implicit finite difference approximation for time fractional diffusion equations time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical brownian motion model in this paper we develop an implicit unconditionally stable numerical method to solve the one dimensional linear time fractional diffusion equation formulated with caputo s fractional derivative on a finite slab several numerical examples of interest are also included

point wise error estimate of a conservative difference scheme for the fractional schrodinger equation in this paper a new conservative difference scheme is proposed for solving the nonlinear space fractional schrodinger equation the riesz space fractional derivative is approximated by the second order accurate weighted and shifted grunwald difference operator building on a careful analysis of this operator the conservation properties including the mass and energy are analyzed and some norm equivalences are established then the numerical solution is shown to be bounded in various norms uniformly with respect to the discretization parameters and optimal order bounds on the global error of the scheme in the l 2 norm semi h 2 norm and l norm are derived the uniform bounds on the numerical solutions as well as the error bounds hold unconditionally in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed numerical tests are performed to support our theoretical results

self similar transport in incomplete chaos particle chaotic dynamics along a stochastic web is studied for three dimensional hamiltonian flow with hexagonal symmetry in a plane two different classes of dynamical motion obtained by different values of a control parameter and corresponding to normal and anomalous diffusion have been considered and compared it is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement flights which are similar to l evy flights approximate trappings of orbits near the boundary layer of islands and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment the main result is related to the self similar properties of different topological and dynamical characteristics of the particle motion this self similarity appears in the weierstrass like random walk process that is responsible for the anomalous transport exponent in the mean moment dependence on t this exponent can be expressed as a ratio of fractal dimensions of space and time sets in the weierstrass like process an explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given

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 fast finite difference method for distributed order space fractional partial differential equations on convex domains abstract fractional partial differential equations pdes provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long range spatial interactions which cannot be modeled accurately by classical second order diffusion equations however numerical methods for space fractional pdes usually generate dense or full stiffness matrices for which a direct solver requires o n 3 computations per time step and o n 2 memory where n is the number of unknowns the significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three dimensional space fractional diffusion equations computationally intractable fast numerical methods were previously developed for space fractional pdes on multidimensional rectangular domains without resorting to lossy compression but rather via the exploration of the tensor product form of the toeplitz like decompositions of the stiffness matrices in this paper we develop a fast finite difference method for distributed order space fractional pdes on a general convex domain in multiple space dimensions the fast method has an optimal order storage requirement and almost linear computational complexity without any lossy compression numerical experiments show the utility of the method

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

a divide and conquer fast finite difference method for spacetime fractional partial differential equation fractional partial differential equations fpdes provide better modeling capabilities for challenging phenomena with long range time memory and spatial interaction than integer order pdes do a conventional numerical discretization of spacetime fpdes requires o n2 mn memory and o mn3 m2n computational work where n is the number of spatial freedoms per time step and m is the number of time steps we develop a fast finite difference method fdm for spacetime fpde i we utilize the toeplitz like structure of the coefficient matrix to develop a matrix free preconditioned fast krylov subspace iterative solver to invert the coefficient matrix at each time step ii we utilize a divide and conquer strategy a recursive direct solver to handle the temporal coupling of the numerical scheme the fast method has an optimal memory requirement of o mn and an approximately linear computational complexity of o nm logn log2m without resorting to any lossy compression numerical experiments show the utility of the method

balancing torques in membrane mediated interactions exact results and numerical illustrations torques on interfaces can be described by a divergence free tensor which is fully encoded in the geometry this tensor consists of two terms one originating in the couple of the stress the other capturing an intrinsic contribution due to curvature in analogy to the description of forces in terms of a stress tensor the torque on a particle can be expressed as a line integral along a contour surrounding the particle interactions between particles mediated by a fluid membrane are studied within this framework in particular torque balance places a strong constraint on the shape of the membrane symmetric two particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime in particular the problem may be solved exactly in the case of two membrane bound parallel cylinders this apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane mediated interactions

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

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

finite element simulation and efficient algorithm for fractional cahn hilliard equation abstract this article is devoted to designing efficient linear finite element algorithm for the fractional cahn hilliard equation an important and newly proposed phase field model combining the advantages of the classic bicg algorithm and the toeplitz like structure of the coefficient matrix we develop a fast bicg fbicg algorithm for the linearized scheme to compute numerically the fractional cahn hilliard equation our theoretical analysis and numerical experiments demonstrate that the proposed fbicg reduces the computation cost and the storage to o m log m and o m possesses the same convergence rates as newton s algorithm does in space and time and preserves the energy dissipation and equality laws the numerical experiments also demonstrate that the fbicg is almost mass conserved recognizes accurately the phase separation by a very clear coarse graining process and the influences of different indices r and s of fractional laplacian and different coefficients k and a on the width of the interfaces

numerical solutions for fractional reaction diffusion equations fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation in a fractional diffusion equation the second derivative in the spatial variable is replaced by a fractional derivative of order less than two the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used fractional reaction diffusion equations combine the fractional diffusion with a classical reaction term in this paper we develop a practical method for numerical solution of fractional reaction diffusion equations based on operator splitting then we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions we also discuss applications to biology where the reaction term models species growth and the diffusion term accounts for movements

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

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

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

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

multigrid methods for space fractional partial differential equations we propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations sfpdes it is shown that our multigrid methods are optimal which means the convergence rates of the methods are independent of the mesh size and mesh level moreover our theoretical analysis and convergence results do not require regularity assumptions of the model problems numerical results are given to support our theoretical findings

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

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

on partially inexact hss iteration methods for the complex symmetric linear systems in space fractional cnls equations the space fractional coupled nonlinear schrdinger cnls equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal plus toeplitz matrix the hermitian and skew hermitian splitting hss method and the partially inexact hss pihss method are employed to solve the discretized linear system in the inner iteration processes of the hss method we only need to solve the linear sub systems associated with the hermitian part inexactly by the conjugate gradient cg method resulting in pihss iteration method theoretical analyses show that both hss and pihss methods are unconditionally convergent numerical examples are given to demonstrate the effectiveness of the hss iteration and the pihss iteration

a note on spectral norms of even order r circulant matrices abstract in this paper we preset some improved techniques for proofs within zhou and jiang 2014 meanwhile focusing on n and r are co prime we generalize those results to investigate formulae of even order r circulant matrices with alternately positive and negative entrywise and we deduce explicit formulae for their spectral norms with the entries in the first row

positive steady states for a prey predator model with some nonlinear diffusion terms abstract this paper discusses a prey predator system with strongly coupled nonlinear diffusion terms we give a sufficient condition for the existence of positive steady state solutions our proof is based on the bifurcation theory some a priori estimates for steady state solutions will play an important role in the proof

a preconditioned numerical solver for stiff nonlinear reaction diffusion equations with fractional laplacians that avoids dense matrices the numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the spatial nonlocality of the fractional differential operators the dense coefficient matrices that arise from spatial discretisation of these operators mean that even one dimensional problems can be difficult to solve using standard methods on grids comprising thousands of nodes or more in this work we address this issue of efficiency for one dimensional nonlinear space fractional reaction diffusion equations with fractional laplacian operators we apply variable order variable stepsize backward differentiation formulas in a jacobian free newton krylov framework to advance the solution in time a key advantage of this approach is the elimination of any requirement to form the dense matrix representation of the fractional laplacian operator we show how a banded approximation to this matrix which can be formed and factorised efficiently can be used as part of an effective preconditioner that accelerates convergence of the krylov subspace iterative solver our approach also captures the full contribution from the nonlinear reaction term in the preconditioner which is crucial for problems that exhibit stiff reactions numerical examples are presented to illustrate the overall effectiveness of the solver

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

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

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

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

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

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

on adi like iteration method for fractional diffusion equations abstract the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of the identity matrix with two diagonal times toeplitz matrices in this paper the alternating direction implicit adi like method based on the classical adi iteration is proposed to solve the discretized linear system theoretical analyses show that the adi like iteration method is convergent each iteration of this method requires the solutions of two linear systems with hessenberg coefficient matrices these two systems can be solved effectively by hessenberg lu factorization numerical examples are presented to illustrate the effectiveness of the adi like iteration method

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

a kronecker product splitting preconditioner for two dimensional space fractional diffusion equations abstract we study preconditioned iterative methods for the linear system arising in the numerical discretization of a two dimensional space fractional diffusion equation our approach is based on a formulation of the discrete problem that is shown to be the sum of two kronecker products by making use of an alternating kronecker product splitting iteration technique we establish a class of fixed point iteration methods theoretical analysis shows that the new method converges to the unique solution of the linear system moreover the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real the basic iteration is accelerated by a krylov subspace method like gmres the corresponding preconditioner is in a form of a kronecker product structure and requires at each iteration the solution of a set of discrete one dimensional fractional diffusion equations we use structure preserving approximations to the discrete one dimensional fractional diffusion operators in the action of the preconditioning matrix numerical examples are presented to illustrate the effectiveness of this approach

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

a fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrodinger equations abstract in this paper a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional schrodinger equations we prove that the scheme preserves both the mass and energy which are defined by virtue of some recursion relationships using the sobolev inequalities and then employing the mathematical induction the discrete scheme is proved to be unconditionally convergent in the sense of l 2 norm and h 2 norm which means that there are no any constraints on the grid ratios then the prior bound of the discrete solution in l 2 norm and l norm are also obtained moreover we propose an iterative algorithm by which the coefficient matrix is independent of the time level and thus it leads to toeplitz like linear systems that can be efficiently solved by krylov subspace solvers with circulant preconditioners this method can reduce the memory requirement of the proposed linearized finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m log m in each iterative step where m is the number of grid nodes finally numerical results are carried out to verify the correction of the theoretical analysis simulate the collision of two solitary waves and show the utility of the fast numerical solution techniques

fast adi method for high dimensional fractional diffusion equations in conservative form with preconditioned strategy in this paper high dimensional two sided space fractional diffusion equations derived from the fractional ficks law and with monotonic variable diffusion coefficients are solved by alternating direction implicit method each linear system corresponding to each spatial direction thus resulted is solved by krylov subspace method the method is accelerated by applying an approximate inverse preconditioner where under certain conditions we showed that the normalized preconditioned matrix is equal to a sum of identity matrix a matrix with small norm and a matrix with low rank such that the preconditioned krylov subspace method converges superlinearly we also briefly present some fast algorithms whose computational cost for solving the linear systems is o nlogn where n is the matrix size the results are illustrated by some numerical examples

fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations abstract the discretizations of two and three dimensional spatial fractional diffusion equations with the shifted finite difference formulas of the grunwald letnikov type can result in discrete linear systems whose coefficient matrices are of the form d t where d is a nonnegative diagonal matrix and t is a block toeplitz with toeplitz block matrix or a block toeplitz with each block being block toeplitz with toeplitz block matrix for these discrete spatial fractional diffusion matrices we construct diagonal and block circulant with circulant block splitting preconditioner for the two dimensional case and diagonal and block circulant with each block being block circulant with circulant block splitting preconditioner for the three dimensional case to further accelerate the convergence rates of krylov subspace iteration methods and we analyze the eigenvalue distributions for the corresponding preconditioned matrices theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1 and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the krylov subspace iteration methods moreover this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant plus diagonal preconditioners in both iteration counts and computing times

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

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

efficient spectral galerkin methods for fractional partial differential equations with variable coefficients efficient spectral galerkin algorithms are developed to solve multi dimensional fractional elliptic equations with variable coefficients in conserved form as well as non conserved form these algorithms are extensions of the spectral galerkin algorithms for usual elliptic pdes developed in 24 more precisely for separable fpdes we construct a direct method by using a matrix diagonalization approach while for non separable fpdes we employ a preconditioned bicgstab method with a suitable separable fpde with constant coefficients as preconditioner the cost of these algorithms is of o n d 1 flops where d is the space dimension we derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries we also present ample numerical results to validate the algorithms and error estimates

a circulant preconditioner for fractional diffusion equations the implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable is employed to discretize fractional diffusion equations the resulting systems are toeplitz like and then the fast fourier transform can be used to reduce the computational cost of the matrix vector multiplication the preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant hence the convergence rate of the proposed iterative algorithm is superlinear numerical experiments are carried out to demonstrate that our circulant preconditioner works very well even though for cases of variable diffusion coefficients

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

fast parareal iterations for fractional diffusion equations abstract numerical methods for fractional pdes is a hot topic recently this work is concerned with the parareal algorithm for system of odes u t a u t f that arising from semi discretizations of time dependent fractional diffusion equations with nonsymmetric riemann liouville fractional derivatives the spatial semi discretization of this kind of fractional derivatives often results in a coefficient matrix a with spectrum a satisfying a s c r where 0 is a measure of dissipativity of the differential equations to accelerate the parareal algorithm we propose a scaled model u t 1 a u t f with 0 to serve the coarse grid correction which is an important component of our parareal algorithm given and we derive a sharp bound of the convergence factor of the parareal iterations moreover by minimizing such a bound we get optimized scaling factor opt it is shown that compared to 1 i e the classical implementation pattern of the coarse grid correction the optimized scaling factor significantly improves the convergence rate numerical examples are presented to support the theoretical finding

domain decomposition methods for space fractional partial differential equations abstract a two level additive schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations sfpdes it is shown that the condition number of the preconditioned system is bounded by c 1 h where h is the maximum diameter of subdomains and is the overlap size among the subdomains numerical results are given to support our theoretical findings

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 generalization of trigonometric transform splitting methods for spatial fractional diffusion equations abstract the spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme with the shifted grunwald formula and the given discretized linear systems have a diagonal plus toeplitz structure in this paper based on trigonometric transformation splitting tts we study efficient iterative method called gtts has two parameters as a focus we give the simple and effective optimal forms of these two parameters respectively we carry out some numerical experiments to illustrate the effectiveness and accuracy of this new algorithm

preconditioned modified hermitian and skew hermitian splitting iteration methods for fractional nonlinear schrodinger equations abstract a variant of preconditioned modified hermitian and skew hermitian splitting pmhss iteration method is proposed for a class of toeplitz like complex linear equations arising from the space fractional coupled nonlinear schrodinger equations theoretical analysis show that the new method is convergent unconditionally moreover the choice of the theoretical optimal parameter is further studied in detail which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the toeplitz matrix and the extreme absolute values of the diagonal matrix in addition the splitting can be used as a preconditioner for krylov subspaces method numerical experiments show that the proposed method is efficient and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem

a fast preconditioned policy iteration method for solving the tempered fractional hjb equation governing american options valuation a fast preconditioned policy iteration method is proposed for the hamiltonjacobibellman hjb equation involving tempered fractional order partial derivatives governing the valuation of american options whose underlying asset follows exponential lvy processes an unconditionally stable upwind finite difference scheme with shifted grnwald approximation is first developed to discretize the established hjb equation under the tempered fractional diffusion models next the policy iteration method as an outer iterative method is utilized to solve the discretized hjb equation and proven to be convergent in finite steps to its numerical solution given the toeplitz like structure of the coefficient matrix in each policy iteration the resulting linear system can be fast solved by the krylov subspace method as an inner iterative method via fast fourier transform fft furthermore a novel preconditioner is proposed to speed up the convergence rate of the inner krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in o nlogn operations under some mild conditions where n is the number of spatial node points numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method