approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

dimension reduction in a semiparametric regression model with errors in covariates we consider a semiparametric estimation method for general regression models when some of the predictors are measured with error the technique relies on a kernel regression of the true covariate on all the observed covariates and surrogates this requires a nonparametric regression in as many dimensions as there are covariates and surrogates the usual theory copes with such higher dimensional problems by using higher order kernels but this is unrealistic for most problems we show that the usual theory is essentially as good as one can do with this technique instead of regression with higher order kernels we propose the use of dimension reduction techniques we assume that the true covariate depends only on a linear combination of the observed covariates and surrogates if this linear combination were known we could apply the one dimensional versions of the semiparametric problem for which standard kernels are applicable we show that if one can estimate the linear directions at the root n rate then asymptotically the resulting estimator of the parameters in the main regression model behaves as if the linear combination were known simulations lend some credence to the asymptotic results

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

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

a finite volume method for two dimensional riemann liouville space fractional diffusion equation and its efficient implementation abstract we develop a finite volume method based on crank nicolson time discretization for the two dimensional nonsymmetric riemann liouville space fractional diffusion equation stability and convergence are then carefully discussed we prove that the finite volume scheme is unconditionally stable and convergent with second order accuracy in time and min 1 1 order in space with respect to a weighted discrete norm here 0 1 are the space fractional order indexes in x and y directions respectively furthermore we rewrite the finite volume scheme into a matrix form and develop a matrix free preconditioned fast krylov subspace iterative method which only requires storage of o n and computational cost of o n log n per iteration without losing any accuracy compared to the direct gaussian elimination method here n is the total number of spatial unknowns consequently the fast finite volume method is particularly suitable for large scale modeling and simulation numerical experiments verify the theoretical results and show strong potential of the fast method

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

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

second order approximation scheme combined with h1 galerkin mfe method for nonlinear time fractional convection diffusion equation abstract in this article a second order approximation scheme combined with an h 1 galerkin mixed finite element mfe method for solving nonlinear convection diffusion equation with time fractional derivative is proposed and analyzed by introducing an auxiliary variable a coupled system is formulated then the spatial direction is approximated by h 1 galerkin mfe method and the temporal fractional derivative and integer derivative are discretized by second order weighted and shifted grunwald difference wsgd formula and linearized second order difference scheme respectively the optimal priori error estimates in l 2 and h 1 norm for the unknown function and the auxiliary variable with second order convergent rate in time are obtained compared to the commonly used l1 approximation with 2 th order convergence rate our method can arrive at the order 2 in time what is more compared with the standard finite element method our method can well approximate the auxiliary variable finally the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis

fast algorithm based on tt m fe system for space fractional allen cahn equations with smooth and non smooth solutions abstract in this article a fast algorithm based on time two mesh tt m finite element fe scheme which aims at solving nonlinear problems quickly is considered to numerically solve the nonlinear space fractional allen cahn equations with smooth and non smooth solutions the implicit second order scheme containing both implicit crank nicolson scheme and second order backward difference method is applied to time direction a fast tt m method is used to increase the speed of calculation and the fe method is developed to approximate the spacial direction the tt m fe algorithm includes the following main computing steps firstly a nonlinear implicit second order fe scheme on the time coarse mesh c is solved by a nonlinear iterative method secondly based on the chosen initial iterative value a linearized fe system on time fine mesh c is solved where some useful coarse numerical solutions are found by the lagrange s interpolation formula the analysis for both stability and a priori error estimates are made in detail finally three numerical examples with smooth and non smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations from which it is easy to find that the computing time can be saved

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

error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives abstract in the current investigation an error estimate has been proposed to solve the two dimensional weakly singular integro partial differential equation with space and time fractional derivatives based on the finite element finite difference scheme the time and space derivatives are based on the riemann liouville and riesz fractional derivatives respectively at first the temporal variable has been discretized by a second order difference scheme and then the space variable has been approximated by the finite element method fem the analytical study shows that the presented scheme is unconditionally stable and convergent finally some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique

high order local discontinuous galerkin method combined with wsgd approximation for a fractional subdiffusion equation abstract in this paper a high order local discontinuous galerkin ldg method combined with weighted and shifted grunwald difference wsgd approximation is developed and discussed for a caputo time fractional subdiffusion equation the time fractional derivative of order 0 1 is approximated by a third order method based on the idea of wsgd operator while the spatial operator is approximated by the ldg method some useful lemmas are first introduced and proved then the analysis of stability and optimal error estimate o t 3 h k 1 are obtained for the ldg method extensive numerical results using p k k 0 1 2 3 elements are presented to validate our theoretical analysis

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

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

approximation of the derivatives of solutions in a normalized domain for 2d solids using the pies method numerical methods for solving boundary value problems despite their popularity are also affected by defects the most troublesome are the necessity of discretization the calculation of singular integrals and large computational complexity the aim of the paper is to develop a general strategy for approximation of the derivatives of solutions the versatility of the approach is ensured by performing approximation in a normalized domain in order to obtain efficient and accurate tool for solving the variety of boundary problems we should combine the mentioned strategy with the method developed by the authors called parametric integral equations system pies the effectiveness of the proposed approach lies in lack of the discretization of both a boundary and a domain less computational complexity possibility of determining the derivatives of solutions in a continuous manner at any point of the boundary and the domain without the necessity of calculating singular integrals the paper presents the different variants of the proposed strategy tests were performed on the examples of elastic bodies with various shapes and boundary conditions the reliability of the proposed approach is shown in comparison to existing analytical solutions the normalized approximation strategy for derivatives in combination with the pies method gives accurate results which are obtained in elementary and efficient way

a fast second order accurate method for a two sided space fractional diffusion equation with variable coefficients in this paper we consider a type of fractional diffusion equation fde with variable coefficients on a finite domain firstly we utilize a second order scheme to approximate the riemannliouville fractional derivative and present the finite difference scheme specifically we discuss the cranknicolson scheme and solve it in matrix form secondly we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second order accuracy of o 2 h2 furthermore we develop a fast accurate iterative method for the cranknicolson scheme which only requires storage of o m and computational cost of o mlogm while retaining the same accuracy and approximation property as gauss elimination where m 1 h is the partition number in space direction finally several numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

an unstructured mesh finite difference finite element method for the three dimensional time space fractional bloch torrey equations on irregular domains abstract in this paper we use the finite element method fem to solve the time space fractional bloch torrey equation on irregular domains in r 3 based on linear lagrange basis functions a space semi discrete fem scheme is given by adopting the l 2 1 approximation for the caputo fractional derivative a fully discrete scheme is presented furthermore we provide the details on how to implement our fem for the space fractional bloch torrey equation also the stability and convergence of the fully discrete scheme is investigated the error estimations with respect to the l 2 and energy norms are given in addition some numerical examples are presented to verify the efficiency of our method

high accuracy analysis of an h1 galerkin mixed finite element method for two dimensional time fractional diffusion equations abstract in this paper an h 1 galerkin mixed finite element approximate scheme is established for a class of two dimensional time fractional diffusion equations by using the bilinear element raviart thomas element and l 1 time stepping method which is unconditionally stable and free of lbb condition and then without the classical ritz projection superclose results for the original variable u in h 1 norm and the flux p u in h d i v norm are derived by means of properties of the elements and l 1 approximation furthermore with the help of the interpolation postprocessing operator the global superconvergence results for the original variable u in h 1 norm are obtained finally numerical simulations verify that the theoretical results are true on both regular meshes and anisotropic meshes

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

galerkin finite element approximation of symmetric space fractional partial differential equations abstract in this paper symmetric space fractional partial differential equations ssfpde with the riesz fractional operator are considered the ssfpde is obtained from the standard advection dispersion equation by replacing the first order and second order space derivatives with the riesz fractional derivatives of order 2 0 1 and 2 1 2 respectively we prove that the variational solution of the ssfpde exists and is unique using the galerkin finite element method and a backward difference technique a fully discrete approximating system is obtained which has a unique solution according to the lax milgram theorem the stability and convergence of the fully discrete schemes are derived finally some numerical experiments are given to confirm our theoretical analysis

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 finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation abstract an efficient numerical technique is proposed to solve one and two dimensional space fractional tempered fractional diffusion wave equations the space fractional is based on the riemann liouville fractional derivative at first the temporal direction is discretized using a second order accurate difference scheme then a classic galerkin finite element is employed to obtain a full discrete scheme furthermore for the time discrete and the full discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method finally two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique

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

schrodinger type equation for subjective identification of supply and demand abstract the present authors have put forward a quantum game theory based model of market prices movements by using fisher information we present a construction of an equation of schrodinger type for probability distributions for relationship between demand and supply various analogies between quantum physics and market phenomena can be found

a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation in this paper a new alternating direction implicit galerkin legendre spectral method for the two dimensional riesz space fractional nonlinear reaction diffusion equation is developed the temporal component is discretized by the crank nicolson method the detailed implementation of the method is presented the stability and convergence analysis is strictly proven which shows that the derived method is stable and convergent of order 2 in time an optimal error estimate in space is also obtained by introducing a new orthogonal projector the present method is extended to solve the fractional fitzhugh nagumo model numerical results are provided to verify the theoretical analysis

numerical algorithm based on adomian decomposition for fractional differential equations in this paper a novel algorithm based on adomian decomposition for fractional differential equations is proposed comparing the present method with the fractional adams method we use this derived computational method to find a smaller efficient dimension such that the fractional lorenz equation is chaotic we also apply this new method to the time fractional burgers equation with initial and boundary value conditions numerical results and computer graphics show that the constructed numerical is efficient

finite element method for two dimensional space fractional advection dispersion equations the backward euler and crank nicolson galerkin fully discrete approximate schemes for two dimensional space fractional advection dispersion equations are established firstly we prove that the corresponding variational problem has a unique solution and the proposed fully discrete schemes are unconditionally stable whose solutions are all unique secondly the optimal error estimates are derived by use of properties of projection operator and fractional derivatives finally numerical examples demonstrate effectiveness of numerical schemes and confirm the theoretical analysis

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