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

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

weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients a class of weighted finite difference methods wfdms for solving a class of initial boundary value problems of space fractional partial differential equations with variable coefficients is presented their stability and convergence properties are considered it is proven that the wfdms are unconditionally stable for email protected email protected 12 and conditionally stable for 12

a difference scheme for a nonlinear partial integrodifferential equation a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in proving consistency the verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature a stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper

the fractional laplacian operator on bounded domains as a special case of the nonlocal diffusion operator we analyze a nonlocal diffusion operator having as special cases the fractional laplacian and fractional differential operators that arise in several applications in our analysis a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem we demonstrate that when sufficient conditions on certain kernel functions hold the solution of the nonlocal equation converges to the solution of the fractional laplacian equation on bounded domains as the nonlocal interactions become infinite we also introduce a continuous galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite volume constraints

a computational method for solving variable order fractional nonlinear diffusion wave equation abstract in this paper we generalize a one dimensional fractional diffusion wave equation to a one dimensional variable order space time fractional nonlinear diffusion wave equation v os tfnd we where the variable order fractional derivatives are considered in the caputo type to solve this introduced equation an easy to follow method is proposed which is based on the chebyshev cardinal functions coupling with the tau and collocation methods to carry out the method an operational matrix of variable order fractional derivative omv ofd is derived for the chebyshev cardinal functions to be employed for expanding the unknown function the proposed method can provide highly accurate approximate solutions by reducing the problem under study to a system of nonlinear algebraic equations which is technically simpler for handling the experimental results confirm the applicability and effectiveness of the method

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

a new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications in the present work first a new fractional numerical differentiation formula called the l1 2 formula to approximate the caputo fractional derivative of order 0 1 is developed it is established by means of the quadratic interpolation approximation using three points t j 2 f t j 2 t j 1 f t j 1 and t j f t j for the integrand f t on each small interval t j 1 t j j 2 while the linear interpolation approximation is applied on the first small interval t 0 t 1 as a result the new formula can be formally viewed as a modification of the classical l1 formula which is obtained by the piecewise linear approximation for f t both the computational efficiency and numerical accuracy of the new formula are superior to that of the l1 formula the coefficients and truncation errors of this formula are discussed in detail two test examples show the numerical accuracy of l1 2 formula second by the new formula two improved finite difference schemes with high order accuracy in time for solving the time fractional sub diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed respectively in addition the application of the new formula into solving fractional ordinary differential equations is also presented several numerical examples are computed the comparison with the corresponding results of finite difference methods by the l1 formula demonstrates that the new l1 2 formula is much more effective and more accurate than the l1 formula when solving time fractional differential equations numerically

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

numerical methods for solving a two dimensional variable order anomalous subdiffusion equation australian research council dp0559807 dp0986766 national natural science foundation of china 10271098 natural science foundation of fujian province 2009j01014

analysis of the spectral meshless radial point interpolation for solving fractional reaction subdiffusion equation abstract the present paper is devoted to the development of spectral meshless radial point interpolation smrpi technique for solving fractional reaction subdiffusion equation in one and two dimensional cases the time fractional derivative is described in the riemann liouville sense the applied approach is based on a combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of smrpi it is proved that the scheme is unconditionally stable with respect to the time variable in h 1 and we show convergence order of the time discrete scheme is o t 0 1 in the current work the thin plate splines tps are used to construct the basis functions the results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

fractional wave equation with a frictional memory kernel of mittag leffler type abstract in this paper we give an analytical treatment of a fractional wave equation with caputo time fractional derivative and frictional memory kernel of mittag leffler type this problem generalizes a recently solved problem 62 of a wave equation for a vibrating string in presence of a fractional friction with power law memory kernel such equations can be used in the context of modeling processes in complex and viscoelastic media

sylvester equations and the numerical solution of partial fractional differential equations we develop a new matrix based approach to the numerical solution of partial differential equations pde and apply it to the numerical solution of partial fractional differential equations pfde the proposed method is to discretize a given pfde as a sylvester equation and parameterize the integral surface using matrix algebra the combination of these two notions results in an algorithm which can solve a general class of pfde efficiently and accurately by means of an o n 3 algorithm for solving the sylvester matrix equation over an m i n grid with m n the proposed parametrization of the integral surface allows for the solution with the more general robin boundary conditions and allows for high order approximations to derivative boundary conditions to achieve our ends we also develop a new matrix based approximation to fractional order derivatives the proposed method is demonstrated by the numerical solution of the fractional diffusion equation with fractional derivatives in both the temporal and spatial directions

compact finite difference scheme and rbf meshless approach for solving 2d rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives abstract this article is devoted to the comparison between a high order difference scheme and radial basis functions rbfs meshless approach for the numerical solution of two dimensional fractional rayleigh stokes problem the fractional derivative of problem is described in the riemann liouville sense in the high order difference scheme we discretize the spaces derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme also in the rbf meshless method we discretize the time fractional derivatives of the mentioned equation by integrating both sides of it then we will use the kansa s approach to approximate the spatial derivatives we prove the stability and convergence for high order difference scheme using fourier analysis and for time discrete scheme in rbf method via energy approach we compare the results of compact finite difference method and rbf meshless approach in terms of accuracy and cpu time also we compare the numerical results of compact difference scheme with other methods in the literature to show the high accuracy and efficiency of the proposed method

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

compact difference schemes for the modified anomalous fractional sub diffusion equation and the fractional diffusion wave equation in this paper compact finite difference schemes for the modified anomalous fractional sub diffusion equation and fractional diffusion wave equation are studied schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two based on the idea of weighted and shifted grunwald difference operator we establish schemes with temporal and spatial accuracy order equal to two and four respectively

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

a computational wavelet method for variable order fractional model of dual phase lag bioheat equation abstract in this study we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue several models with different levels of simplifications have been proposed one of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time is the 2d transient dual phase lag dpl variable order fractional energy equation for finding the numerical solution of this general case we propose an efficient semi discrete method based on the two dimensional legendre wavelets 2d lws precisely the variable order fractional derivatives of the model are discretized in the first stage and then the response of the model is expanded by the 2d lws consequently the main problem is transformed into an equivalent system of algebraic equations which can be simply tackled the stability of the proposed method is examined theoretically and experimentally also the procedure is described for one example to examine the computational efficiency of method the experimental results show the stability and spectral accuracy of the proposed method according to the achieved results increasing the fractional order from 0 1 to 1 0 leads to increment of maximum tissue temperatures by about 29 near the center of the targeted region

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

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

element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi type equation with robin boundary condition the traditional element free galerkin efg approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares mls approximation in the current paper we propose a new version of the efg method based on the shape functions of reproducing kernel particle method rkpm in other words based on the developed approach in han and meng 2001 the fractional tricomi type equation will be solved using the new technique the fractional derivative has been introduced in the caputos sense and is approximated by a finite difference plan of order o 3 1 2 we use the efg rkpm to discrete the spatial direction we illustrate some numerical results on non rectangular domains the unconditional stability and convergence of the new technique have been proved numerical examples display the theoretical results and the efficiency of the proposed approach also the numerical results are compared with the finite element method fem and efg mls procedure

asymptotic green s functions for time fractional diffusion equation and their application for anomalous diffusion problem abstract asymptotic green s functions for short and long times for time fractional diffusion equation derived by simple and heuristic method are provided in case if fractional derivative is presented in caputo sense the applicability of the asymptotic green s functions for solving the anomalous diffusion problem on a semi infinite rod is demonstrated the initial value problem for longtime solution of the time fractional diffusion equation by green s function approach is resolved

levy flights in a steep potential well levy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial monomodal to a terminal bimodal distribution occurs chechkin et al phys rev e 67 010102 r 2003 in this paper we present a detailed study of levy flights in potentials of the type u x x c with c 2 apart from the bifurcation into bimodality we find the interesting result that for c 4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial condition and the building up of the two emerging side peaks which are characteristic for the stationary state thus for certain system parameters there exists a transient trimodal distribution of the levy flight these properties of levy flights in external potentials of the power law type can be represented by certain phase diagrams we also present details about the proof of multimodality and the numerical procedures to establish the probability distribution of the process

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

a fourth order compact solution of the two dimensional modified anomalous fractional sub diffusion equation with a nonlinear source term this work is concerned to the study of high order difference scheme for the solution of a two dimensional modified anomalous sub diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses the space fractional derivatives are described in the riemann liouville sense in the proposed scheme we discretize the space derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme we prove the stability and convergence of proposed scheme using the fourier analysis the convergence order of the proposed method is o t h x 4 h y 4 comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme

a fourth order extrapolated compact difference method for time fractional convection reaction diffusion equations with spatially variable coefficients abstract this paper is concerned with numerical methods for a class of time fractional convection reaction diffusion equations the convection and reaction coefficients of the equation may be spatially variable based on the weighted and shifted grunwald letnikov formula for the time fractional derivative and a compact finite difference approximation for the spatial derivative we establish an unconditionally stable compact difference method the local truncation error and the solvability of the resulting scheme are discussed in detail the stability of the method and its convergence of third order in time and fourth order in space are rigorously proved by the discrete energy method combining this method with a richardson extrapolation we present an extrapolated compact difference method which is fourth order accurate in both time and space a rigorous proof for the convergence of the extrapolation method is given numerical results confirm our theoretical analysis and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method

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

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

error estimate for the numerical solution of fractional reaction subdiffusion process based on a meshless method in this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction subdiffusion equation firstly we obtain a time discrete scheme based on a finite difference scheme then we use the meshless galerkin method to approximate the spatial derivatives and obtain a full discrete scheme in the proposed scheme some integrals appear over the boundary and the domain of problem which will be approximated using gauss legendre quadrature rule then we prove that the time discrete scheme is unconditionally stable and convergent using the energy method we show convergence order of the time discrete scheme is o the aim of this paper is to obtain an error estimate and to show convergence for the meshless galerkin method based on the radial basis functions numerical examples confirm the efficiency and accuracy of the proposed scheme

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

application of the operational matrix of fractional order legendre functions for solving the time fractional convection diffusion equation the operational matrix of fractional order legendre functions method are considered the fractional order convection diffusion problem is solved the problem can be used extensively in science and engineering as in oil reservoir simulations problems which an initially discontinuous profile is propagated by diffusion and convection in this paper the application of the operational matrix of fractional order legendre functions flfs to solve the time fractional convection diffusion equation has been investigated fractional calculus has been applied to model the engineering and physical processes which are best described with other mathematical tools the time variable of the time fractional convection diffusion equation and its space variable have been approximated by flfs and shifted legendre polynomials respectively the fractional derivatives together with product matrices of flfs are employed to convert the solution of this problem to the solution of a system of algebraic equations

numerical simulation for the variable order galilei invariant advection diffusion equation with a nonlinear source term in this paper we consider the variable order galilei advection diffusion equation with a nonlinear source term a numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation the stability and convergence of the numerical scheme are analyzed besides another numerical scheme for improving temporal accuracy is also developed finally some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

orthogonal spline collocation methods for the subdiffusion equation we develop two kinds of numerical schemes to efficiently solve the subdiffusion equation which is used to describe anomalous subdiffusive transport processes the time fractional derivative is first discretized by l1 approximation and the grunwald letnikov approximation respectively then we use the orthogonal spline collocation method to approximate the two semi discretized subdiffusion equations the stability and convergence of time semi discretization and full discretization schemes are both established strictly for the two schemes both of them are unconditionally stable numerically the convergent orders in space including the solution and its first derivative are four for the hermite cubic spline approximation and theoretically we get that at least the solution itself has a fourth order convergent rate extensive numerical results are presented to show the convergent order and robustness of the numerical schemes

a discrete time random walk model for anomalous diffusion the continuous time random walk introduced in the physics literature by montroll and weiss has been widely used to model anomalous diffusion in external force fields one of the features of this model is that the governing equations for the evolution of the probability density function in the diffusion limit can generally be simplified using fractional calculus this has in turn led to intensive research efforts over the past decade to develop robust numerical methods for the governing equations represented as fractional partial differential equations here we introduce a discrete time random walk that can also be used to model anomalous diffusion in an external force field the governing evolution equations for the probability density function share the continuous time random walk diffusion limit thus the discrete time random walk provides a novel numerical method for solving anomalous diffusion equations in the diffusion limit including the fractional fokker planck equation this method has the clear advantage that the discretisation of the diffusion limit equation which is necessary for numerical analysis is itself a well defined physical process some examples using the discrete time random walk to provide numerical solutions of the probability density function for anomalous subdiffusion including forcing are provided

