a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

positive solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x u p 1 u x r 3 k x u 2 x r 3 with p 3 5 assuming that a r 3 r and k r 3 r are nonnegative functions such that lim x a x a 0 lim x k x 0 and satisfying suitable assumptions but not requiring any symmetry property on them we prove the existence of positive solutions

discovering variable fractional orders of advection dispersion equations from field data using multi fidelity bayesian optimization abstract the fractional advection dispersion equation fade can describe accurately the solute transport in groundwater but its fractional order has to be determined a priori here we employ multi fidelity bayesian optimization to obtain the fractional order under various conditions and we obtain more accurate results compared to previously published data moreover the present method is very efficient as we use different levels of resolution to construct a stochastic surrogate model and quantify its uncertainty we consider two different problem set ups in the first set up we obtain variable fractional orders of one dimensional fade considering both synthetic and field data in the second set up we identify constant fractional orders of two dimensional fade using synthetic data we employ multi resolution simulations using two level and three level gaussian process regression models to construct the surrogates

anomalous diffusion with ballistic scaling a new fractional derivative abstract anomalous diffusion with ballistic scaling is characterized by a linear spreading rate with respect to time that scales like pure advection ballistic scaling may be modeled with a symmetric riesz derivative if the spreading is symmetric however ballistic scaling coupled with a skewness is observed in many applications including hydrology nuclear physics viscoelasticity and acoustics the goal of this paper is to find a governing equation for anomalous diffusion with ballistic scaling and arbitrary skewness to address this problem we propose a new operator called the zolotarev derivative which is valid for all orders 0 2 the fourier symbol of this operator is related to the characteristic function of a stable random variable in the zolotarev m parameterization in the symmetric case the zolotarev derivative reduces to the well known riesz derivative for 1 the zolotarev derivative is a linear combination of riemann liouville fractional derivatives and a first derivative for 1 the zolotarev derivative is a non local operator that models ballistic anomalous diffusion we prove that this operator is continuous with respect to we derive generator caputo and riemann liouville forms of this operator and provide two examples the solutions of diffusion equations utilizing the zolotarev derivative with an impulse initial condition are shifted and scaled stable densities in the m parameterization

the e 2 a sections of mathrm sl 2 a in this paper we characterize the normal subgroups of sl 2 a for commutative rings a in a class which includes the arithmetic dedekind domains with infinitely many units and all rings satisfying the first stable range condition the characterization works as follows given an ideal j of a a group u of units in a j and an additive subgroup p of a closed under certain operations a structure we have dubbed a radix there is a canonically defined normal subgroup g j u p of sl 2 a these subgroups are generic in the sense that the commutators sl 2 a g j u p are all of the commutators of sl 2 a with normal subgroups of sl 2 a and a subgroup n is normal in sl 2 a if and only if sl 2 a g j u p c n c g j u p for one of these generic groups this is proved in the main theorem 3 1 3 the proof is accomplished by using results of bass milnor and serre 4 vaserstein 37 and liehl 17 in conjunction with the theory developed in sections 1 and 2 also invaluable is a refinement of the method of reduction introduced in 8 and 9 it appears in the proof of theorem 3 1 2 and is what makes it possible to reduce the question for arithmetic dedekind domains to the case of proper homomorphic images all of which are finite rings and satisfy the

on ground state solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x f u in r 3 k x u 2 in r 3 and prove the existence of ground state solutions for system sp under certain assumptions on the linear and nonlinear terms some recent results from different authors are extended

infinitely many positive solutions for a schrodinger poisson system abstract we are interested in the existence of infinitely many positive solutions of the schrodinger poisson system u u v x u u p 1 u x r 3 v x u 2 x r 3 where v x is a positive bounded function 1 p 5 and v r r x has the following decay property v r a r m o 1 r m with a 0 m 3 2 0 the solutions obtained are non radial

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

regularizing effect for some p laplacian systems abstract we study existence and regularity of weak solutions for the following p laplacian system p u a 1 u r 2 u f u w 0 1 p p u r w 0 1 p where is an open bounded subset of r n n 2 p v div v p 2 v is the p laplacian operator for 1 p n a 0 r 1 0 p 1 and f belongs to a suitable lebesgue space in particular we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions

the existence and concentration of positive solutions for a nonlinear schrodinger poisson system with critical growth we consider the schrodinger poisson system e2 u v x u x u f u u2 in r3 where the nonlinear term f is of critical growth in this paper we construct a solution u of the above elliptic system which concentrates at an isolated component of positive locally minimum points of v as 0 under certain conditions on f in particular the monotonicity of f s s3 and the so called ambrosetti rabinowitz condition are not required

on the normal subgroups ofgl 2 a in this paper we determine the normal subgroups of gl 2 a for a a commutative local ring the papers of klingenberg a lacroix 3 and our paper 1 constitute the previous work on this problem as a consequence of our results we are able to describe the normal subgroups of gl 2 a for all artinian rings a hence in particular for all finite rings a if n is a subset of gl 2 a we denote by i n the level ideal of n that is the smallest ideal of a such that n consists of scalar matrices modulo i n for j an ideal of a gc 2 a j te gl 2 a 1 i t e j is the normal subgroup consisting of all matrices which are scalars modulo j while sl 2 a j te gl 2 a 1 det t 1 and tz imod j is the principal congruence subgroup for j klingenberg proved that if i e a and the residue field of a is not g8 3 then a subgroup n of gl 2 a is normal if and only if sl 2 a z n en equivalently n is normal if and only if sl 2 a j e nc gc 2 a j for some ideal j in which case j z n his paper also shows that this criterion for normality holds in gl n a for n 3 without any hypotheses on the local ring a our paper 1 characterizes normal subgroups of gl 2 a for any ring a with 6 a since every local ring is an sr ring this leaves only the case of local rings in which 2 is not a unit i e in which the residue field has characteristic 2 lacroix in 3 deals with local rings in which the residue field has characteristic 2 but is not gi 2 obtaining the same normality criterion as klingenberg this leaves open the case of local rings with residue field ge 2 we shall use the method of reduction employed in cl to give a normality criterion for subgroups of gl 2 a for a a local ring with gj 2 as residue field moreover the method will allow us to recapture the results of klingenberg and lacroix with great efficiency thus we will be able to 395 0021 8693 90 3 00

fourth order accurate scheme for the space fractional diffusion equations because of the nonlocal properties of fractional operators higher order schemes play a more important role in discretizing fractional derivatives than classical ones the striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first order schemes but greatly improve the accuracy nowadays there are already two types of second order discretization schemes for space fractional derivatives the first type is given and discussed in sousa and li arxiv 1109 2345v1 2011 chen and deng appl math model 38 2014 pp 3244 3259 chen deng and wu appl numer math 70 2013 pp 22 41 and the second type is a class of schemes presented in tian zhou and deng math comp to appear also available online from arxiv 1201 5949 2012 the core object of this paper is to derive a class of fourth order approximations called the weighted and shifted lubich difference operators for space fractional derivatives then we use the derived schemes t

the thomas fermi von weizsacker theory of atoms and molecules we place the thomas fermi von weizsacker model of atoms on a firm mathematical footing we prove existence and uniqueness of solutions of the thomas fermi von weizsacker equation as well as the fact that they minimize the thomas fermi von weizsacker energy functional moreover we prove the existence of binding for two very dissimilar atoms in the frame of this model

solving time fractional advection dispersion equation by variable weights particle tracking method particle tracking method is an efficient and reliable method to solve time fractional advection dispersion equation which can describe anomalous transport in heterogeneous porous media however this method will lead to severe fluctuation or disappearance of solutions if the concentration value is small a variable weights method is developed to conquer the shortcoming of particle tracking method then one dimensional and two dimensional time fractional advection dispersion equations are solved by the variable weights particle tracking method compared to traditional particle tracking method the variable weights version may eliminate the fluctuation and improve the accuracy by orders of magnitude without more computational cost

multiplicity of small negative energy solutions for a class of nonlinear schrodinger poisson systems this paper deals with the following nonlinear schrodinger poisson email protected v x u k x f x u h x f x u inr 3 email protected f k x u 2 inr 3 where v x k x and h x are nonnegative continuous functions under appropriate assumptions on v x k x h x and f x u we prove the existence of infinitely many small negative energy solutions by using the variant fountain theorem established by zou recent results from the literature are extended

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

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

tempered fractional calculus fractional derivatives and integrals are convolutions with a power law multiplying by an exponential factor leads to tempered fractional derivatives and integrals tempered fractional diffusion equations where the usual second derivative in space is replaced by a tempered fractional derivative govern the limits of random walk models with an exponentially tempered power law jump distribution the limiting tempered stable probability densities exhibit semi heavy tails which are commonly observed in finance tempered power law waiting times lead to tempered fractional time derivatives which have proven useful in geophysics the tempered fractional derivative or integral of a brownian motion called a tempered fractional brownian motion can exhibit semi long range dependence the increments of this process called tempered fractional gaussian noise provide a useful new stochastic model for wind speed data a tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations and it also provides a useful new correlation model in time series

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

an asymptotically linear schrodinger poisson system on r3 abstract in this paper we are concerned with the following nonlinear schrodinger poisson equation u u u f x u u x r 3 u 2 lim x x 0 where u r 3 r are positive functions the parameters 0 f x s tend to p x and q x l r 3 respectively as s 0 and s under some simple assumptions on f x s we give existence and nonexistence results depending on the parameters and

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

multi bump solutions for the nonlinear schrodinger poisson system in this paper we study a kind of nonlinear schrodinger poisson system with a parameter e for any positive integer m we prove that there exists e m 0 such that for 0 e e m the equation has an m bump positive solution under some suitable conditions as a consequence the equation has more and more multi bump positive solutions as e 0

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

backward fractional advection dispersion model for contaminant source prediction the forward fractional advection dispersion equation fade provides a useful model for non fickian transport in heterogeneous porous media the space fade captures the long leading tail skewness and fast spreading typically seen in concentration profiles from field data this paper develops the corresponding backward fade model to identify source location and release time the backward method is developed from the theory of inverse problems and then explained from a stochastic point of view the resultant backward fade differs significantly from the traditional backward advection dispersion equation ade because the fractional derivative is not self adjoint and the probability density function for backward locations is highly skewed finally the method is validated using tracer data from a well known field experiment where the peak of the backward fade curve predicts source release time while the median or a range of percentiles can be used to determine the most likely source location for the observed plume the backward ade cannot reliably identify the source in this application since the forward ade does not provide an adequate fit to the concentration data