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

spectral methods for the time fractional diffusion wave equation in a semi infinite channel in this paper we consider the numerical approximation of the time fractional diffusion wave equation in a semi infinite channel the time fractional derivative is described in caputo sense with order 1 2 a fully discrete spectral scheme based on a finite difference method in the time direction and a laguerre legendre spectral method in the space direction is proposed we also propose an alternating direction implicit adi spectral scheme in order to reduce the amount of computation the stability and convergence of both schemes are rigorously established numerical results are presented to support our theoretical analysis

finite difference finite element method for two dimensional space and time fractional bloch torrey equations in this paper a class of two dimensional space and time fractional bloch torrey equations 2d stfbtes are considered some definitions and properties of fractional derivative spaces are presented by finite difference method and galerkin finite element method a semi discrete variational formulation for 2d stfbtes is obtained the stability and convergence of the semi discrete form are discussed then a fully discrete scheme of 2d stfbtes is derived and the convergence is investigated finally some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis

donaldson thomas invariants of 2 dimensional sheaves inside threefolds and modular forms abstract motivated by the s duality conjecture we study the donaldson thomas invariants of the 2 dimensional gieseker stable sheaves on a threefold these sheaves are supported on the fibers of a nonsingular threefold x fibered over a nonsingular curve in the case where x is a k3 fibration we express these invariants in terms of the euler characteristic of the hilbert scheme of points on the k3 fiber and the noether lefschetz numbers of the fibration we prove that a certain generating function of these invariants is a vector modular form of weight 3 2 as predicted in s duality

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

parameters estimation for a new anomalous thermal diffusion model in layered media in this paper we study an inverse problem of parameters estimation for a new time fractional heat conduction model in multilayered medium in the anomalous thermal diffusion model we consider the fractional derivative boundary conditions and the conduction obeys modified fourier law with riemannliouville fractional operator of different order in each layer for the direct problem we construct an effective finite difference scheme by using the balance method to deal with the discontinuity interface for the inverse problem we apply the nonlinear conjugate gradient ncg method with different conjugated coefficients to simultaneously identify the fractional exponent in each layer finally we use experimental data to verify the effectiveness of the proposed technique in which the jacobian matrix is achieved by a derivative free approach we analyze the sensitivity coefficients and the convergence behaviors of the ncg algorithm the simulation results confirm that the fractional heat conduction model with estimated parameters gives a more accurate fitting than the classical counterpart and the ncg method is a feasible and effective technique for the inverse problem of parameters estimation in fractional model

crank nicolson adi galerkin finite element method for two dimensional fractional fitzhugh nagumo monodomain model in this paper a two dimensional fractional fitzhugh nagumo monodomain model 2d ffhnmm with zero dirichlet boundary condition is considered the model consists of a coupled two dimensional space fractional nonlinear reaction diffusion model 2d sfnrdm and an ordinary differential equation the 2d sfnrdm and ordinary differential equation are decoupled at each time step a new crank nicolson alternating direction implicit adi galerkin finite element method for the 2d sfnrdm is developed the stability and convergence of the numerical method are discussed finally some numerical examples on 2d sfnrdm and 2d ffhnmm are given for verification of our theoretical analysis

spectral iterative method and convergence analysis for solving nonlinear fractional differential equation abstract in this study a new spectral iterative method sim based on fractional interpolation is presented for solving nonlinear fractional differential equations fdes involving caputo derivative this method is equipped with a pre algorithm to find the singularity index of solution of the problem this pre algorithm gives us a real parameter as the index of the fractional interpolation basis for which the sim achieves the highest order of convergence in comparison with some recent results about the error estimates for fractional approximations a more accurate convergence rate has been attained we have also proposed the order of convergence for fractional interpolation error under the l 2 norm finally general error analysis of sim has been considered the numerical results clearly demonstrate the capability of the proposed method

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

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

stability of solutions of a class of nonlinear fractional laplacian parabolic problems abstract in this paper we study the forward and the backward in time problems for a class of nonlinear partial differential equations of fractional laplace operator with nonlinear source function the equations are generalized from lots of well known equations such as the burger equation the cahn hilliard equation herein we investigate the stability of the solution of the forward problem depended on the fractional order of the operator and the instability of the backward problem and then we establish a regularization method some numerical experiments are performed to verify efficiency and accuracy of our method

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local 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 with 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 in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

time dependent fractional advection diffusion equations by an implicit mls meshless method recently many new applications in engineering and science are governed by a series of fractional partial differential equations fpdes unlike the normal partial differential equations pdes the differential order in a fpde is with a fractional order which will lead to new challenges for numerical simulation because most existing numerical simulation techniques are developed for the pde with an integer differential order the current dominant numerical method for fpdes is finite difference method fdm which is usually difficult to handle a complex problem domain and also hard to use irregular nodal distribution this paper aims to develop an implicit meshless approach based on the moving least squares mls approximation for numerical simulation of fractional advection diffusion equations fade which is a typical fpde the discrete system of equations is obtained by using the mls meshless shape functions and the meshless strong forms the stability and convergence related to the time discretization of this approach are then discussed and theoretically proven several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation it is concluded that the present meshless formulation is very effective for the modeling and simulation of the fade

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

a novel compact adi scheme for two dimensional riesz space fractional nonlinear reaction diffusion equations abstract this paper is concerned with the construction and analysis of a novel linearized compact adi scheme for the two dimensional riesz space fractional nonlinear reaction diffusion equations convergence of the proposed scheme is proved the highlight is that the time discretization is achieved by applying a second order one step and linearized method the time discretization requires only one starting value which is sharp contrast to the extrapolated crank nicolson method or the usual second order linearized schemes numerical examples on several fractional models are presented to confirm our theoretical results

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

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

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

new explicit and accelerated techniques for solving fractional order differential equations abstract this paper proposes new direct and acceleration numerical methods for solving fractional order differential equations fdes for the caputo differential operator with fractional order 0 o h 2 and o h 3 respectively in order to overcome a strong singular issue as 0 we propose the explicit predictor corrector scheme with perturbation technique in case of 1 the truncation errors are reduced by o h and o h 2 in order to accelerate the convergence rate we decompose 2 2 convert the fde into the system of fdes and apply the predictor corrector scheme numerical tests for linear nonlinear variable order and time fractional sub diffusion problems demonstrate that the proposed methods give a prominent performance we also compare the numerical results with other high order explicit and implicit schema

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

variational formulation of time fractional parabolic equations abstract we consider initial boundary value problems for parabolic pde t u u f with fractional caputo derivative t of order 1 2 1 as time derivative and the usual laplacian as space derivative also called fractional diffusion equations in the literature we prove well posedness of corresponding variational formulations based entirely on fractional sobolev bochner spaces and extend existing results for possible choices of the initial value for u at t 0

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

global and exploding solutions for nonlocal quadratic evolution problems nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered the inertial type nonlinearity is quadratic and nonlocal while the diffusive term also nonlocal is anomalous and fractal i e represented by a fractional power of the laplacian conditions for global in time existence versus finite time blow up are studied self similar solutions are constructed for certain homogeneous initial data

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

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