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

high order methods and numerical boundary conditions abstract in this paper we present high order difference schemes for convection diffusion problems when we apply high order numerical methods to problems where physical boundary conditions are not periodic there is a need to choose adequate numerical boundary conditions in order to preserve the high order accuracy next to the boundary we do not usually have enough discrete points to apply the high order scheme and therefore at these nodes we must consider different approximations named the numerical boundary conditions the choice of numerical boundary conditions can influence the overall accuracy of the scheme and most of the times do influence the stability here we discuss which orders of accuracy are reasonable to be considered at the numerical boundary conditions such that we do not pay a high price in accuracy and stability

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

a parallel cgs block centered finite difference method for a nonlinear time fractional parabolic equation abstract in this article a parallel conjugate gradient squared cgs block centered finite difference scheme is introduced and analyzed to cast about the numerical solution of a nonlinear time fractional parabolic equation with the neumann condition and a nonlinear reaction term the unconditionally stable result which just depends on initial value and source item is derived some a priori estimates of discrete l 2 norm with optimal order of convergence o t 2 h 2 k 2 with pressure and velocity are established on both uniform and non uniform rectangular grids where t is the time step h and k are maximal mesh sizes of x and y directional grids in our model because the simulation is iterated over a series of time steps it is most beneficial if all the calculation units in a time step can be run separately in cgs algorithm adding openmp instructions to the circulation calculations in an iterative step can realize parallel computing to examine the efficiency and accuracy of the proposed method numerical experiments using the schemes are studied the results clearly show the benefit of using the proposed approach in terms of execution time reduction and speedup with respect to the sequential running in a single thread

generalization of alpha beta sss search procedures abstract search procedures such as alpha beta and sss are used to solve minimax game trees with a notable exception of b most of these procedures assume the static model i e the computation is done solely on the basis of static values given to terminal nodes the first goal of this paper is to generalize these to the informed model which permits the usage of heuristic information pertaining to nonterminal nodes such as upper and lower bounds and estimates of the exact values realizable from the corresponding game positions we provide a general framework within which various conventional procedures including alpha beta and sss can be naturally generalized to the informed model for the static model it is known that sss surpasses alpha beta in the sense that it explores only a subset of the nodes which are explored by alpha beta the second goal of this paper is assuming the informed model to develop a precise characterization of the class of search procedures that surpass alpha beta it turns out that the class contains many search procedures other than sss even for the static model finally some computational comparison among these search procedures is made by solving the 4 4 othello game

multigrid method for fractional diffusion equations the fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted grunwald formula the scheme is unconditionally stable and the coefficient matrix possesses the toeplitz like structure a multigrid method is proposed to solve the resulting system meanwhile the fast toeplitz matrix vector multiplication is utilized to lower the computational cost with only o nlogn complexity where n is the number of the grid points numerical experiments are given to demonstrate the efficiency of the method

a radial basis functions method for fractional diffusion equations one of the ongoing issues with fractional diffusion models is the design of an efficient high order numerical discretization this is one of the reasons why fractional diffusion models are not yet more widely used to describe complex systems in this paper we derive a radial basis functions rbf discretization of the one dimensional space fractional diffusion equation in order to remove the ill conditioning that often impairs the convergence rate of standard rbf methods we use the rbf qr method 1 33 by using this algorithm we can analytically remove the ill conditioning that appears when the number of nodes increases or when basis functions are made increasingly flat the resulting rbf qr based method exhibits an exponential rate of convergence for infinitely smooth solutions that is comparable to the one achieved with pseudo spectral methods we illustrate the flexibility of the algorithm by comparing the standard rbf and rbf qr methods for two numerical examples our results suggest that the global character of the rbfs makes them well suited to fractional diffusion equations they naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from a second order to a fractional order diffusion model as such they should be considered as one of the methods of choice to discretize fractional diffusion models of complex systems

a novel equivalent definition of caputo fractional derivative without singular kernel and superconvergent analysis in this article we present a new second order finite difference discrete scheme for a fractal mobile immobile transport model based on equivalent transformative caputo formulation the new transformative formulation takes the singular kernel away to make the integral calculation more efficient furthermore this definition is also effective where is a positive integer besides the t caputo derivative also helps us to increase the convergence rate of the discretization of the order 0 1 caputo derivative from o 2 to o 3 where is the time step for numerical analysis a crank nicolson finite difference scheme to solve the fractal mobile immobile transport model is introduced and analyzed the unconditional stability and a priori estimates of the scheme are given rigorously moreover the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis

least squares spectral method for the solution of a fractional advection dispersion equation fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields this article describes a least squares spectral method for solving advection dispersion equations using caputo or riemann liouville fractional derivatives a gauss lobatto jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition exponential convergence rate of the operator is verified when increasing the order of the approximation solutions are calculated for fractional time and fractional space differential equations comparisons with finite difference schemes are included a significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate

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

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

a second order explicit finite difference method for the fractional advection diffusion equation we develop a numerical method for fractional advection diffusion problems with source terms in domains with homogeneous boundary conditions the numerical method is derived by using a lax wendroff type time discretization procedure it is explicit and second order accurate the convergence of the numerical method is studied and numerical results are presented

a two grid mixed finite element method for a nonlinear fourth order reaction diffusion problem with time fractional derivative in this article we develop a two grid algorithm based on the mixed finite element mfe method for a nonlinear fourth order reaction diffusion equation with the time fractional derivative of caputo type we formulate the problem as a nonlinear fully discrete mfe system where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the mfe method to solve the nonlinear mfe system more efficiently we propose a two grid algorithm which is composed of two steps we first solve a nonlinear mfe system on a coarse grid by nonlinear iterations then solve the linearized mfe system on the fine grid by newton iteration numerical stability and optimal error estimate o k 2 h r 1 h 2 r 2 in l 2 norm are proved for our two grid scheme where k h and h are the time step size coarse grid mesh size and fine grid mesh size respectively we implement the two grid algorithm and present the numerical results justifying our theoretical error estimate the numerical tests also show that the two grid method is much more efficient than solving the nonlinear mfe system directly

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

a second order crank nicolson scheme for fractional cattaneo equation based on new fractional derivative abstract recently caputo and fabrizio introduce a new derivative with fractional order which has the ability to describe the material heterogeneities and the fluctuations of different scales in this article a crank nicolson finite difference scheme to solve fractional cattaneo equation based on the new fractional derivative is introduced and analyzed some a priori estimates of discrete l l2 errors with optimal order of convergence rate o 2 h 2 are established on uniform partition moreover the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis

smooth topological spaces abstract in 1986 badard introduced the concept of a smooth topological space in this paper we give some links between smooth topological spaces and the corresponding results in chang s fuzzy topological spaces the concepts of subspace continuity and compactness are studied

an explicit high order method for fractional advection diffusion equations we propose a high order explicit finite difference method for fractional advection diffusion equations these equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order with 1 2 this operator is defined by a combination of the left and right riemann liouville fractional derivatives we study the convergence of the numerical method through consistency and stability the order of convergence varies between two and three and for advection dominated flows is close to three although the method is conditionally stable the restrictions allow wide stability regions the analysis is confirmed by numerical examples

finite difference finite element method for a nonlinear time fractional fourth order reaction diffusion problem abstract in this article a finite difference finite element algorithm which is based on a finite difference approximation in time direction and finite element method in spatial direction is presented and discussed to cast about for the numerical solutions of a time fractional fourth order reaction diffusion problem with a nonlinear reaction term to avoid the use of higher order elements the original problem with spatial fourth order derivative need to be changed into a second order coupled system by introducing an intermediate variable u then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction the unconditionally stable result in the norm which just depends on initial value and source item is derived some a priori estimates of l 2 norm with optimal order of convergence o t 2 h m 1 where t and h are time step length and space mesh parameter respectively are obtained to confirm the theoretical analysis some numerical results are provided by our method

the use of finite difference element approaches for solving the time fractional subdiffusion equation in this paper two finite difference element approaches for the time fractional subdiffusion equation with dirichlet boundary conditions are developed in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method the two methods are unconditionally stable and convergent of order o tau q h r 1 in the l 2 norm where q 2 beta or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth beta 0 beta 1 is the order of the fractional derivative tau and h are the step sizes in time and space respectively and r is the degree of the polynomial space the corresponding schemes for the subdiffusion equation with neumann boundary conditions are presented as well where the stability and convergence are shown numerical examples are provided to verify the theoretical analysis comparisons between the algorithms derived in this paper and the existing algorithms are given

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

the kolmogorov forward fractional partial differential equation for the cgmy process with applications in option pricing abstract in this paper we derive the kolmogorov forward fractional partial differential equation fpde for the cgmy process the resulting fpde is solved numerically with a second order method in space and backward differentiation formula of order two in time we price options by integrating the resulting probability density function multiplied by the pay off function of the option hence we only have to solve one fpde to price several options this is useful in practical applications where it is common to price many options simultaneously for the same underlying diffusion model the traditional way to price options requires the solution of one fpde per option since the fpdes are of similar type and complexity the advantage of our suggested method is obvious in the case of pricing multiple european type options on the same underlying asset when only the pay off function differs the numerical experiments verify that the extra cost of pricing several options compared to only one option is very small with our suggested method

finite difference approximations for a fractional advection diffusion problem the use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed fractional space derivatives are used to model anomalous diffusion or dispersion where a particle plume spreads at a rate inconsistent with the classical brownian motion model when a fractional derivative replaces the second derivative in a diffusion or dispersion model it leads to enhanced diffusion also called superdiffusion we consider a one dimensional advection diffusion model where the usual second order derivative gives place to a fractional derivative of order a with 1 a 2 we derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection diffusion equation we present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions in the end we present a test problem

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed