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

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

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

a preconditioned fast quadratic spline collocation method for two sided space fractional partial differential equations abstract quadratic spline collocation methods for the one and two dimensional fractional diffusion equations are proposed by carefully exploring the mathematical structure of the coefficient matrix we propose a matrix free fast krylov subspace iterative solver for the corresponding quadratic spline collocation scheme and then preconditioning technique is applied to further accelerate the convergence of the fast krylov subspace iterative method it shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from o n 2 to o n log n per iteration and memory requirement from o n 2 to o n while still well approximates the fractional diffusion equations without any accuracy lost numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method

a finite volume method for two sided fractional diffusion equations on non uniform meshes we derive a finite volume method for two sided fractional diffusion equations with riemannliouville derivatives in one spatial dimension the method applies to non uniform meshes with arbitrary nodal spacing the discretisation utilises the integral definition of the fractional derivatives and we show that it leads to a diagonally dominant matrix representation and a provably stable numerical scheme being a finite volume method the numerical scheme is fully conservative and the ability to locally refine the mesh can produce solutions with more accuracy for the same number of nodes compared to a uniform mesh as we demonstrate numerically

stock loan valuation based on the finite moment log stable process abstract the empirical test suggests that the log return series of stock price in us market reject the normal distribution and admit instead a subclass of the asymmetric distribution in this paper we investigate the stock loan problem under the assumption that the return of stock follows the finite moment log stable process fmls in this case the pricing model of stock loan can be described by a space fractional partial differential equation with time varying free boundary condition firstly a penalty term is introduced to change the original problem to be defined on a fixed domain and then a fully implicit difference scheme has been developed secondly based on the fully implicit scheme we prove that the stock loan value generated by the penalty method cannot fall below the value obtained when the stock loan is exercised early thirdly the numerical experiments are carried out to demonstrate differences of stock loan model under the fmls and the standard normal distribution optimal redemption strategy of stock loan has been achieved furthermore the impact of key parameters in our model on the stock loan evaluation are analyzed and some reasonable explanation are given

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

local discontinuous galerkin method for a nonlinear time fractional fourth order partial differential equation abstract in this article a fully discrete local discontinuous galerkin ldg method with high order temporal convergence rate is presented and developed to look for the numerical solution of nonlinear time fractional fourth order partial differential equation pde in the temporal direction for approximating the fractional derivative with order 0 1 the weighted and shifted grunwald difference wsgd scheme with second order convergence rate is introduced and for approximating the integer time derivative two step backward euler method with second order convergence rate is used for the spatial direction the ldg method is used for the numerical theories the stability is derived and a priori error results are proved further some error results and convergence rates are calculated by numerical procedure to illustrate the effectiveness of proposed 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

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

preconditioned iterative methods for space time fractional advection diffusion equations in this paper we propose practical numerical methods for solving a class of initial boundary value problems of space time fractional advection diffusion equations first we propose an implicit method based on two sided grunwald formulae and discuss its stability and consistency then we develop the preconditioned generalized minimal residual preconditioned gmres method and preconditioned conjugate gradient normal residual preconditioned cgnr method with easily constructed preconditioners importantly because resulting systems are toeplitz like fast fourier transform can be applied to significantly reduce the computational cost we perform numerical experiments to demonstrate the efficiency of our preconditioners even in cases with variable coefficients

a fast semi implicit difference method for a nonlinear two sided space fractional diffusion equation with variable diffusivity coefficients in this paper we derive a new nonlinear two sided space fractional diffusion equation with variable coefficients from the fractional fick s law a semi implicit difference method sidm for this equation is proposed the stability and convergence of the sidm are discussed for the implementation we develop a fast accurate iterative method for the sidm by decomposing the dense coefficient matrix into a combination of toeplitz like matrices this fast iterative method significantly reduces the storage requirement of o n 2 and computational cost of o n 3 down to n and o n log n where n is the number of grid points the method retains the same accuracy as the underlying sidm solved with gaussian elimination finally some numerical results are shown to verify the accuracy and efficiency of the new method

high order finite element methods for time fractional partial differential equations the aim of this paper is to develop high order methods for solving time fractional partial differential equations the proposed high order method is based on high order finite element method for space and finite difference method for time optimal convergence rate o dt 2 a n r is proved for the r 1 th order finite element method r 2

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

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

a preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh numerical methods for fractional differential equations generate full stiffness matrices which were traditionally solved via gaussian type direct solvers that require o n 3 of computational work and o n 2 of memory to store where n is the number of spatial grid points in the discretization we develop a preconditioned fast krylov subspace iterative method for the efficient and faithful solution of finite volume schemes defined on a locally refined composite mesh for fractional differential equations to resolve boundary layers of the solutions numerical results are presented to show the utility of the method

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

reply to comment on self organized criticality and absorbing states lessons from the ising model in r dickman m a mu noz a vespignani and s zapperi braz j phys 30 27 2000 dickman et al suggested that self organized criticality can be produced by coupling the activity of an absorbing state model to a dissipation mechanism and adding an external drive we analyzed the proposed mechanism in g pruessner and o peters phys rev e 73 025106 r 2006 and found that if this mechanism is at work the finite size scaling found in self organized criticality will depend on the details of the implementation of dissipation and driving in the preceding paper m j alava l laurson a vespignani and s zapperi phys rev e 77 048101 2008 alava et al show that one avalanche exponent in the absorbing state approach becomes independent of dissipation and driving in our reply we clarify their findings and put them in the context of the original paper

an efficient matrix splitting preconditioning technique for two dimensional unsteady space fractional diffusion equations abstract we utilize the matrix splitting iteration method based on the structure of the coefficient matrix to construct the preconditioner for the finite difference discretization of two dimensional time dependent space fractional diffusion equation with variable diffusivity coefficients the spectral radius of the preconditioned matrix is shown to be clustered around one as we prove that it can be bounded by the euclidean norm of a sum of three matrices where either the eigenvalues of the component matrices are clustered around one or their euclidean norms decrease as we refine the spatial mesh size numerical comparisons are presented to demonstrate the effectiveness and efficiency of the proposed preconditioner

stability and convergence analysis of the quadratic spline collocation method for time dependent fractional diffusion equations abstract a quadratic spline collocation method combined with the crank nicolson time discretization is proposed for time dependent two sided fractional diffusion equations by carefully analyzing the mathematical properties of the coefficient matrix the new scheme is proved to be unconditionally stable in the sense of discrete l2 norm for 2 where is the order of the space fractional derivative of the fractional diffusion equation and 1 2576 see lemma 3 1 furthermore the fractional order spline interpolation error over the collocation points is studied and subsequently we show that the spline collocation solution of the fractional diffusion equation converges to the exact one with order o h 3 2 under the discrete l2 norm where and h are the temporal and spatial step sizes respectively finally numerical experiments are given to verify the theoretical results

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

parameter estimation for the fractional fractal diffusion model based on its numerical solution in this paper we mainly consider a problem of parameter estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media the bayesian method is proposed to estimate parameters for the fractional fractal diffusion model based on its numerical solution firstly the center box difference method is employed to solve the fractional fractal diffusion equation subject to the initial boundary conditions then we apply the bayesian method to estimate three parameters for the model simultaneously that is the fractional order the fractal dimension d f and the structure parameter to testify the validity of the results for the model s parameter estimation experimental data from the fast desorption process of methane in coal are used it is shown that the numerical results modeled with parameters given by the bayesian method coincide well with the desorption experimental data indicating that the bayesian method is efficient and valid in identifying multi parameter for the fractional fractal diffusion model besides the fractional fractal diffusion model is demonstrated to be more capable than the classical fick s model to describe the anomalous diffusion behavior of gas in coal to clarify the parameters effect on the relative diffusion amount of methane for the fractional fractal diffusion model parameter sensitivity analysis is also carried out this paper provides a specific and efficient parameter estimation method for the fractional fractal diffusion model

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

stability and convergence of a new finite volume method for a two sided space fractional diffusion equation in this paper we consider a two sided space fractional diffusion equation with variable coefficients on a finite domain firstly based on the nodal basis functions we present a new fractional finite volume method for the two sided space fractional diffusion equation and derive the implicit scheme and solve it in matrix form secondly we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent finally some numerical examples are given to show the effectiveness of the new numerical method and the results are in excellent agreement with theoretical analysis

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

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

high order algorithms for riesz derivative and their applications ii in this paper we firstly develop two high order approximate formulas for the riesz fractional derivative secondly we propose a temporal second order numerical method for a fractional reaction dispersion equation where we discretize the riesz fractional derivative by using two numerical schemes we prove that the numerical methods for a spatial riesz fractional reaction dispersion equation are both unconditionally stable and convergent and the orders of convergence are o 2 h 6 and o 2 h 8 in which and h are spatial and temporal step sizes respectively finally we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis

a stable explicitly solvable numerical method for the riesz fractional advection dispersion equations abstract in this paper we present a finite difference scheme for solving the riesz fractional advection dispersion equations rfades the scheme is obtained by using asymmetric discretization technique and modify the shifted grunwald approximation to fractional derivative by calculating the unknowns in differential nodal point sequences at the odd and even time levels the discrete solution of the scheme can be obtained explicitly the computational cost for the scheme at each time step can be o klog k by using the fast matrix vector multiplication with the help of toeplitz structure where k is the number of unknowns we prove that the scheme is solvable and unconditionally stable we derive the error estimates in discrete l2 norm which is optimal in some cases numerical examples are presented to verify our theoretical results

high order numerical methods for the riesz space fractional advection dispersion equations abstract in this paper we propose high order numerical methods for the riesz space fractional advection dispersion equations rsfade on a finite domain the rsfade is obtained from the standard advection dispersion equation by replacing the first order and second order space derivative with the riesz fractional derivatives of order 0 1 and 1 2 respectively firstly we utilize the weighted and shifted grunwald difference operators to approximate the riesz fractional derivative and present the finite difference method for the rsfade specifically we discuss the crank nicolson scheme and solve it in matrix form secondly we prove that the scheme is unconditionally stable and convergent with the accuracy of o 2 h 2 thirdly we use the richardson extrapolation method rem to improve the convergence order which can be o 4 h 4 finally some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent with theoretical analysis

direct meshless local petrov galerkin dmlpg method for time fractional fourth order reaction diffusion problem on complex domains abstract a new numerical scheme has been developed based on the fast and efficient meshless local weak form i e direct meshless local petrov galerkin dmlpg method for solving the fractional fourth order partial differential equation on computational domains with complex shape the fractional derivative is the riemann liouville fractional derivative at first a finite difference scheme with the second order accuracy has been employed to discrete the time variable then the dmlpg technique is employed to achieve a full discrete scheme the time discrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the l 2 space also some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and non regular grid points

finite volume element method and its stability analysis for analyzing the behavior of sub diffusion problems in this paper we analyze the spatially semi discrete piecewise linear finite volume element method for the time fractional sub diffusion problem in two dimensions and give an approximate solution of this problem at first we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods furthermore we use the standard finite element ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method finally some numerical examples are included to illustrate the effectiveness of the new technique

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

high order unconditionally stable difference schemes for the riesz space fractional telegraph equation in this paper a class of unconditionally stable difference schemes based on the pade approximation is presented for the riesz space fractional telegraph equation firstly we introduce a new variable to transform the original differential equation to an equivalent differential equation system then we apply a second order fractional central difference scheme to discretize the riesz space fractional operator finally we use 1 1 2 2 and 3 3 pade approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations matrix analysis is used to show the unconditional stability of the proposed algorithms two examples with known exact solutions are chosen to assess the proposed difference schemes numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space fractional hyperbolic equation