport in a two dimensional domain for different expressions for the memory term the parameters defining these expressions are measurable quantities the calculations demonstrate long tailing arising principally from the memory term and the effects on arrival times that are controlled largely by the generalized concentration flux term

tt m finite element algorithm for a two dimensional space fractional gray scott model abstract in this article a fast time two mesh tt m finite element fe method for the two dimensional space fractional gray scott model is studied and discussed to get the numerical solutions effectively the method mainly includes three steps firstly one uses an iterative method for solving the coupled nonlinear system on the time coarse grid secondly by an interpolation formula one can get any coarse values on the time fine mesh finally based on the computed coarser solutions a linear fe system on time fine mesh can be constructed by using the two variables taylor s formula here some theoretical results which include stability and a priori error for the fully discrete scheme are analyzed and proved furthermore the computing data are given to verify the correctness of the theoretical results and to illustrate that the tt m fe algorithm can reduce the computing time

fractional advection dispersion equation a classical mass balance with convolution fickian flux the classical form of the partial differential equation governing advective dispersive transport of a solute in idealized porous media is the advection dispersion equation ade this equation is built on a fickian constitutive theory that gives the local dispersive flux as the inner product of a constant dispersion tensor and the spatial gradient of the solute concentration over the past decade investigators in subsurface transport have been increasingly focused on anomalous i e non fickian dispersion in natural geologic formations it has been recognized that this phenomenon involves spatial and possibly temporal nonlocality in the constitutive theory describing the dispersive flux most recently several researchers have modeled anomalous dispersion using a constitutive theory that relies on fractional as opposed to integer derivatives of the concentration field the resulting ade itself is expressed in terms of fractional derivatives and they are described as fractional ades here we show that the fractional ade is obtained as a special case of the authors convolution fickian nonlocal ade cushman and ginn 1993 to obtain the fractional ade from the convolution fickian model requires only a judicial choice of the wave vector and frequency dependent dispersion tensor

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

moments for tempered fractional advection diffusion equations this paper develops moment formulas for exponentially tempered fractional advection diffusion equations tfades that transition from anomalous to asymptotic diffusion limits over time exact analytical expressions or series representations for spatial moments up to the fourth order are derived by integral transform or asymptotic expansion approach a fully lagrangian solver cross verified by an implicit eulerian approach is also developed to calculate numerically the complete evolution of moments for the tfades with complex initial and boundary conditions moment analysis identifies the diffusion equation that attracts the tempered anomalous diffusion in the long time limit fitting of moments measured at two end members of alluvial systems checks the applicability of moment analysis in understanding real diffusion

numerical solution of burgers equation with high order splitting methods in this work high order splitting methods have been used for calculating the numerical solutions of burgers equation in one space dimension with periodic dirichlet neumann and robin boundary conditions however splitting methods with real coefficients of order higher than two necessarily have negative coefficients and cannot be used for time irreversible systems such as burgers equations due to the time irreversibility of the laplacian operator therefore the splitting methods with complex coefficients and extrapolation methods with real and positive coefficients have been employed if we consider the system as the perturbation of an exactly solvable problem or one that can be easily approximated numerically it is possible to employ highly efficient methods to approximate burgers equation the numerical results show that both the methods with complex time steps having one set of coefficients real and positive say a i r and b i c and high order extrapolation methods derived from a lower order splitting method produce very accurate solutions of burgers equation

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

quantitative characterization of pore scale disorder effects on transport in homogeneous granular media breakthrough curves btc of a passive tracer in macroscopically homogeneous granular materials well sorted unconsolidated sands or glass beads were measured in a series of column experiments the early and late arrival times are observed to differ systematically from theoretical predictions based on solution of the advective dispersion equation for uniform porous media we propose that subtle and residual pore scale disorder effects in the porous media can account for these observations we determine an ensemble averaged distribution of particle transfer rates based on a master equation for the local flux averaged concentration which incorporates these effects and utilize it to calculate btc that are in excellent agreement with the entire series of observations theoretical prediction of the dependence of the effective macroscopic parameters on measurable quantities is also in excellent agreement with the observations

