the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

convergence analysis of moving finite element methods for space fractional differential equations most existent papers have focused on the fixed mesh methods for solving space fractional differential equations however since some classes of space fractional differential equations may have singular or even finite time blowup solutions it is highly needed to develop adaptive mesh methods to solve these problems in this paper the moving finite element methods are studied for a class of time dependent space fractional differential equations the convergence theories of the methods are derived with l 2 norm and numerical examples are provided to support the theoretical results to simplify the analysis a fractional ritz projection operator is introduced and the error estimation of the projection is derived under the moving mesh setting

spectral methods in time for a class of parabolic partial differential equations in this paper we introduce a fully spectral solution for the partial differential equation u t uu x vu x x mu x x x lu x x x x 0 for periodic boundary conditions in space the use of fourier expansion in x admits of a particularly efficient algorithm with respect to expansion of the time dependence in a chebyshev series boundary conditions other than periodic may still be treated with reasonable though lesser efficiency for all cases very high accuracy is attainable at moderate computational cost relative to the expense of variable order finite difference methods in time

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

galerkin spectral method for nonlinear time fractional cable equation with smooth and nonsmooth solutions abstract in this work we study the numerical solutions of the time fractional cable equations with nonlinear term where the fractional derivatives are described in riemann liouville sense an explicit scheme is constructed based upon finite difference method in time and legendre spectral method in space stability and convergence of scheme are proved rigorously moreover an improved algorithm for the problem with nonsmooth solutions is proposed by adding correction terms to the approximations of first order derivative fractional derivatives and nonlinear term numerical examples are given to support theoretical analysis

stable multi domain spectral penalty methods for fractional partial differential equations we propose stable multi domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order first a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials the approximation order is analyzed and verified through numerical examples based on the discrete fractional derivative we introduce stable multi domain spectral penalty methods for solving fractional advection and diffusion equations the equations are discretized in each sub domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes

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

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

finite difference methods for the time fractional diffusion equation on non uniform meshes since fractional derivatives are integrals with weakly singular kernel the discretization on the uniform mesh may lead to poor accuracy the finite difference approximation of caputo derivative on non uniform meshes is investigated in this paper the method is applied to solve the fractional diffusion equation and a semi discrete scheme is obtained the unconditional stability and h 1 norm convergence are proved a fully discrete difference scheme is constructed with space discretization by compact difference method the error estimates are established for two kinds of nonuniform meshes numerical tests are carried out to support the theoretical results and comparing with the method on uniform grid shows the efficiency of our methods moreover a moving local refinement technique is introduced to improve the temporal accuracy of numerical solution

an efficient algorithm for the evaluation of convolution integrals we propose an algorithm for the numerical evaluation of convolution integrals of the form 0 xk x y f y x dy for x 0 x our method is especially suitable in situations where the fundamental interval 0 x is very long and the kernel function k is expensive to calculate separate versions are provided where the forcing function f is known in advance and where it must be determined step by step along the solution path these methods are efficient with respect to both run time and memory requirements

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

error estimates of generalized spectral iterative methods with accurate convergence rates for solving systems of fractional two point boundary value problems abstract the main purpose of this study is to provide an efficient spectral iterative method based on fractional interpolants for solving a class of linear and nonlinear fractional two point boundary value problems involving left and right fractional derivatives in order to achieve this goal by composition of iterative method and spectral method in each step the unknown solution of system is expanded by using two classes of generalized jacobi functions to obtain numerically coefficients for this system of fractional differential equations on the interval 1 1 the singularities of the solution are considered of types 1 t and 1 x where 0 l 2 norm finally some numerical results are given to demonstrate the effectiveness and applicability of the proposed method and accuracy of the presented convergence rates

stability and convergence analysis of the quadratic spline collocation method for time dependent fractional diffusion equations abstract a quadratic spline collocation method combined with the crank nicolson time discretization is proposed for time dependent two sided fractional diffusion equations by carefully analyzing the mathematical properties of the coefficient matrix the new scheme is proved to be unconditionally stable in the sense of discrete l2 norm for 2 where is the order of the space fractional derivative of the fractional diffusion equation and 1 2576 see lemma 3 1 furthermore the fractional order spline interpolation error over the collocation points is studied and subsequently we show that the spline collocation solution of the fractional diffusion equation converges to the exact one with order o h 3 2 under the discrete l2 norm where and h are the temporal and spatial step sizes respectively finally numerical experiments are given to verify the theoretical results

trapezoidal scheme for time space fractional diffusion equation with riesz derivative abstract in this paper a finite difference scheme is proposed to solve time space fractional diffusion equation which has second order accuracy in both time and space direction the time and space fractional derivatives are considered in the senses of caputo and riesz respectively first the centered difference approach is used to approximate the riesz fractional derivative in space then the obtained fractional ordinary differential equations are transformed into equivalent volterra integral equations and then the trapezoidal rule is utilized to approximate the volterra integral equations the stability and convergence of our scheme are proved via mathematical induction method finally numerical experiments are performed to confirm the high accuracy and efficiency of our scheme

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

numerical solution of nonlinear fractional integro differential equations with weakly singular kernels via a modification of hat functions abstract in the present paper a modification of hat functions mhfs has been considered for solving a class of nonlinear fractional integro differential equations with weakly singular kernels numerically the fractional order operational matrix of integration is introduced we provide an error estimation for the approximation of a function by a series of mhfs to suggest a numerical method the main problem is converted to an equivalent volterra integral equation of the second kind and operational matrices of mhfs are used to reduce the problem to the solution of bivariate polynomial equations finally illustrative examples are provided to confirm the accuracy and validity of the proposed method

a pde approach to space time fractional parabolic problems we study solution techniques for parabolic equations with fractional diffusion and caputo fractional time derivative the latter being discretized and analyzed in a general hilbert space setting the spatial fractional diffusion is realized as the dirichlet to neumann map for a nonuniformly elliptic problem posed on a semi infinite cylinder in one more spatial dimension we write our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition we propose and analyze an implicit fully discrete scheme first degree tensor product finite elements in space and an implicit finite difference discretization in time we prove stability and error estimates for this scheme

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

nmr flow velocity mapping in random percolation model objects evidence for a power law dependence of the volume averaged velocity on the probe volume radius two or three dimensional lacunar objects were fabricated using computer simulated random sitepercolation networks as templates the flow of water pumped through the pore space was studied with the aid of nuclear magnetic resonance nmr microscopy modified for mapping of the velocity vector field the percolation backbones of the objects were determined by exclusion of all pixels or voxels of the spin density images with velocities below the noise level an evaluation procedure was established which reliably renders the fractal dimensions of the whole cluster and of its backbone the volume averaged velocity magnitude as a function of the probe volume radius r was found to obey a power lawv v r 2l in the range a r jv where a is the voxel edge length and jv the velocity correlation length the exponents turned out to be l50 3260 04 and l50 8260 03 for the two and three dimensional objects respectively in order to test the time dependence of the mean squared displacement on fractals r 2 5at 2 dw expected for random walks of a fractal dimension dw self diffusion of gaseous methane was examined in the pore spaces of the same objects with the aid of field gradient nmr diffusometry the results are in accordance with the theoretical predictions for anomalous diffusion on percolation clusters this finding is supported by studies of incoherent water flow in the percolation network s1063 651x 96 06510 5

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

incorporation of linear multipoint constraints in domain decomposition based iterative solvers part ii blending feti dp and mortar methods and assembling floating substructures two approaches are presented for extending the feti dp domain decomposition based iterative method to the solution of problems with linear multipoint constraints lmpcs both cases of arbitrary lmpcs and lmpcs generated by the mortar method for enforcing a weak continuity of the solution across non matching finite element interfaces are considered in the first approach the lmpcs are addressed during the preconditioning step but enforced only at convergence in the second approach the lmpcs are enforced at each iteration through the solution of an auxiliary coarse problem both extended feti dp solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems the obtained results reveal that both solvers are numerically scalable and complement each other from the memory and cpu performance viewpoints

space time spectral element methods for one dimensional nonlinear advection diffusion problems abstract the following space time galerkin spectral element methods are developed and applied to solve the burgers equation with small viscosity a coupled methods consisting of an explicit method for hyperbolic dominated equations and an implicit method for parabolic dominated equations b two splitting methods which solve the hyperbolic substep explicitly and the parabolic one implicitly one uses spectral elements in the explicit part and the other uses the adams bashforth multistep method a subcycling technique in which several convective steps are taken for each implicit viscous step was also investigated for the two splitting methods a stability analysis of the four methods is performed and subsequent results are debated a convergence study and a comparison of computer execution time for the four methods is made and the results are discussed comparative study leads to the conclusion that the space time spectral element splitting method with subcycling is superior to the other methods presented in terms of robustness and computer execution time the number of subcycles should be kept low 2 3 in order to avoid significant loss of accuracy the coupled explicit method is also applied to the solution of the one dimensional coupled continuity momentum and energy equations for non isothermal flow of an ideal gas with temperature dependent properties in a cylindrical duct of variable radius

tempered fractional sturm liouville eigenproblems continuum time random walk is a general model for particle kinetics which allows for incorporating waiting times and or non gaussian jump distributions with divergent second moments to account for levy flights exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered stable levy processes with finite moments where the fluid continuous limit leads to the tempered fractional diffusion equation the development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions in this study we introduce two classes of regular and singular tempered fractional sturm liouville problems of two kinds tfslp i and tfslp ii of order nu in 0 2 in the regular case the corresponding tempered differential operators are associated with tempering functions p i x exp 2 tau and p ii x exp 2 tau tau geq 0 respectively in the regular tfslp i

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

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 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

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

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

alternating direction implicit euler method for the two dimensional fractional evolution equation in this paper we present a fast and efficient numerical method to solve a two dimensional fractional evolution equation on a finite domain this numerical method combines the alternating direction implicit adi approach with the second order difference quotient in space the backward euler in time and order one convolution quadrature approximating the integral term by using the discrete energy method we prove that the adi scheme is unconditionally stable and the numerical solution converges to the exact one with order o k h x 2 h y 2 where k is the temporal grid size and h x h y are spatial grid sizes in the x and y directions respectively two numerical examples with known exact solution are also presented and the behavior of the error is analyzed to verify the order of convergence of the adi euler method

spectral direction splitting methods for two dimensional space fractional diffusion equations a numerical method for a kind of time dependent two dimensional two sided space fractional diffusion equations is developed in this paper the proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization the direction splitting approach renders the underlying two dimensional equation into a set of one dimensional space fractional diffusion equations at each time step then these one dimensional equations are solved by using the spectral method based on weak formulations a time error estimate is derived for the semi discrete solution and the unconditional stability of the fully discretized scheme is proved some numerical examples are presented to validate the proposed method

a multi domain spectral method for time fractional differential equations this paper proposes an approach for high order time integration within a multi domain setting for time fractional differential equations since the kernel is singular or nearly singular two main difficulties arise after the domain decomposition how to properly account for the history memory part and how to perform the integration accurately to address these issues we propose a novel hybrid approach for the numerical integration based on the combination of three term recurrence relations of jacobi polynomials and high order gauss quadrature the different approximations used in the hybrid approach are justified theoretically and through numerical examples based on this we propose a new multi domain spectral method for high order accurate time integrations and study its stability properties by identifying the method as a generalized linear method numerical experiments confirm hp convergence for both time fractional differential equations and time fractional partial differential equations

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

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

space time spectral element method for solution of second order hyperbolic equations abstract a two field mixed hybrid spectral element formulation in time and space is presented explicit stsee and implicit stsei algorithms are developed and used to solve a second order scalar hyperbolic equation the algorithms were analyzed and their characteristics in terms of truncation error and stability are presented this analysis was confirmed by numerical results the improvement in accuracy in comparison with semi discrete schemes is remarkable

finite difference spectral approximation for a time space fractional equation on two and three space dimensions abstract we consider the numerical method for solving a time space fractional equation whose solution has a weak initial singularity l 1 scheme on graded mesh is used to capture the initial weak singularity it is shown that the method can recover order 2 convergence by choosing some appropriate grading parameters where 0 1 is the order of temporal caputo fractional derivative a fully discrete scheme is proposed by combining spectral approximation for the discretization of space in 2d and 3d cases stability and convergence of the fully discrete scheme are rigorously established numerical results are presented to show the sharpness of the error estimate

numerical solution of fractional differential equations with a collocation method based on muntz polynomials this paper presents a computational technique based on the collocation method and muntz polynomials for the solution of fractional differential equations an appropriate representation of the solution via the muntz polynomials reduces its numerical treatment to the solution of a system of algebraic equations the main advantage of the present method is its superior accuracy and exponential convergence consequently one can obtain good results even by using a small number of collocation points the accuracy and performance of the proposed method are examined by means of some numerical experiments

spectral methods for tempered fractional differential equations in this paper we first introduce fractional integral spaces which possess some features i when 0 1 functions in these spaces are not required to be zero on the boundary ii the tempered fractional operators are equivalent to the riemann liouville operator in the sense of the norm spec tral galerkin and petrov galerkin methods for tempered fractional advection problems and tempered fractional diffusion problems can be developed as the classical spectral galerkin and petrov galerkin methods error analysis is provided and numerically confirmed for the tempered fractional advection and diffusion problems

optimal error estimates of spectral petrov galerkin and collocation methods for initial value problems of fractional differential equations we present optimal error estimates for spectral petrov galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval we also develop laguerre spectral petrov galerkin methods and collocation methods for fractional equations on the half line numerical results confirm the error estimates

well conditioned fractional collocation methods using fractional birkhoff interpolation basis the purpose of this paper is twofold firstly we provide explicit and compact formulas for computing both caputo and modified riemann liouville rl fractional pseudospectral differentiation matrices f psdms of any order at general jacobi gauss lobatto jgl points we show that in the caputo case it suffices to compute f psdm of order 0 1 to compute that of any order k with integer k 0 while in the modified rl case it is only necessary to evaluate a fractional integral matrix of order 0 1 secondly we introduce suitable fractional jgl birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties i the matrix generated from the new basis yields the exact inverse of f psdm at interior jgl points ii the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal and iii the resulted linear system is well conditioned in the caputo case while in the modified rl case the eigenvalues of the coefficient matrix are highly concentrated in both cases the linear systems of the collocation schemes using the new basis can be solved by an iterative solver within a few iterations notably the inverse can be computed in a very stable manner so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations fdes it is also noteworthy that the choice of certain special jgl points with parameters related to the order of the equations can ease the implementation we highlight that the use of the bateman s fractional integral formulas and fast transforms between jacobi polynomials with different parameters is essential for our algorithm development

finite difference spectral approximations for the fractional cable equation fujian nsf s0750017 national nsf of china 10531080 ministry of education of china 973 high performance scientific computation research program 2005cb321703

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

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

a fast finite volume method for conservative space time fractional diffusion equations discretized on space time locally refined meshes abstract a fast finite volume method was developed for conservative space time fractional diffusion partial differential equations in two space dimensions in the method a locally refined composite spatial mesh is used to resolve the boundary layers of the solutions and a graded temporal mesh is used to resolve the singularity of the solutions near the initial time a fast numerical method was developed by fully exploring the structure of the stiffness matrices numerical results show the utility of the method

certain new models of the multi space fractional gardner equation abstract in this paper we study an accurate numerical method for solving the multi space fractional gardner equation msfge with the caputo fabrizio cf and atangana baleanu ab fractional derivatives where the space fractional terms are under the sense of caputo to the best knowledge of the reader we are first to use the spectral collocation method based on the third chebyshev approximations to reduced the multi space fractional gardner equation to a system of ordinary differential equations by using the properties of chebyshev polynomials and then solved them via the finite difference method fdm by computing the absolute errors we present the effectiveness and accuracy of the proposed methods the present paper investigates the dynamics of gardner equation by considering two fractional operators that is the caputo fabrizio and atangana baleanu this is entirely new idea by using the two operators on a gardner equation our results prove that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non local singular kernels

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

a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

a compact adi method and its extrapolation for time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions a compact alternating direction implicit adi finite difference method is proposed for two dimensional time fractional sub diffusion equations with nonhomogeneous neumann boundary conditions the unconditional stability and convergence of the method is proved the error estimates in the weighted l 2 l 2 and l l norms are obtained the proposed method has the fourth order spatial accuracy and the temporal accuracy of order min 2 1 min 2 1 where 0 1 0 1 is the order of the fractional derivative in order to further improve the temporal accuracy two richardson extrapolation algorithms are presented numerical results demonstrate the accuracy of the compact adi method and the high efficiency of the extrapolation algorithms

central local discontinuous galerkin method for the space fractional diffusion equation abstract this paper provides the semi discrete scheme by the central local discontinuous galerkin method for space fractional diffusion equation on two sets of overlapping cells and then the stability analysis and error estimates of the scheme are presented lastly we verify the effectiveness of the proposed scheme by the one and two dimensional numerical experiments

a new difference scheme for the time fractional diffusion equation abstract in this paper we construct a new difference analog of the caputo fractional derivative called the l 2 1 formula the basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered stability of the suggested schemes and also their convergence in the grid l 2 norm with the rate equal to the order of the approximation error are proved the obtained results are supported by the numerical calculations carried out for some test problems

the method of approximate particular solutions for the time fractional diffusion equation with a non local boundary condition in this paper we consider the numerical solution of the time fractional diffusion equation with a non local boundary condition the method of approximate particular solutions maps using multiquadric radial basis function mq rbf is employed for this equation due to the accuracy of the mq based meshless methods is severely influenced by the shape parameter we adopt a leave one out cross validation loocv algorithm proposed by rippa 34 to enhance the performance of the maps the numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time fractional diffusion equation with a non local boundary condition

parameter estimation for the fractional fractal diffusion model based on its numerical solution in this paper we mainly consider a problem of parameter estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media the bayesian method is proposed to estimate parameters for the fractional fractal diffusion model based on its numerical solution firstly the center box difference method is employed to solve the fractional fractal diffusion equation subject to the initial boundary conditions then we apply the bayesian method to estimate three parameters for the model simultaneously that is the fractional order the fractal dimension d f and the structure parameter to testify the validity of the results for the model s parameter estimation experimental data from the fast desorption process of methane in coal are used it is shown that the numerical results modeled with parameters given by the bayesian method coincide well with the desorption experimental data indicating that the bayesian method is efficient and valid in identifying multi parameter for the fractional fractal diffusion model besides the fractional fractal diffusion model is demonstrated to be more capable than the classical fick s model to describe the anomalous diffusion behavior of gas in coal to clarify the parameters effect on the relative diffusion amount of methane for the fractional fractal diffusion model parameter sensitivity analysis is also carried out this paper provides a specific and efficient parameter estimation method for the fractional fractal diffusion model

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

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

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

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

a high order nodal discontinuous galerkin method for a linearized fractional cahnhilliard equation in this paper we develop and analyze a nodal discontinuous galerkin method for the linearized fractional cahnhilliard equation containing derivatives of fractional order in space the caputo derivative is chosen as the representation of spatial derivative the linearized fractional cahnhilliard problem has been expressed as a system of low order differential integral equations we adopt the nodal discontinuous galerkin methods for the full spatial discretization using a high order nodal basis set of orthonormal lagrangelegendre polynomials of arbitrary order in space on each element of computational domain moreover we prove the stability and optimal order of convergence n 1 for the linearized fractional cahnhilliard problem when polynomials of degree n are used numerical experiments are displayed to verify the theoretical results

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

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

a discontinuous petrov galerkin method for time fractional diffusion equations we propose and analyze a time stepping discontinuous petrov galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time fractional subdiffusion problems we prove the existence uniqueness and stability of approximate solutions and derive error estimates to achieve high order convergence rates from the time discretizations the time mesh is graded appropriately near t 0 to compensate for the singular temporal behavior of the exact solution near t 0 caused by the weakly singular kernel but the spatial mesh is quasi uniform in the l infty 0 t l 2 omega norm 0 t is the time domain and omega is the spatial domain for sufficiently graded time meshes a global convergence of order k m alpha 2 h r 1 is shown where 0 alpha 1 is the fractional exponent k is the maximum time step h is the maximum diameter of the elements of the spatial mesh and m and r are the degrees of approximate solutions in time an

crank nicolson quasi wavelets method for solving fourth order partial integro differential equation with a weakly singular kernel in this paper we study a novel numerical scheme for the fourth order partial integro differential equation with a weakly singular kernel in the time direction a crank nicolson time stepping is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term and the quasi wavelets numerical method for space discretization our interest in the present paper is a continuation of the investigation in yang et al 33 where we study discretization in time by using the forward euler scheme the comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order partial integro differential equation with a weakly singular kernel we also tested the method proposed on several one and two dimensional problems with very promising results besides in order to demonstrate the power of the quasi wavelets method in comparison with standard discretization methods we also consider the high frequency oscillation problems with the integro differential term

a generalized laguerre spectral petrov galerkin method for the time fractional subdiffusion equation on the semi infinite domain abstract in this paper a generalized laguerre spectral petrov galerkin method is proposed to solve a class of time fractional subdiffusion equations on the semi infinite domain we use the generalized associated laguerre functions of the first kind and the generalized laguerre functions as basis functions in time and space directions separately the respective projection error estimates can be obtained as well we derive the projection error estimates of the solutions in time space directions the approximation results of the fully discrete spectral scheme are introduced in this paper some numerical results are presented to illustrate the efficiency of this method

a simple finite element method for boundary value problems with a riemann liouville derivative we consider a boundary value problem involving a riemann liouville fractional derivative of order 3 2 2 on the unit interval 0 1 the standard galerkin finite element approximation converges slowly due to the presence of singularity term x 1 in the solution representation in this work we develop a simple technique by transforming it into a second order two point boundary value problem with nonlocal low order terms whose solution can reconstruct directly the solution to the original problem the stability of the variational formulation and the optimal regularity pickup of the solution are analyzed a novel galerkin finite element method with piecewise linear or quadratic finite elements is developed and l 2 d error estimates are provided the approach is then applied to the corresponding fractional sturm liouville problem and error estimates of the eigenvalue approximations are given extensive numerical results fully confirm our theoretical study

alternating direction implicit galerkin finite element method for the two dimensional fractional diffusion wave equation new numerical techniques are presented for the solution of the two dimensional fractional diffusion wave equation with a time fractional derivative of order 1 2 in these methods galerkin finite element is used for the spatial discretization and for the time stepping new alternating direction implicit adi method based on the crank nicolson method are considered the unconditional stability and l 2 norm convergence of the adi scheme are proved rigorously numerical results are presented to support our theoretical analysis

numerical approximation of an interface problem for fractional in time diffusion equation an initial boundary value problem for fractional in time diffusion equation with interface is considered its well posedness in the corresponding sobolev spaces is proved some finite difference schemes approximating the problem are proposed and their stability and convergence are investigated

first passage time moments of asymmetric levy flights we investigate the first passage dynamics of symmetric and asymmetric l evy flights in a semi infinite and bounded intervals by solving the space fractional diffusion equation we analyse the fractional order moments of the first passage time probability density function for different values of the index of stability and the skewness parameter a comparison with results using the langevin approach to l evy flights is presented for the semi infinite domain in certain special cases analytic results are derived explicitly and in bounded intervals a general analytical expression for the mean first passage time of l evy flights with arbitrary skewness is presented these results are complemented with extensive numerical analyses

time fractional diffusion equation for signal smoothing the time fractional diffusion equation is used for signal smoothing compared to the classical diffusion equation the time fractional diffusion equation has another adjustable time fractional derivative order to control the diffusion process therefore some simulated signals are used to compare the smoothing performance between the time fractional diffusion equation and the classical diffusion equation as well as between classical smoothing methods regularization method savitzky golay method and wavelet method in the end the time fractional diffusion filtering is applied in an nmr spectrum smoothing results indicate that the time fractional diffusion filtering is advantage over the classical diffusion filtering and their smoothing performance is better than that of classical smoothing methods

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

θ schemes for finite element discretization of the space time fractional diffusion equations the numerical solution of a space time fractional diffusion equation used to model the anomalous diffusion is considered spatial discretization is effected using a finite element method whereas the scheme is used for temporal discretization the fully discrete scheme is analyzed for all 0 1 to determine conditional and unconditional stability regimes for the scheme and also to obtain error estimates for the approximate solution the analysis is facilitated by making use of a variational formulation of the equations that is based on a recently developed nonlocal calculus one dimensional numerical examples are provided that illustrate the theoretical stability and convergence results

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

fractional spectral collocation methods for linear and nonlinear variable order fpdes while several high order methods have been developed for fractional pdes fpdes with fixed order there are no such methods for fpdes with field variable order these equations allow multiphysics simulations seamlessly e g from diffusion to sub diffusion or from wave dynamics transitioning to diffusion by simply varying the fractional order as a function of space or time we develop an exponentially accurate fractional spectral collocation method for solving linear nonlinear fpdes with field variable order following the spectral theory developed in 1 for fractional sturm liouville eigenproblems we introduce a new family of interpolants called left right sided and central fractional lagrange interpolants we employ the fractional derivatives of left right sided riemann liouville and riesz type and obtain the corresponding fractional differentiation matrices by collocating the field variable fractional orders we solve several fpdes including time and space fractional advection equation time and space fractional advection diffusion equation and finally the space fractional burgers equation to demonstrate the performance of the method in addition we develop a spectral penalty method for enforcing inhomogeneous initial conditions our numerical results confirm the exponential like convergence of the proposed fractional collocation methods

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 unified spectral method for fpdes with two sided derivatives part ii stability and error analysis abstract we present the stability and error analysis of the unified petrov galerkin spectral method developed in 1 for linear fractional partial differential equations with two sided derivatives and constant coefficients in any 1 d dimensional space time hypercube d 1 2 3 subject to homogeneous dirichlet initial boundary conditions specifically we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method finally we perform several numerical simulations to compare the theoretical and computational rates of convergence

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

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space