the high order compact numerical algorithms for the two dimensional fractional sub diffusion equation in this paper performing the average operators on the space variables a numerical scheme with third order temporal convergence for the two dimensional fractional sub diffusion equation is considered for which the unconditional stability and convergence in l1 l norm are strictly analyzed for 0 0 9569347 by using the discrete energy method therewith adding small perturbation terms we construct a compact alternating direction implicit difference scheme for the two dimensional case finally some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all 0 1

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

extending the bruhat order and the length function from the weyl group to the weyl monoid abstract for a symmetrizable kac moody algebra the category of admissible representations of the category o is an analogue of the category of finite dimensional representations of a semisimple lie algebra the monoid which is associated to this category and the category of restricted duals by a tannaka krein reconstruction contains the kac moody group as open dense unit group it has similar properties as a reductive algebraic monoid in particular there are bruhat and birkhoff decompositions the weyl group replaced by the weyl monoid c mokler an analogue of a reductive algebraic monoid whose unit group is a kac moody group arxiv math ag 0204246 mem amer math soc in press we determine the closure relations of the bruhat and birkhoff cells which give extensions of the bruhat order from the weyl group to the weyl monoid we show that the bruhat and birkhoff cells are irreducible and principal open in their closures we give product decompositions of the bruhat and birkhoff cells we define extended length functions which are compatible with the extended bruhat orders we show a generalization of some of the tits axioms for twin bn pairs

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

an implicit midpoint difference scheme for the fractional ginzburg landau equation this paper proposes and analyzes an efficient difference scheme for the nonlinear complex ginzburg landau equation involving fractional laplacian the scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted grunwald difference operator for the spatial fractional laplacian by virtue of a careful analysis of the difference operator some useful inequalities with respect to suitable fractional sobolev norms are established then the numerical solution is shown to be bounded and convergent in the l h 2 norm with the optimal order o 2 h 2 with time step and mesh size h the a priori bound as well as the convergence order holds unconditionally in the sense that no restriction on the time step in terms of the mesh size h needs to be assumed numerical tests are performed to validate the theoretical results and effectiveness of the scheme

stable extensions of homeomorphisms on the pseudo arc we prove the following theorem if p is a proper subcontinuum of the pseudoarc p h is a homeomorphism from p onto itself and 0 is an open set in p that contains p then there is a homeomorphism h from p onto itself such that h p h and h x x for x 0 a pseudoarc can be characterized as a chainable hereditarily indecomposable continuum although the main result here extends a result of w lewis lw and answers in the affirmative a question he asked in that same paper stable homeomorphisms of the pseudoarc this paper was motivated by a conversation the author had with professor marcy barge who works in the area of topological dynamics professor barge is interested in knowing what sort of self homeomorphisms the pseudoarc admits and in particular what sort of homeomorphisms it admits with the property that some of the points of the space have periodic orbits that pseudoarcs admit many such homeomorphisms will follow from a much earlier result of g lehner lg and the result here it will also follow by combining the theorem proven here g lehner s theorem and a theorem in kj that any continuum with the property that all its proper nondegenerate subcontinua are pseudoarcs admits many such homeomorphisms examples of continua with this property include pseudocircles b pseudosolenoids f some examples due to t ingram i and h cook c as well as the pseudoarc itself we will prove the following theorem theorem if p is a proper nondegenerate subcontinuum of the pseudoarc p p c o an open set in p and h is a homeomorphism from p onto itself then h can be extended to a homeomorphism h of p onto itself such that h x x for x o a continuum is indecomposable if no proper subcontinuum of it has interior and it is hereditarily indecomposable if each subcontinuum is indecomposable a chain is a collection of sets c co cy cn such that ci f c3 0 if and only if i j 1 a continuum is chainable if it has open chain covers of arbitrarily small mesh as mentioned previously the pseudoarc p is a hereditarily indecomposable chainable continuum kb this continuum also has the fixed point property h received by the editors june 9 1987 this research was presented at the annual spring topology conference march 1 1987 university of alabama at birmingham this conference was supported by the nsf 1980 mathematics subject classification 1985 revision primary 54f20 54f50

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

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

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 note on efficient preconditioner of implicit runge kutta methods with application to fractional diffusion equations abstract this paper is concerned with the construction of efficient preconditioning method for systems arising from implicit runge kutta discretizations of time dependent pdes a polynomial preconditioner is proposed based on the kronecker product splitting iteration method introduced by chen bit 54 607 721 2014 some useful properties on the spectrum of the preconditioned matrix are established numerical experiments concerning discrete solution of fractional diffusion equations are presented to show the effectiveness of the proposed polynomial preconditioner

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

design and analysis of a dissipative scheme to solve a generalized multi dimensional higgs boson equation in the de sitter space time abstract in this work we design a numerically efficient finite difference technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with riesz fractional derivatives of orders in 0 1 1 2 which considers a generalized potential and a time dependent diffusion factor an energy integral for the mathematical model is readily available and we propose an explicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties as the continuous model a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the 3 1 dimensional de sitter space time and find results qualitatively similar to those available in the literature the present work is the first paper to report on a hamiltonian discretization of the higgs boson equation both fractional and non fractional in the de sitter space time and its numerical analysis more precisely the present manuscript is the first paper of the literature in which a dissipation preserving scheme to solve the multi dimensional fractional higgs boson equation in the de sitter space time is proposed and thoroughly analyzed indeed it is worth pointing out that previous efforts used techniques based on the runge kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed

a compact difference scheme for a two dimensional fractional klein gordon equation with neumann boundary conditions in this paper a high order finite difference scheme for a two dimensional fractional klein gordon equation subject to neumann boundary conditions is proposed the difficulty induced by the nonlinear term and the neumann conditions is carefully handled in the proposed scheme the stability and convergence of the finite difference scheme are analyzed using the matrix form of the scheme numerical examples are provided to demonstrate the theoretical results

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

an implicit difference scheme with the kps preconditioner for two dimensional time space fractional convection diffusion equations abstract this paper deals with the numerical computation and analysis for a class of two dimensional time space fractional convection diffusion equations an implicit difference scheme is derived for solving this class of equations it is proved under some suitable conditions that the derived difference scheme is stable and convergent moreover the convergence orders of the scheme in time and space are also given in order to accelerate the convergence rate by combining the kronecker product splitting kps preconditioner with the generalized minimal residual gmres method a preconditioning strategy for implementing the difference scheme is introduced finally several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods

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

maximum norm error analysis of difference schemes for fractional diffusion equations in this study we present several numerical approximation methods for solving the fractional diffusion equations with the riesz space fractional derivative based on the fractional centered difference approximation to the riesz fractional derivative an explicit and an implicit finite difference schemes for solving the space fractional diffusion equations are obtained the stability and global convergence of these two schemes in the maximum norm are analyzed rigorously the convergence order is o h 2 where is the temporal grid size and h is spatial grid size respectively furthermore a numerical scheme for the one dimensional multi term time space fractional diffusion equations is obtained the unconditionally stability and convergence of the scheme in maximum norm are established the convergence orders are 2 in the temporal direction and two in the spatial direction where is the maximum time fractional derivative orders in the equation in addition the extension to the two dimensional case is also discussed finally several numerical examples are provided to show the effectiveness and accuracy of our methods

second order numerical methods for multi term fractional differential equations smooth and non smooth solutions abstract starting with the asymptotic expansion of the error equation of the shifted grunwald letnikov formula we derive a new modified weighted shifted grunwald letnikov wsgl formula by introducing appropriate correction terms we then apply one special case of the modified wsgl formula to solve multi term fractional ordinary and partial differential equations and we prove the linear stability and second order convergence for both smooth and non smooth solutions when the regularity of the solutions is known we show theoretically and numerically that numerical solutions with good accuracy can be obtained with only a few correction terms moreover the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions numerical simulations verify the theoretical results and demonstrate that the new wsgl formula leads to better performance compared to other known numerical approximations with similar resolution

local discontinuous galerkin methods for the time tempered fractional diffusion equation abstract in this article we consider high order discrete schemes for solving the time tempered fractional diffusion equation we present a semi discrete scheme by using the local discontinuous galerkin ldg discretization in the spatial variable we prove that the semi discrete scheme is unconditionally stable in l2 norm and convergence with optimal convergence rate o h k 1 where h is the spatial step size we develop a class of fully discrete ldg schemes by combining the weighted and shifted lubich difference operators with respect to the time variable and establish the error estimates finally numerical experiments are presented to verify the theoretical results

crank nicolson galerkin spectral method for solving two dimensional time space distributed order weakly singular integro partial differential equation abstract the fractional pdes based upon the distributed order fractional derivative have several applications in physics the two dimensional time space distributed order weakly singular integro partial differential model is investigated by a combination of finite difference and galerkin spectral methods a second order finite difference formula is employed to approximate the temporal variable in this stage the stability and convergence of the semi discrete scheme are proved then the galerkin spectral method based on the modified jacobi polynomials is applied to discrete the space variable also in this step the error estimate of the full discrete scheme is studied finally two test problems have been presented to confirm the theoretical results

space fractional advection diffusion and reflective boundary condition anomalous diffusive transport arises in a large diversity of disordered media stochastic formulations in terms of continuous time random walks ctrws with transition probability densities showing space and or time diverging moments were developed to account for anomalous behaviors a broad class of ctrws was shown to correspond on the macroscopic scale to advection diffusion equations involving derivatives of noninteger order in particular ctrws with l evy distribution of jumps and finite mean waiting time lead to a space fractional equation that accounts for superdiffusion and involves a nonlocal integral differential operator within this framework we analyze the evolution of particles performing symmetric l evy flights with respect to a fluid moving at uniform speed v the particles are restricted to a semi infinite domain limited by a reflective barrier we show that the introduction of the boundary condition induces a modification in the kernel of the nonlocal operator thus the macroscopic space fractional advection diffusion equation obtained is different from that in an infinite medium

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

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

a limited memory block bi diagonal toeplitz preconditioner for block lower triangular toeplitz system from time space fractional diffusion equation abstract a block lower triangular toeplitz system arising from the time space fractional diffusion equation is discussed for efficient solutions of such the linear system the preconditioned biconjugate gradient stabilized method and the flexible general minimal residual method are exploited the main contribution of this paper has two aspects i a block bi diagonal toeplitz preconditioner is developed for the block lower triangular toeplitz system whose storage is of o n with n being the spatial grid number ii a new skew circulant preconditioner is designed to accelerate the inverse of the block bi diagonal toeplitz preconditioner multiplying a vector numerical experiments are given to demonstrate the effectiveness of our two proposed preconditioners

fast numerical simulation of a new time space fractional option pricing model governing european call option abstract when the fluctuation of option price is regarded as a fractal transmission system and the stock price follows a levy distribution a time space fractional option pricing model tsfopm is obtained then we discuss the numerical simulation of the tsfopm a discrete implicit numerical scheme with a second order accuracy in space and a 2 order accuracy in time is constructed where is a transmission exponent the stability and convergence of the obtained numerical scheme are analyzed moreover a fast bi conjugate gradient stabilized method is proposed to solve the numerical scheme in order to reduce the storage space and computational cost then a numerical example with exact solution is presented to demonstrate the accuracy and effectiveness of the proposed numerical method finally the tsfopm and the above numerical technique are applied to price european call option the characteristics of the fractional option pricing model are analyzed in comparison with the classical black scholes b s model

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

linearized crank nicolson scheme for the nonlinear time space fractional schrodinger equations abstract in this paper a crank nicolson difference scheme is first derived for solving the nonlinear time space fractional schrodinger equations the truncation error and stability of the scheme are discussed in detail the existence of the numerical solution is shown by the brouwer fixed point theorem for improving the calculating efficiency a three level linearized difference scheme is also proposed and analyzed both schemes are subsequently extended to the nonlinear coupled equations and some similar results are given and proved several numerical experiments are included to verify the accuracy and efficiency of the two types of schemes and comparison with the related work is presented

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

fourth order finite difference schemes for time space fractional sub diffusion equations in this paper we devote to the study of high order finite difference schemes for one and two dimensional time space fractional sub diffusion equations a fourth order finite difference scheme is invoked for the spatial fractional derivatives and the l 1 approximation is applied to the temporal fractional parts for the two dimensional case an alternating direction implicit scheme based on l 1 approximation is proposed the stability and convergence of the proposed methods are studied numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes

a novel finite element method for the distributed order time fractional cable equation in two dimensions abstract in this paper the unstructured mesh galerkin finite element method with a weighted and shifted grunwald difference approximation and composite trapezoid formula is presented to solve the nonhomogeneous two dimensional distributed order time fractional cable equation on irregular convex domains the crank nicolson type discretization of the finite element scheme is implemented to obtain the numerical solution the stability and convergence of the numerical scheme are discussed and derived finally some numerical examples on irregular convex domains are given to confirm our theoretical results

adaptive finite element method for fractional differential equations using hierarchical matrices abstract a robust and fast solver for the fractional differential equation fdes involving the riesz fractional derivative is developed using an adaptive finite element method it is based on the utilization of hierarchical matrices h matrices for the representation of the stiffness matrix resulting from the finite element discretization of the fdes we employ a geometric multigrid method for the solution of the algebraic system of equations we combine it with an adaptive algorithm based on a posteriori error estimation a posteriori error estimation based adaptive algorithm is used to deal with general type singularities arising in the solution of the fdes through various test examples we demonstrate the efficiency of the method and the high accuracy of the numerical solution even in the presence of singularities the proposed technique has been verified effectively through fundamental examples including riesz left right riemann liouville fractional derivative and furthermore it can be readily extended to more general fractional differential equations with different boundary conditions and low order terms

fourth order compact scheme for space fractional advection diffusion reaction equations with variable coefficients abstract in this work a new fourth order compact approximation is derived for riemann liouville space fractional derivatives modified wave numbers are obtained for various approximations of fractional derivatives considering these modified wave numbers fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations a novel fourth order compact scheme is developed for space fractional advection diffusion reaction equations with variable coefficients in one and two dimensions in the proposed compact scheme the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations this splitting of second derivative approximations allows us to obtain a similar system of linear equations as in zhao and tian 2017 which presents only second order accurate finite difference method for advection diffusion reaction equations in one dimension the stability of the proposed compact scheme is demonstrated numerically furthermore the proposed compact scheme is employed to space fractional black scholes equation for pricing european options as an application of fractional derivatives in mathematical finance numerical illustrations are presented to validate the theoretical claims

spectral analysis and multigrid preconditioners for two dimensional space fractional diffusion equations abstract fractional diffusion equations fdes are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance in this paper we focus on a two dimensional space fde problem discretized by means of a second order finite difference scheme obtained as combination of the crank nicolson scheme and the so called weighted and shifted grunwald formula by fully exploiting the toeplitz like structure of the resulting linear system we provide a detailed spectral analysis of the coefficient matrix at each time step both in the case of constant and variable diffusion coefficients such a spectral analysis has a very crucial role since it can be used for designing fast and robust iterative solvers in particular we employ the obtained spectral information to define a galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped jacobi as smoother and to prove the linear convergence rate of the corresponding two grid method the theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the v cycle method and the geometric multigrid on this basis we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for krylov methods several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate

a fast preconditioned iterative method for two dimensional options pricing under fractional differential models abstract in recent years fractional partial differential equation fpde has been widely applied in options pricing problems which better explains many important empirical facts of financial markets however the vast majority of the literatures focus on pricing the single asset option under the fpde framework in this paper a two dimensional fpde governing the valuation of rainbow options is established where two underlying assets are assumed to follow independent exponential levy processes and its boundary conditions are determined by solving one dimensional fpdes a second order accurate finite difference scheme is proposed to discretize the two dimensional fpde given the block toeplitz with toeplitz block structure of the coefficient matrix a fast preconditioned krylov subspace method is developed for solving the resulting linear system with o n log n computational complexity per iteration where n is the matrix size the proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods

linearized adi schemes for two dimensional space fractional nonlinear ginzburg landau equation abstract space and time approximations for two dimensional space fractional complex ginzburg landau equation are examined the schemes under consideration are discreted by the second order backward differential formula bdf2 in time and two classes of the fractional centered finite difference methods in space a linearized technique is employed by the extrapolation we prove the unique solvability and stability for both numerical methods the convergence of both numerical methods is analyzed at length utilizing the energy argument and the convergence orders under the optimal step size ratio are o 2 h 2 and o 2 h 4 in the sense of the discrete l 2 norm where is the time step size h max h x h y and h x h y are spatial grid sizes in the x direction and y direction respectively in addition we construct a multistep alternating direction implicit adi scheme and a multistep compact adi scheme based on bdf2 for the efficiently numerical implementation finally numerical examples are carried out to verify our theoretical results

dissipation preserving galerkin legendre spectral methods for two dimensional fractional nonlinear wave equations abstract in this paper we consider implicit and linearized dissipation preserving galerkin legendre spectral methods to solve space fractional nonlinear damped wave equation in two dimensions the full discrete schemes preserve dissipation of energy in damped case and conservation of energy in undamped case moreover the stability and convergence analysis of the full discrete schemes are rigorously given furthermore we get that two methods are convergent with second order accuracy in time and optimal error estimates in space in order to reduce the computational cost we adopt matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation numerical results are presented to validate the theoretical analysis

finite difference approximation of space fractional diffusion problems a mathematical analysis is presented to establish the convergence of the matrix transformation or matrix transfer method for the finite difference approximation of space fractional diffusion problems combined this with an implicit euler time discretization the optimal order convergence is proved with respect to the discrete l2 and the maximum norm the analysis is performed on general two and three dimensional domains with homogeneous boundary conditions the corresponding error estimates are illustrated with some numerical experiments

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 fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

high order approximation to caputo derivatives and caputo type advection diffusion equations iii in this paper a series of new high order numerical approximations to th 0 1 order caputo derivative is constructed by using r th degree interpolation approximation for the integral function where r 4 is a positive integer as a result the new formulas can be viewed as the extensions of the existing jobs cao et al 2015 li et al 2014 the convergence orders are o r 1 where is the time stepsize two test examples are given to demonstrate the efficiency of these schemes then we adopt the derived schemes to solve the caputo type advection diffusion equation with dirichlet boundary conditions the local truncation error of the derived difference scheme is o r 1 h 2 where is the time stepsize and h the space one the stability and convergence of the proposed schemes for r 4 are also considered without loss of generality we only display the numerical examples for r 4 5 which support the numerical algorithms

a fourth order scheme for space fractional diffusion equations abstract a weighted and shifted difference formula is constructed based on the lubich operators which gives a forth order and unconditionally stable difference scheme for the cauchy problem of space fractional diffusion equations the novelty of the proposed method here is that only four weighted parameters are required compared to eight parameters used in the previous work to achieve the fourth order accuracy and to ensure the stability at the same time to verify the efficiency of the proposed scheme several numerical experiments for both one dimensional and two dimensional fractional diffusion problems are provided

second order accuracy finite difference methods for space fractional partial differential equations abstract space fractional partial differential equations are used for simulations of for example diffusion of radioactive materials and financial and other models which are characterized by heavy tailed distributions a number of first order accuracy finite difference methods have been proposed in the present paper we introduce second order accuracy finite difference methods with dirichlet boundary conditions these methods have a parameter in these schemes and the parameter stabilizes the schemes this means that there exist various schemes with second order accuracy but the stability of each scheme is different in the present paper we introduce the most stable scheme for any fractional calculus order by choosing the optimal parameter in addition we describe a phenomenon whereby the expected accuracy cannot be obtained if the analytical solution can be expanded to a series having less than second order around boundaries this also happens in both existing methods and the proposed methods in the present paper we develop the stability conditions for the proposed schemes and numerical examples of second order accuracy and accuracy decay are shown

the implicit midpoint method for the modified anomalous sub diffusion equation with a nonlinear source term in this paper the implicit midpoint method is used to solve the semi discrete modified anomalous sub diffusion equation with a nonlinear source term and the weighted and shifted grunwald letnikov difference operator and the compact difference operator are applied to approximate the riemann liouville fractional derivative and space partial derivative respectively then the new numerical scheme is constructed the stability and the convergence of this method are analyzed numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results

split step alternating direction implicit difference scheme for the fractional schrodinger equation in two dimensions in this paper we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space fractional schrodinger equation with the riesz fractional derivative the scheme is constructed by combining the split step method for handling the nonlinearity with the alternating direction implicit adi method for resolving the multi dimensions difficulty the riesz space fractional derivative is approximated by the second order accurate fractional centered difference based on matrix analysis we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems the unconditional convergence is proved rigorously in the linear case numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme

fast numerical solution for fractional diffusion equations by exponential quadrature rule after spatial discretization to the fractional diffusion equation by the shifted grunwald formula it leads to a system of ordinary differential equations where the resulting coefficient matrix possesses the toeplitz like structure an exponential quadrature rule is employed to solve such a system of ordinary differential equations the convergence by the proposed method is theoretically studied in practical computation the product of a toeplitz like matrix exponential and a vector is calculated by the shift invert arnoldi method meanwhile the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift invert arnoldi method numerical results are given to demonstrate the efficiency of the proposed method the matrix of the spatial discretization of the fractional diffusion equation is toeplitz like an exponential quadrature rule is employed to solve the system of ordinary differential equations the toeplitz like matrix exponential is calculated by the shift invert arnoldi method the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift invert arnoldi method numerical results show the efficiency of the exponential quadrature rule

a fast second order implicit scheme for non linear time space fractional diffusion equation with time delay and drift term abstract in this paper a second order accurate implicit scheme based on the l2 1 formula for temporal variable and the fractional centered difference formula for spatial discretization is established to solve a class of time space fractional diffusion equations with time drift term and non linear delayed source function the stability of this scheme is proved rigorously by the discrete energy method under several auxiliary assumptions then we theoretically and numerically show that the proposed scheme converges in the l2 norm with the order o t 2 h 2 with time step t and mesh size h moreover it finds that the discreted linear systems are symmetric toeplitz systems in order to solve these systems efficiently the conjugate gradient method with suitable circulant preconditioners is designed in each iterative step the storage requirements and the computational complexity of the resulting equations are o n and o n log n respectively where n is the number of grid nodes numerical experiments are carried out to demonstrate the effectiveness of our proposed circulant preconditioners

finite difference spectral approximations for the distributed order time fractional reaction diffusion equation on an unbounded domain the numerical approximation of the distributed order time fractional reaction diffusion equation on a semi infinite spatial domain is discussed in this paper a fully discrete scheme based on finite difference method in time and spectral approximation using laguerre functions in space is proposed the scheme is unconditionally stable and convergent with order o 2 2 n 1 m 2 where n and m are the time step size step size in distributed order variable polynomial degree and regularity in the space variable of the exact solution respectively a pseudospectral scheme is also proposed and analyzed some numerical examples are presented to demonstrate the efficiency of the proposed scheme

the monte carlo markov chain method for solving the modified anomalous fractional sub diffusion equation abstract in this paper the monte carlo markov chain method for solving the modified anomalous fractional sub diffusion equation is studied most of the previous methods are low in temporal and spatial accuracy order based on the idea of monte carlo markov chain method and compact finite difference schemes a probability model for solving the modified anomalous fractional sub diffusion equation is established numerical examples are given to show the feasibility of the proposed scheme compared with the compact finite difference method the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order and it is also applied to solve partial differential equation in irregular domains

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

high order numerical approximation formulas for riemann liouville riesz tempered fractional derivatives construction and application i abstract in this paper we develop a new numerical algorithm for solving the riesz tempered space fractional diffusion equation the stability and convergence of the numerical scheme are discussed via the technique of matrix analysis finally numerical experiments are performed to confirm the effectiveness of our numerical algorithm

the accuracy and stability of cn wsgd schemes for space fractional diffusion equation abstract in this paper we discuss the accuracy and the stability of the cn wsgd when it is applied to the initial boundary value problem for a space fractional diffusion equation in the cn wsgd scheme the weighted and shifted grunwald difference wsgd scheme is used to discretize the space fractional derivative and the crank nicolson cn scheme is used to discretize the temporal derivative there is a free parameter in a cn wsgd scheme that affects the stability and the accuracy of the scheme in general a cn wsgd scheme has temporally and spatially second order accuracy and there exists a scheme that has spatially third order accuracy however the third order accuracy scheme is not always stable in this paper we present and prove some theoretical results old and new on the stability of several cn wsgd schemes and consider choosing a value for the parameter such that the corresponding cn wsgd scheme is unconditioned stable and has optimal upper bound for the accuracy numerical results are presented to verify the theoretical results

efficient numerical schemes for fractional water wave models in this paper efficient numerical schemes are proposed for solving the fractional water wave models that describe the propagation of surface water wave by using the weighted and shifted grunwald letnikov wsgl formula to approximate the nonlocal fractional operators we design a series of second order accurate difference schemes for the considered models the existence stability and convergence of numerical solutions of the proposed numerical schemes are established rigorously the analysis shows that the proposed numerical schemes are unconditionally stable with second order accuracy for both temporal and spatial discretizations several numerical results are provided to verify the efficiency and accuracy of our theoretical analysis

some high order difference schemes for the distributed order differential equations two difference schemes are derived for both one dimensional and two dimensional distributed order differential equations it is proved that the schemes are unconditionally stable and convergent in an l 1 l norm with the convergence orders o 2 h 2 2 and o 2 h 4 4 respectively where h and are the step sizes in time space and distributed order variables several numerical examples are given to confirm the theoretical results

positivity and boundedness preserving schemes for space time fractional predator prey reaction diffusion model the semi implicit schemes for the nonlinear predator prey reaction diffusion model with the space time fractional derivatives are discussed where the space fractional derivative is discretized by the fractional centered difference and wsgd scheme the stability and convergence of the semi implicit schemes are analyzed in the l norm we theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model and show that the numerical solutions preserve the positivity and boundedness the presented numerical examples confirm the theoretical results and convergence orders

alternating direction implicit spectral element method adi sem for solving multi dimensional generalized modified anomalous sub diffusion equation abstract the main aim of the current paper is to solve the multi dimensional generalized modified anomalous sub diffusion equation by using a new spectral element method at first the time variable has been discretized by a finite difference scheme with second order accuracy the stability and convergence of the time discrete scheme have been investigated we show that the time discrete scheme is unconditionally stable and the convergence order is o 2 in the temporal direction secondly the galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable the unconditional stability and convergence of the full discrete scheme have been proved by developing the proposed scheme we need to calculate one dimensional integrals for two dimensional problems and two dimensional integrals for three dimensional problems thus the used cpu time for the presented numerical procedure is lower than the two and three dimensional galerkin spectral element methods also the proposed method is suitable for computational domains obtained from the tensor product finally two examples are analyzed to check the theoretical results

a multigrid method for linear systems arising from time dependent two dimensional space fractional diffusion equations abstract in this paper we study a v cycle multigrid method for linear systems arising from time dependent two dimensional space fractional diffusion equations the coefficient matrices of the linear systems are structured such that their matrix vector multiplications can be computed efficiently the main advantage using the multigrid method is to handle the space fractional diffusion equations on non rectangular domains and to solve the linear systems with non constant coefficients more effectively the main idea of the proposed multigrid method is to employ two banded splitting iteration schemes as pre smoother and post smoother the pre smoother and the post smoother take banded splitting of the coefficient matrix under the x dominant ordering and the y dominant ordering respectively we prove the convergence of the proposed two banded splitting iteration schemes in the sense of infinity norm results of numerical experiments for time dependent two dimensional space fractional diffusion equations on rectangular l shape and u shape domains are reported to demonstrate that both computational time and iteration number required by the proposed method are significantly smaller than those of the other tested methods