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

a semi analytical fourier spectral method for the allen cahn equation in recent years fourier spectral methods have been widely used as a powerful tool for solving phase field equations to improve its effectiveness many researchers have employed stabilized semi implicit fourier spectral sifs methods which allow a much larger time step than a usual explicit scheme our mathematical analysis and numerical experiments however suggest that an effective time step is smaller than a time step specified in the sifs schemes in consequence the sifs scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes in order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step we present a first and a second order semi analytical fourier spectral safs methods for solving the allen cahn equation the core idea of the methods is to decompose the original equation into linear and nonlinear subequations which have closed form solutions in the fourier and physical spaces respectively both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi implicit fourier spectral method

local polynomial regression and simulation extrapolation summary the paper introduces a new local polynomial estimator and develops supporting asymptotic theory for nonparametric regression in the presence of covariate measurement error we address the measurement error with cook and stefanski s simulation extrapolation simex algorithm our method improves on previous local polynomial estimators for this problem by using a bandwidth selection procedure that addresses simex s particular estimation method and considers higher degree local polynomial estimators we illustrate the accuracy of our asymptotic expressions with a monte carlo study compare our method with other estimators with a second set of monte carlo simulations and apply our method to a data set from nutritional epidemiology simex was originally developed for parametric models although simex is in principle applicable to nonparametric models a serious problem arises with simex in nonparametric situations the problem is that smoothing parameter selectors that are developed for data without measurement error are no longer appropriate and can result in considerable undersmoothing we believe that this is the first paper to address this difficulty

multi scale orthogonal basis method for nonlinear fractional equations with fractional integral boundary value conditions abstract in this paper we investigate the multi scale orthogonal basis method for fractional integral boundary value problems we apply the newton iteration method to linearize the nonlinear problems and employees the idea of collocation method to determine the coefficients of multi scale orthogonal basis then the approximation solution is obtained the error estimation and stable analysis are presented in detailed the final numerical experiments verify that the accuracy of our method

spectral methods for partial differential equations in irregular domains the spectral smoothed boundary method in this paper we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no flux boundary conditions the idea is to embed the domain into a box and use a smoothing term to encode the boundary conditions into a modified equation that can be approached by standard spectral methods the main features of this method are its capability to deal with domains of arbitrary shape and its easy implementation via fast fourier transform routines we discuss several examples of practical interest and test the results against standard numerical methods

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

a second order operator splitting fourier spectral method for fractional in space reaction diffusion equations abstract fractional differential equations have been proved to be valuable tools for modeling diffusive processes associated with anomalous diffusion or spatial heterogeneity however many numerical methods for these equations have limitations in terms of computational efficiency due to the nonlocal nature of the fractional operator which leads to large dense matrices the aim of this paper is to propose a second order operator splitting fourier spectral method as an accurate and efficient approach for solving fractional in space reaction diffusion equations this approach gives a full diagonal representation of the fractional operator and achieves spectral convergence regardless of the fractional power in the problem in order to achieve second order time accuracy we decompose the original equation into linear and nonlinear subequations and combine a half time linear solver and a full time second order nonlinear solver followed by a final half time linear solver we numerically demonstrate the accuracy and efficiency of the proposed method and apply the proposed method to investigate the effect of the fractional power in fractional in space reaction diffusion equations including the allen cahn fitzhugh nagumo and gray scott models

a superfast preconditioned iterative method for steady state space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second order diffusion equations because of the nonlocal property of fractional differential operators the corresponding numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 for a problem of size n we develop a superfast preconditioned conjugate gradient squared method for the efficient solution of steady state space fractional diffusion equations the method reduces the computational work from o n 2 to o nlogn per iteration and reduces the memory requirement from o n 2 to o n furthermore the method significantly reduces the number of iterations to be mesh size independent preliminary numerical experiments for a one dimensional steady state diffusion equation with 2 1 3 nodes show that the fast method reduces the overall cpu time from 3h and 27min for the gaussian elimination to 0 39s for the fast method while retaining the accuracy of gaussian elimination in contrast the regular conjugate gradient squared method diverges after 2days of simulations and more than 20 000 iterations

a semi alternating direction method for a 2 d fractional fitzhugh nagumo monodomain model on an approximate irregular domain a fitzhugh nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue in this paper we consider a two dimensional fractional fitzhugh nagumo monodomain model on an irregular domain the model consists of a coupled riesz space fractional nonlinear reaction diffusion model and an ordinary differential equation describing the ionic fluxes as a function of the membrane potential second we use a decoupling technique and focus on solving the riesz space fractional nonlinear reaction diffusion model a novel spatially second order accurate semi implicit alternating direction method siadm for this model on an approximate irregular domain is proposed third stability and convergence of the siadm are proved finally some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two dimensional fractional fitzhugh nagumo model on both an approximate circular and an approximate irregular domain

nonlinear interest rate dynamics and implications for the term structure this paper explores nonlinear dynamics in the time series of the short term interest rate in the united states the proposed model is an autoregressive threshold model augmented by conditional heteroskedasticity the performance of the model is evaluated by considering its implications for the term structure of interest rates the nonlinear dynamics imply a form of nonlinearity in the levels relation between the long and the short rate

n th order operator splitting schemes and nonreversible systems this paper is concerned with partitioned nth order accurate split operator schemes built using m distinct solution operators for semidiscrete equations of the form frac d dt u j f j 1 u k cdots f j m u k which arise among others from constant coefficient parabolic partial differential equations we prove that for every n geq 3 and m geqslant 2 each solution operator must be applied for at least one backward fractional time step during each complete time step this result has important consequences for applications to the complex ginzburg landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components furthermore for the special case of n 3 and m 2 we analytically determine all possible schemes

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

lattice boltzmann method for the fractional advection diffusion equation mass transport such as movement of phosphorus in soils and solutes in rivers is a natural phenomenon and its study plays an important role in science and engineering it is found that there are numerous practical diffusion phenomena that do not obey the classical advection diffusion equation ade such diffusion is called abnormal or superdiffusion and it is well described using a fractional advection diffusion equation fade the fade finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems however solution to the fade is difficult and the existing numerical methods are complicated and inefficient in this study a fresh lattice boltzmann method is developed for solving the fractional advection diffusion equation labfade the fade is transformed into an equation similar to an advection diffusion equation and solved using the lattice boltzmann method the labfade has all the advantages of the conventional lattice boltzmann method and avoids a complex solution procedure unlike other existing numerical methods the method has been validated through simulations of several benchmark tests a point source diffusion a boundary value problem of steady diffusion and an initial boundary value problem of unsteady diffusion with the coexistence of source and sink terms in addition by including the effects of the skewness the fractional order and the single relaxation time the accuracy and convergence of the method have been assessed the numerical predictions are compared with the analytical solutions and they indicate that the method is second order accurate the method presented will allow the fade to be more widely applied to complex mass transport problems in science and engineering

moment analysis for spatiotemporal fractional dispersion 1 the evolution of the first five nonnegative integer order spatial moments corresponding to the mass mean variance skewness and kurtosis are investigated systematically for spatiotemporal nonlocal fractional dispersion three commonly used fractional order transport equations including the time fractional advection dispersion equation time fade the fractal mobile immobile mim equation and the fully fractional advection dispersion equation ffade are considered analytical solutions verify our numerical results and reveal the anomalous evolution of the moments following adams and gelhar s 1992 work on the classical ade we find that a simultaneous analysis of all moments is critical in discriminating between different nonlocal models the evolution of dispersion among the subdiffusive to superdiffusive rates is then further explored numerically by a non markovian random walk particle tracking method that can be used for any heterogeneous boundary or initial value problem in three dimensions both the analytical and the numerical results also show the similarity at the early time and the difference at the late time of moment growth for solutes in different phases mobile versus total described by the mim models further simulations of the 1 d bromide snapshots measured at the made experiments using all three models with parameters fitted by the observed zeroth to fourth moments indicate that 1 both the time and space nonlocality strongly affect the solute transport at the made site 2 all five spatial moments should be considered in transport model selection and calibration because those up to the variance cannot effectively discriminate between nonlocal models and 3 the log concentration should be used when evaluating the plume leading edge and the effects of space nonlocality