l1 fourier spectral methods for a class of generalized two dimensional time fractional nonlinear anomalous diffusion equations abstract in this paper l1 fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two dimensional time fractional nonlinear anomalous diffusion equations involving caputo fractional derivative firstly we establish the l1 fourier galerkin full discrete and l1 fourier collocation schemes with fourier spectral discretization in spatial direction and l1 difference method in temporal direction secondly stability and convergence for both galerkin and collocation approximations are proved it is shown that the proposed methods are convergent with spectral accuracy in space and 2 order accuracy in time for implementation the equivalence between pseudospectral method and collocation method is discussed furthermore a numerical algorithm of fourier pseudospectral method is developed based on two dimensional fast fourier transform fft2 technique finally numerical examples are provided to test the theoretical claims as is shown in the numerical experiments fourier spectral methods are powerful enough with excellent efficiency and accuracy

validation of a simple method for representing spheres and slender bodies in an immersed boundary method for stokes flow on an unbounded domain we test the efficacy of using a single lagrangian point to represent a sphere and a one dimensional array of such points to represent a slender body in a new immersed boundary method for stokes flow a numerical parameter the spacing of the eulerian grid is used to determine the effective radius of the immersed sphere or slender body such representations are much less expensive computationally than those with two or three dimensional meshes of lagrangian points to perform this test we develop a numerical method to solve the discretized stokes equations on an unbounded eulerian grid which contains an arbitrary configuration of lagrangian points that apply force to the fluid and that move with the fluid we compare results computed with this new immersed boundary method to known results for spheres and rigid cylinders in stokes flow in r 3 we find that for certain choices of parameters the interactions with the fluid of a single lagrangian point accurately replicate those of a sphere of some particular radius independent of the location of the point with respect to the eulerian grid the interactions of a linear array of lagrangian points for certain choices of parameters accurately replicate those of a cylinder of some particular radius independent of the position and orientation of the array with respect to the eulerian grid the effective radius of the sphere and the effective radius of the cylinder turn out to be related in a simple and natural way our results suggest recipes for choosing parameters that should be useful to practitioners one surprising result is that one must not use too many lagrangian points in an array another is that the approximate delta functions traditionally used in the immersed boundary method perform much better than higher order delta functions with the same support

a mixed finite element method for a time fractional fourth order partial differential equation in this paper a numerical theory based on the mixed finite element method for a time fractional fourth order partial differential equation pde is presented and analyzed an auxiliary variable email protected is introduced then the fourth order equation can be split into the coupled system of two second order equations the time caputo fractional derivative is discretized by a finite difference method and the spatial direction is approximated by the mixed finite element method the stabilities based on a priori analysis for two variables are discussed and some a priori error estimates in l 2 norm for the scalar unknown u and the variable email protected are derived respectively moreover an a priori error result in h 1 norm for the scalar unknown u also is proved for verifying the theoretical analysis a numerical test is made by using matlab procedure

on finite difference methods for fourth order fractional diffusion wave and subdiffusion systems abstract in this paper firstly the finite difference method is explored for the fourth order fractional diffusion wave system the method is proved to be uniquely solvable stable and convergent in l norm by the energy method then we examine a subdiffusion system and present the numerical analysis using a different method numerical experiments are provided to demonstrate the accuracy and efficiency of the proposed schemes

numerical solution of two sided space fractional wave equation using finite difference method in this paper a class of finite difference method for solving two sided space fractional wave equation is considered the stability and consistency of the method are discussed by means of gerschgorin theorem and using the stability matrix analysis numerical solutions of some wave fractional partial differential equation models are presented the results obtained are compared to exact solutions

local rbf fd technique for solving the two dimensional modified anomalous sub diffusion equation abstract the main aim of this paper is to propose an efficient and suitable numerical procedure based on the local meshless collocation method for solving the two dimensional modified anomalous sub diffusion equation the fractional derivative is based on the riemann liouville fractional integral firstly a finite difference scheme with o has been employed to discrete the time variable and also the local radial basis finite difference lrbf fd method is used to discrete the spatial direction for the presented numerical technique we prove the unconditional stability and also obtain an error bound we employ a test problem to show the accuracy of the proposed technique also we solve the mentioned model on irregular domain to show the efficincy of the developed technique

finite difference methods for fractional dispersion equations abstract in this paper the fractional weighted average finite difference method for space fractional advection dispersion equation is proposed which is based on shifted grunwald formula this method is unconditionally stable consistent and convergent a numerical example is given and the numerical results verify the theoretical conclusions

blow up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition abstract in this paper we consider a semilinear heat equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove global existence theorem for max p l 1 some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given

numerical approximations for fractional diffusion equations via splines a one dimensional fractional diffusion model is considered where the usual second order derivative gives place to a fractional derivative of order a with 1 email protected 2 we consider the caputo derivative as the space derivative which is a form of representing the fractional derivative by an integral operator an implicit numerical method is derived which uses a spline approximation for the caputo derivative the consistency and stability of the method are examined and numerical results are presented

effects of a fractional friction with power law memory kernel on string vibrations in this paper we give an analytical treatment of a wave equation for a vibrating string in the presence of a fractional friction with power law memory kernel the exact solution is obtained in terms of the mittag leffler type functions and a generalized integral operator containing a four parameter mittag leffler function in the kernel the method of separation of the variables laplace transform method and sturm liouville problem are used to solve the equation analytically the asymptotic behaviors of the solution of a special case fractional differential equation are obtained directly from the analytical solution of the equation and by using the tauberian theorems the proposed model may be used for describing processes where the memory effects of complex media could not be neglected

numerical treatment for solving the perturbed fractional pdes using hybrid techniques in this article an accurate numerical approach is introduced in this approach we mixed between the fractional finite difference method and the restrictive taylor approximation rta the proposed method is implemented to solve numerically the perturbed fractional partial differential equations fpdes special attention is given to study the stability and consistency of the method by means of gerschgorin theorem and using the stability matrix analysis two numerical examples are given and compared with the exact solution

superconvergence analysis of nonconforming finite element method for time fractional nonlinear parabolic equations on anisotropic meshes abstract in this paper we prove a novel result of the consistency error estimate with order o h 2 for e q 1 r o t element see lemma 2 on anisotropic meshes then a linearized fully discrete galerkin finite element method fem is studied for the time fractional nonlinear parabolic problems and the superclose and superconvergent estimates of order o h 2 in broken h 1 norm on anisotropic meshes are derived by using the proved character of e q 1 r o t element which improve the results in the existing literature numerical results are provided to confirm the theoretical analysis

a class of shifted high order numerical methods for the fractional mobile immobile transport equations abstract in this article we apply the generalized bdf2 to the fractional mobile immobile transport equations for its temporal discretization and the finite element method in the spatial direction to derive the stability estimates and obtain the optimal error convergence rate some properties of the convolution weights are proved based on which we show the scheme is unconditionally stable with an error of o 2 h r 1 where and h represent the temporal and spatial mesh size respectively we conduct exhaustive numerical tests to further confirm our theoretical analysis and to overcome the initial singularity of the time fractional derivative we adopt the generalized bdf2 with starting parts in accordance with the framework of the shifted convolution quadrature scq

a meshless method for solving the time fractional advection diffusion equation with variable coefficients abstract in this paper an efficient and accurate meshless method is proposed for solving the time fractional advection diffusion equation with variable coefficients which is based on the moving least square mls approximation in the proposed method firstly the time fractional derivative is approximated by a finite difference scheme of order o t 2 0 1 and then the mls approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the caputo sense also the validity of the proposed method is investigated in error analysis discussion the main aim is to show that the meshless method based on the mls shape functions is highly appropriate for solving fractional partial differential equations fpdes with variable coefficients the efficiency and accuracy of the proposed method are verified by solving several examples

a cubic trigonometric b spline collocation approach for the fractional sub diffusion equations an algorithm for fractional sub diffusion model using cubic trigonometric b spline approach is presented the proposed scheme is unconditionally stable using von neumann approach error norms confirm the accuracy and efficiency of presented scheme the proposed method provides more accurate results as compare to cubic b spline method a cubic trigonometric b spline collocation approach for the numerical solution of fractional sub diffusion equation is presented in this paper the approach is based on the usual finite difference scheme to discretize the time derivative while the approximation of the second order derivative with respect to space is obtained by the cubic trigonometric b spline functions with the help of grunwald letnikov discretization of the riemann liouville derivative the scheme is shown to be stable using the fourier method and the accuracy of the scheme is tested by application to a test problem the results of the numerical test verify the accuracy and efficiency of the proposed algorithm

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

a compact lod method and its extrapolation for two dimensional modified anomalous fractional sub diffusion equations a crank nicolson type compact locally one dimensional lod finite difference method is proposed for a class of two dimensional modified anomalous fractional sub diffusion equations with two time riemann liouville fractional derivatives of orders 1 and 1 0 1 the resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one dimensional problems this property evidently enhances the simplicity of programming and makes the computations more easy the unconditional stability and convergence of the scheme are rigorously proved the error estimates in the standard h 1 and l 2 norms and the weighted l norm are obtained and show that the proposed compact lod method has the accuracy of the order 2 min in time and 4 in space a richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min 4 min if and min 1 4 if a comparison study of the compact lod method with the other existing methods is given to show its superiority numerical results confirm our theoretical analysis and demonstrate the accuracy and the effectiveness of the compact lod method and the extrapolation algorithm

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

a hybridizable discontinuous galerkin method for fractional diffusion problems we study the use of the hybridizable discontinuous galerkin hdg method for numerically solving fractional diffusion equations of order alpha with 1 alpha 0 1 0 for exact time marching we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular thus if for each time t in 0 t t 0 t the approximations are taken to be piecewise polynomials of degree k ge 0 k 0 on the spatial domain varomega the approximations to u u in the l infty bigr 0 t l 2 varomega bigr l 0 t l2 norm and to nabla u u in the l infty bigr 0 t mathbf l 2 varomega bigr l 0 t l2 norm are proven to converge with the rate h k 1 hk 1 where h h is the maximum diameter of the elements of the mesh moreover for k ge 1 k 1 and quasi uniform meshes we obtain a superconvergence result which allows us to compute in an elementwise manner a new approximation for u u converging with a rate of sqrt log t h 2 alpha 1 h k 2 log th 2 1 hk 2

compact alternating direction implicit scheme for the two dimensional fractional diffusion wave equation in this paper we consider the numerical method for solving the two dimensional fractional diffusion wave equation with a time fractional derivative of order alpha 1 alpha 2 a difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit adi method in the time stepping is proposed and analyzed the unconditional stability and h 1 norm convergence of the scheme are proved rigorously the convergence order is mathcal o tau 3 alpha h 1 4 h 4 2 where tau is the temporal grid size and h 1 h 2 are spatial grid sizes in the x and y directions respectively in addition a crank nicolson adi scheme is presented and the corresponding error estimates are also established numerical results are presented to support our theoretical analysis and indicate that the compact adi scheme reduces the storage requirement and cpu time successfully

diffusive approximation of a time fractional burger s equation in nonlinear acoustics a fractional time derivative is introduced into burger s equation to model losses of nonlinear waves this term amounts to a time convolution product which greatly penalizes the numerical modeling a diffusive representation of the fractional derivative is adopted here replacing this nonlocal operator by a continuum of memory variables that satisfy local in time ordinary differential equations then a quadrature formula yields a system of local partial differential equations well suited to numerical integration the determination of the quadrature coefficients is crucial to ensure both the well posedness of the system and the computational efficiency of the diffusive approximation for this purpose optimization with constraint is shown to be a very efficient strategy strang splitting is used to solve successively the hyperbolic part by a shock capturing scheme and the diffusive part exactly numerical experiments are proposed to assess the efficiency of the numerical modeling and to illustrate the effect of the fractional attenuation on the wave propagation

a modified memory based mathematical model describing fluid flow in porous media abstract this study presents a modified memory based mathematical model describing fluid flow in porous media for the first time such model is derived employing the grunwald letnikov g l definition of the riemann liouville r l time fractional operator via a generalized darcy s equation the proposed mathematical model is suitable to describe the anomalous diffusion behavior observed in a medium of fractal geometry as well as in disordered and highly heterogeneous porous media a numerical scheme based on existing discretization method is employed to handle the modified memory based mathematical model the accuracy of the numerical model is validated through the analytical solution for a simplified problem in addition numerical experiments are presented to demonstrate the effect of the anomalous diffusion exponent on the predicted reservoir pressure wellbore pressure and volumetric flux considering the flow of an under saturated oil in a hydrocarbon reservoir as an example results show that decrease in magnitude of the anomalous diffusion exponent results in larger pore and wellbore pressure this study would open a door to apply the g l interpretation of the time fractional operator in numerical modeling of fluid flow through porous media

analysis of a new finite difference local discontinuous galerkin method for the fractional diffusion wave equation in this paper a finite difference local discontinuous galerkin method for the fractional diffusion wave equation is presented and analyzed we first propose a new finite difference method to approximate the time fractional derivatives and give a semidiscrete scheme in time further we develop a fully discrete scheme for the fractional diffusion wave equation and prove that the method is unconditionally stable and convergent with order o hk 1 t 3 where k is the degree of piecewise polynomial extensive numerical examples are carried out to confirm the theoretical convergence rates

a tau approach for solution of the space fractional diffusion equation fractional differentials provide more accurate models of systems under consideration in this paper approximation techniques based on the shifted legendre tau idea are presented to solve a class of initial boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain the fractional derivatives are described in the caputo sense the technique is derived by expanding the required approximate solution as the elements of shifted legendre polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from the computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

numerical analysis of nonlinear subdiffusion equations we present a general framework for the rigorous numerical analysis of time fractional nonlinear parabolic partial differential equations with a fractional derivative of order alpha in 0 1 in time it relies on three technical tools a fractional version of the discrete gronwall type inequality discrete maximal regularity and regularity theory of nonlinear equations we establish a general criterion for showing the fractional discrete gronwall inequality and verify it for the l1 scheme and convolution quadrature generated by backward difference formulas further we provide a complete solution theory e g existence uniqueness and regularity for a time fractional diffusion equation with a lipschitz nonlinear source term together with the known results of discrete maximal regularity we derive pointwise l 2 omega norm error estimates for semidiscrete galerkin finite element solutions and fully discrete solutions which are of order o h 2 up to a logarithmic factor and o tau alpha

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 novel high order space time spectral method for the time fractional fokker planck equation the fractional fokker planck equation is an important physical model for simulating anomalous diffusions with external forces because of the nonlocal property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations in this paper a space time spectral method is presented for the numerical solution of the time fractional fokker planck initial boundary value problem the proposed method employs the jacobi polynomials for the temporal discretization and fourier like basis functions for the spatial discretization due to the diagonalizable trait of the fourier like basis functions this leads to a reduced representation of the inner product in the galerkin analysis we prove that the time fractional fokker planck equation attains the same approximation order as the time fractional diffusion equation developed in x li and c xu siam j numer anal 47 2009 pp 2108 2131 by using the present method that indicates an e

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

compact alternating direction implicit method for two dimensional time fractional diffusion equation high order compact finite difference scheme with operator splitting technique for solving two dimensional time fractional diffusion equation is considered in this paper a grunwald letnikov approximation is used for the riemann liouville time derivative and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme alternating direction implicit adi method is used to split the original problem into two separate one dimensional problems the local truncation error is analyzed and the stability is discussed by the fourier method the proposed scheme is suitable when the order of the time fractional derivative c lies in the interval 12 1 a correction term is added to maintain high accuracy when c 0 12 numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

the bem for numerical solution of partial fractional differential equations a numerical method is presented for the solution of partial fractional differential equations fdes arising in engineering applications and in general in mathematical physics the solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape the method is based on the concept of the analog equation which in conjunction with the boundary element method bem enables the spatial discretization and converts a partial fde into a system of coupled ordinary multi term fdes then this system is solved using the numerical method for the solution of such equations developed recently by katsikadelis the method is illustrated by solving second order partial fdes and its efficiency and accuracy is validated