fractional gray scott model well posedness discretization and simulations abstract the gray scott gs model represents the dynamics and steady state pattern formation in reaction diffusion systems and has been extensively studied in the past in this paper we consider the effects of anomalous diffusion on pattern formation by introducing the fractional laplacian into the gs model first we prove the well posedness of the fractional gs model we then introduce the crank nicolson c n scheme for time discretization and weighted shifted grunwald difference operator for spatial discretization we perform stability analysis for the time semi discrete numerical scheme and furthermore we analyze numerically the errors with benchmark solutions that show second order convergence both in time and space we also employ the spectral collocation method in space and c n scheme in time to solve the gs model in order to verify the accuracy of our numerical solutions we observe the formation of different patterns at different values of the fractional order which are quite different from the patterns of the corresponding integer order gs model and quantify them by using the radial distribution function rdf finally we discover the scaling law for steady patterns of the rdfs in terms of the fractional order

nonlinear hamiltonian equations with fractional damping existence and uniqueness are proved for a class of hamiltonian nonlinear differential equations of second order with a caputo fractional derivative of order 0 2 for noninteger the rate of convergence of solutions to a critical point is algebraic for 0 1 a non negative dissipation function is constructed for 2 the equation is hamiltonian the solution is an analytic function of for 0 1 and for 1 2 numerical studies of nonlinear pendulum and the duffing equation with a fractional derivative term replacing the usual damping show striking differences between ordinary differential and fractional 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

on the theory of space time codes for psk modulation the design of space time codes to achieve full spatial diversity over fading channels has largely been addressed by handcrafting example codes using computer search methods and only for small numbers of antennas the lack of more general designs is in part due to the fact that the diversity advantage of a code is the minimum rank among the complex baseband differences between modulated codewords which is difficult to relate to traditional code designs over finite fields and rings we present general binary design criteria for psk modulated space time codes for linear bpsk qpsk codes the rank of binary projections of the unmodulated codewords as binary matrices over the binary field is a sufficient design criterion full binary rank guarantees full spatial diversity this criterion accounts for much of what is currently known about psk modulated space time codes we develop new fundamental code constructions for both quasi static and time varying channels these are perhaps the first general constructions other than delay diversity schemes that guarantee full spatial diversity for an arbitrary number of transmit antennas

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

monte carlo simulation of superdiffusion and subdiffusion in macroscopically heterogeneous media 1 monte carlo simulations are developed to approximate one dimensional superdiffusion and subdiffusion in macroscopically heterogeneous media with discontinuous or continuous transport parameters for superdiffusion characterized by a space fractional order derivative model one empirical reflection scheme is built to track particle trajectory across an interface with discontinuous dispersion coefficient d where the reflection probability depends on both and the ratio of d different from the superdiffusive case anomalous diffusion described by a time fractional derivative model can be decomposed into a motion component and a hitting time process where the discontinuity affects only the motion process implying an efficient monte carlo simulation of decoupled continuous time random walks the discontinuity of effective porosity n is also discussed and results show the influence of the ratio of n on solute particle dynamics in addition for anomalous superdiffusion and subdiffusion in heterogeneous media with spatially continuous d and n langevin analysis reveals that the corresponding particle dynamics contain three independent stable levy noises scaled by d the gradient of d and the gradient of ln n a new implicit eulerian finite difference method is also developed to solve the spatiotemporal fractional derivative models and then extensively cross verify the lagrangian solutions further testing against one field example of mixed superdiffusion and subdiffusion reveals the applicability and flexibility of the novel monte carlo approach in simulating realistic plumes in macroscopically heterogeneous media with locally variable transport parameters

efficient solution of two sided nonlinear space fractional diffusion equations using fast poisson preconditioners we develop a fast poisson preconditioner for the efficient numerical solution of a class of two sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines using the shifted gr unwald finite difference formulas to approximate the two sided i e the left and right riemann liouville fractional derivatives the resulting semi discrete nonlinear systems have dense jacobian matrices owing to the non local property of fractional derivatives we employ a modern initial value problem solver utilising backward differentiation formulas and jacobian free newton krylov methods to solve these systems for efficient performance of the jacobianfree newton krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver the key contribution of our work is to generalise the fast poisson preconditioner widely used for integer order diffusion equations so that it applies to the two sided space fractional diffusion equation a number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy