improving performance of fxlms algorithm for active noise control of impulsive noise this paper concerns active noise control anc of impulsive noise modeled using non gaussian stable processes the most famous filtered x least mean square fxlms algorithm for anc systems is based on minimization of variance of error signal for the impulse noise the fxlms algorithm becomes unstable as second order moments do not exist for non gaussian processes among the existing algorithms for anc of impulsive noise one is based on minimizing least mean p power lmp of the error signal resulting in fxlmp algorithm the other is based on modifying on the basis of statistical properties the reference signal in update of fxlms algorithm in this paper the proposed algorithm is an extension of the later approach extensive simulations are carried out which demonstrate the effectiveness of the proposed algorithm it achieves the best performance among the existing algorithms and at the same computational complexity as that of fxlms algorithm

numerical analysis of a new space time variable fractional order advection dispersion equation many physical processes appear to exhibit fractional order behavior that may vary with time and or space the continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable in this paper we consider a new space time variable fractional order advection dispersion equation on a finite domain the equation is obtained from the standard advection dispersion equation by replacing the first order time derivative by coimbra s variable fractional derivative of order x 0 1 x 0 1 and the first order and second order space derivatives by the riemann liouville derivatives of order x t 0 1 x t 0 1 and x t 1 2 x t 1 2 respectively we propose an implicit euler approximation for the equation and investigate the stability and convergence of the approximation finally numerical examples are provided to show that the implicit euler approximation is computationally efficient

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

a continuous galerkin method for pseudo hyperbolic equations with variable coefficients abstract this paper we study a space time continuous galerkin stcg method with mesh change for two dimensional 2 d pseudo hyperbolic equations with variable coefficients which gives the same treatment for time and space discretizations namely both time and space variables are discretized via finite element fe method in addition it allows the change of time step and space mesh which are necessary for adaptive algorithms we prove the existence and uniqueness of approximation solution and give a prior error estimate without any conditions attached to the space time grid finally a numerical example is given to confirm the feasibility to the scheme constructed here

a priori estimates for mixed finite element approximations of second order hyperbolic equations with absorbing boundary conditions optimal order l infty in time l 2 in space a priori error estimates are derived for mixed finite element approximations for both displacement and stress for a second order hyperbolic equation with first order absorbing boundary conditions continuous in time explicit in time and implicit in time procedures are formulated and analyzed

on the time fractional navier stokes equations abstract this paper is concerned with the navier stokes equations with time fractional derivative of order 0 1 this type of equations can be used to simulate anomalous diffusion in fractal media we establish the existence and uniqueness of local and global mild solutions in h q meanwhile we also give local mild solutions in j q moreover we prove the existence and regularity of classical solutions for such equations in j q

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

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

analysis of ciarlet raviart mixed finite element methods for solving damped boussinesq equation abstract in this paper we consider the numerical solution of damped boussinesq equation using ciarlet raviart mixed finite element method an implicit finite difference scheme is used for the time discretization a priori error estimates are analyzed and stability analysis of the method is shown we obtain an optimal error estimate in l 2 norm with quadratic or higher order element for both semi and fully discrete finite element approximations finally numerical examples are given to verify the theoretical results

second order approximations for variable order fractional derivatives fractional calculus allows variable order of fractional operators which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and or time in this paper we derive two second order approximation formulas for the variable order fractional time derivatives involved in anomalous diffusion and wave propagation we then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub diffusion and super diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable order function finally we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super diffusion and also simulations of the burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and or time taken together our results demonstrate that variable order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long standing issues of outflow boundary conditions and monotonicity of integer order pdes

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

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

self similarity versus self affinity the sierpinski gasket revisited a hilbert curve is constructed on the sierpinski gasket it is shown that despite the fact that the sierpinski gasket is rigorously self similar the hilbert curve is merely self affine

a new robust variable step size nlms algorithm a new framework for designing robust adaptive filters is introduced it is based on the optimization of a certain cost function subject to a time dependent constraint on the norm of the filter update particularly we present a robust variable step size nlms algorithm which optimizes the square of the a posteriori error we also show the link between the proposed algorithm and another one derived using a robust statistics approach in addition a theoretical model for predicting the transient and steady state behavior and a proof of almost sure filter convergence are provided the algorithm is then tested in different environments for system identification and acoustic echo cancelation applications

nonlinear system identification in impulsive environments nonlinear system identification has been studied under the assumption that the noise has finite second and higher order statistics in many practical applications impulsive measurement noise severely weakens the effectiveness of conventional methods in this paper spl alpha stable noise is used as a noise model in such case the minimum mean square error mmse criterion is no longer an appropriate metric for estimation error due to the lack of finite second order statistics of the noise therefore we adopt minimum dispersion criterion which in turn leads to the adaptive least mean pth power lmp algorithm it is shown that the lmp algorithm under the spl alpha stable noise model converges as long as the step size satisfies certain conditions the effect of p on the performance is also investigated compared with conventional methods the proposed method is more robust to impulsive noise and has better performance

finite element methods for nonlinear sobolev equations with nonlinear boundary conditions abstract we derive optimal l 2 error estimates for semi discrete finite element methods for nonlinear sobolev equations with nonlinear boundary conditions a new projection is introduced and used in the error analysis

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

second order approximation scheme combined with h1 galerkin mfe method for nonlinear time fractional convection diffusion equation abstract in this article a second order approximation scheme combined with an h 1 galerkin mixed finite element mfe method for solving nonlinear convection diffusion equation with time fractional derivative is proposed and analyzed by introducing an auxiliary variable a coupled system is formulated then the spatial direction is approximated by h 1 galerkin mfe method and the temporal fractional derivative and integer derivative are discretized by second order weighted and shifted grunwald difference wsgd formula and linearized second order difference scheme respectively the optimal priori error estimates in l 2 and h 1 norm for the unknown function and the auxiliary variable with second order convergent rate in time are obtained compared to the commonly used l1 approximation with 2 th order convergence rate our method can arrive at the order 2 in time what is more compared with the standard finite element method our method can well approximate the auxiliary variable finally the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis

fractal sequences derived from the self similar extensions of the sierpinski gasket the well known sierpinski gasket has been generalised by the authors into the generalised pascal sierpinski gaskets gpsg of orders k l where both k and l are or 2 it has been shown here that families of self similar sequences can be derived from these extensions of the sierpinski gasket when k 2 and l is a prime number

high order finite difference methods for a second order dual phase lagging models of microscale heat transfer two adi solvers are given for 2d and 3d heat conduct equations at microscale they are uniquely solvable it is shown that they are of order 2 in time and order 4 in space numerical results exhibit the efficiency of the solvers in this paper a compact alternating direction implicit adi method which combines the fourth order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators is firstly presented for solving two dimensional 2d second order dual phase lagging models of microscale heat transfer by the discrete energy method it is shown that it is second order accurate in time and fourth order accurate in space with respect to l2 norms additionally the compact adi method is successfully generalized to solve corresponding three dimensional 3d problem also the convergence result of the solver for 3d case is given rigorously finally numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results

a high order exponential scheme for solving 1d unsteady convection diffusion equations in this paper a high order exponential hoe scheme is developed for the solution of the unsteady one dimensional convection diffusion equation the present scheme uses the fourth order compact exponential difference formula for the spatial discretization and the 2 2 pade approximation for the temporal discretization the proposed scheme achieves fourth order accuracy in temporal and spatial variables and is unconditionally stable numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature the results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems

numerical algorithm of two mixed finite element schemes for a fourth order diffusion equation two numerical algorithms based on h 1 galerkin mixed finite element gmfe methods are presented and analyzed for a fourth order diffusion equation by introducing three variables with two different ways the two first order system of four equations are formulated the semi discrete optimal error estimates are derived for a numerical scheme and the fully discrete analysis of optimal error results for the other one are made moreover the numerical procedures for two numerical schemes are made to confirm the theoretical results of theorems

on the spatial fourier transforms of the pascal sierpinski gaskets it has been shown here that the fourier transform which can be obtained experimentally of an evolving fractal can be manipulated to yield the fractal dimension although only the pascal sierpinski gaskets have been considered here because of their generality it is expected that this technique can be utilized to identify other planar fractals as well

expanded mixed finite element method for second order hyperbolic equations abstract in this paper an expanded mixed finite element method is developed to approximate the solution of second order hyperbolic equations we discuss and theoretically prove the existence and uniqueness of solution for the semidiscrete formulation and obtain the optimal order error estimates in the l l 2 norm in addition we also construct the fully discrete approximations and provide the corresponding error analysis some numerical examples are presented to demonstrate the theoretical analysis

h1 galerkin mixed finite element methods for pseudo hyperbolic equations h 1 galerkin mixed finite element methods are discussed for a class of second order pseudo hyperbolic equations depending on the physical quantities of interest two methods are discussed optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension an extension to problems in two and three space variables is also discussed the existence and uniqueness are derived and it is showed that the h 1 galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the lbb consistency condition

adaptive volterra filter with continuous lp norm using a logarithmic cost for nonlinear active noise control abstract the filtered x least mean lp norm fxlmp algorithm is proven to be useful for nonlinear active noise control nanc systems however its performance deteriorates when the impulsive noises are presented in nanc systems to surmount this shortcoming a new nonlinear adaptive algorithm based on volterra expansion model vfxloglmp is developed in this paper which is derived by minimizing the lp norm of logarithmic cost it is found that the fxlmp and vfxloglmp require to select an appropriate value of p according to the prior information on noise characteristics which prohibit their practical applications based on vfxloglmp algorithm we proposed a continuous lp norm algorithm with logarithmic cost vfxlogclmp which does not need the parameter selection and thresholds estimation benefiting from the various error norms for 1 p 2 it remains the robustness of vfxloglmp moreover the convergence behavior of vfxlogclmp for moving average secondary paths and stochastic input signals is performed compared to the existing algorithms two versions of the proposed algorithms have much better convergence and stability in impulsive noise environments

an improved tau method for the multi dimensional fractional rayleigh stokes problem for a heated generalized second grade fluid abstract we develop efficient algorithms based on the legendre tau approximation for one and two dimensional fractional rayleigh stokes problems for a generalized second grade fluid the time fractional derivative is described in the riemann liouville sense discussions on the l 2 convergence of the proposed method are presented numerical results for one and two dimensional examples with smooth and nonsmooth solutions are provided to verify the validity of the theoretical analysis and to illustrate the efficiency of the proposed algorithms

comparative numerical analysis using reduced order modeling strategies for nonlinear large scale systems we perform a comparative analysis using three reduced order strategies missing point estimation mpe method gappy pod method and discrete empirical interpolation method deim applied to a biological model describing the spatio temporal dynamics of a predator prey community the comparative study is focused on the efficiency of the reduced order approximations and the complexity reduction of the nonlinear terms different variants are discussed related to the projection based model reduction framework combined with selective spatial sampling to efficiently perform the online computations numerical results are presented

on a new class of planar fractals the pascal sierpinski gaskets a new class of planar fractals called the pascal sierpinski gaskets is described of which the well known sierpinski gasket is a special case some of these gaskets are true mandelbrot fractals possessing non integral dimensions as well as self similarity the remaining ones are not self similar but appear to have nonintegral dimensions

a high order exponential adi scheme for two dimensional time fractional convection diffusion equations in this paper a high order compact exponential alternating direction implicit adi scheme for two dimensional fractional convection diffusion equations is proposed with o t 3 c h 1 4 h 2 4 accuracy where t h 1 h 2 are the temporal and spatial step sizes respectively the convergence of the finite difference scheme is studied using its matrix form by the energy method difficulty arising from the convection term is overcome by an analysis based on the eigenvalue decomposition of nonsymmetric tridiagonal matrices

active noise control with on line estimation of non gaussian noise characteristics abstract active noise control anc is a methodology for attenuating noise based on adaptive signal processing algorithms anc is well assessed for the attenuation of gaussian noise but the rejection of non gaussian impulsive noise signals represents a much more critical task that may even impair algorithm convergence to overcome this problem the adaptive filter weight update process must be modified by discarding or discounting samples associated with impulsive noise this can be done either by modeling the impulsive noise with a non gaussian distribution such as the symmetric stable s s distribution or by applying an outlier detection method with both approaches the accuracy in the noise description appears to be crucial for effective noise reduction this paper proposes two novel approaches for the attenuation of impulsive noise both for invariant and time varying noise distributions the first one is based on the on line estimation of an s s model of the noise probabilistic description the second relies on a simple on line recursive procedure that reliably estimates amplitude thresholds for outlier detection both methods compare favorably with competitor approaches while maintaining a sufficiently low algorithm complexity several examples are shown to demonstrate the algorithms effectiveness

a high order compact exponential scheme for the fractional convection diffusion equation a high order compact exponential finite difference scheme for solving the fractional convection diffusion equation is considered in this paper the convection and diffusion terms are approximated by a compact exponential finite difference scheme with a high order approximation for the caputo time derivative for this fully discrete implicit scheme the local truncation error is analyzed and the fourier method is used to discuss the stability the error estimate is given by the discrete energy method numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

highly accurate numerical schemes for multi dimensional space variable order fractional schrdinger equations as a natural generalization of the fractional schrdinger equation the variable order fractional schrdinger equation has been exploited to study fractional quantum phenomena in this paper we develop an exponentially accurate jacobigausslobatto collocation jgl c method to solve the variable order fractional schrdinger equations in one dimension 1d and two dimensions 2d in this method the aforementioned problem is reduced to a system of ordinary differential equations odes in the time variable as a result we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations one based on the implicit rungekutta irk method of fourth order and the other based on jacobigaussradau collocation jgr c method the validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1d and 2d the convergence of the methods is graphically analyzed the results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable order nonlinear partial differential equations

analysis of mixed finite element method mfem for solving the generalized fractional reaction diffusion equation on nonrectangular domains abstract in the current manuscript we consider a generalized fractional reaction diffusion equation the considered model is based on the time fractional derivative the developed scheme is based on two procedures at first we obtain a semi discrete scheme for the temporal direction the time fractional derivative is discretized by using a difference scheme with second order accuracy thus the final time discrete formula has second order accuracy in the temporal direction then we obtain a fully discrete scheme by applying the mixed finite element method mfem for the constructed numerical technique we prove the unconditional stability and also obtain an error bound for the full discrete scheme by using the energy method we employ some test problems to show the accuracy of the proposed technique

fractal dimension from the back scattering cross section the feasibility of recovering the fractal dimension of self similar fractals from the back scattering measurements is demonstrated for this purpose the well known sierpinski gasket is modelled as an ensemble of lossless dielectric beads and the scattered field computations are carried out using the coupled dipole approximation procedure

high accuracy analysis of an h1 galerkin mixed finite element method for two dimensional time fractional diffusion equations abstract in this paper an h 1 galerkin mixed finite element approximate scheme is established for a class of two dimensional time fractional diffusion equations by using the bilinear element raviart thomas element and l 1 time stepping method which is unconditionally stable and free of lbb condition and then without the classical ritz projection superclose results for the original variable u in h 1 norm and the flux p u in h d i v norm are derived by means of properties of the elements and l 1 approximation furthermore with the help of the interpolation postprocessing operator the global superconvergence results for the original variable u in h 1 norm are obtained finally numerical simulations verify that the theoretical results are true on both regular meshes and anisotropic meshes

improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper the methods of analysis currently available in literature assume that the input signals to the filter are white this restriction is removed for gaussian signals in our analysis expressions for the second moment of the coefficient vector and the steady state error power are also derived simulation results are presented and the theoretical and empirical curves show a very good match

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

adaptive volterra filters for active control of nonlinear noise processes this paper presents a volterra filtered x least mean square lms algorithm for feedforward active noise control the research has demonstrated that linear active noise control anc systems can be successfully applied to reduce the broadband noise and narrowband noise specifically such linear anc systems are very efficient in reduction of low frequency noise however in some situations the noise that comes from a dynamic system may he a nonlinear and deterministic noise process rather than a stochastic white or tonal noise process and the primary noise at the canceling point may exhibit nonlinear distortion furthermore the secondary path estimate in the anc system which denotes the transfer function between the secondary source secondary speaker and the error microphone may have nonminimum phase and hence the causality constraint is violated if such situations exist the linear anc system will suffer performance degradation an implementation of a volterra filtered x lms vfxlms algorithm based on a multichannel structure is described for feedforward active noise control numerical simulation results show that the developed algorithm achieves performance improvement over the standard filtered x lms algorithm for the following two situations 1 the reference noise is a nonlinear noise process and at the same time the secondary path estimate is of nonminimum phase 2 the primary path exhibits the nonlinear behavior in addition the developed vfxlms algorithm can also be employed as an alternative in the case where the standard filtered x lms algorithm does not perform well

least squares galerkin procedures for pseudohyperbolic equations abstract two least squares galerkin finite element schemes are formulated to solve pseudohyperbolic equations the advantage of this method is that it is not subject to the lbb condition the convergence analysis shows that the methods both yield the approximate solutions with optimal accuracy in l 2 2 l 2 moreover the two methods get the approximate solutions with first order and second order accuracy in time increment respectively numerical example shows that the two schemes are effective

fractal structures derivable from the generalisations of the pascal triangle generalisations of order k or 2 of the pascal triangle are used to construct generalised pascal sierpinski gaskets of orders k l or 2 it is shown that all such gaskets are self affine fractals but when k 2 and l is prime then the gasket is rigorously self similar and possesses a similarity dimension the evolutionary morphology of the gaskets of orders k l prime bears a resemblance to the growth of pyrolitic graphite films and other material structures

an h 1 galerkin mixed finite element method for parabolic partial differential equations in this paper an h1 galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts compared to the standard h1 galerkin procedure c1 continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method moreover it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method but without lbb consistency condition and quasiuniformity requirement on the finite element mesh finally a better rate of convergence for the flux in l2 norm is derived using a modified h1 galerkin mixed method in two and three space dimensions which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution

numerical solution of distributed order fractional differential equations by hybrid functions in this paper a new numerical method for solving the distributed fractional differential equations is presented the method is based upon hybrid functions approximation the properties of hybrid functions consisting of block pulse functions and bernoulli polynomials are presented the riemann liouville fractional integral operator for hybrid functions is introduced this operator is then utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

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

domain decomposition procedures combined with h1 galerkin mixed finite element method for parabolic equation abstract non overlapping domain decomposition procedures are considered for parabolic equation these procedures are combined with using h 1 galerkin mixed finite element method in the sub domains to approximate the primary variable u and its flux simultaneously explicit calculations are built by using integral mean methods to present the inter domain boundary conditions for the flux thus the parallelism can be achieved by these procedures two approximation schemes are established time step constraints are proved necessary to preserve stability which are less severe than that of fully explicit galerkin finite element method the mixed finite element spaces are allowed to be of different polynomial degrees and not subject to the lbb consistency condition new nonstandard elliptic projections are defined and analyzed optimal error estimates for the variable u in h 1 norm and its flux in l 2 norm and are derived for these schemes numerical experiments are presented to confirm the theoretical results

an unconditionally stable compact adi method for three dimensional time fractional convection diffusion equation abstract a high order compact finite difference method is presented for solving the three dimensional 3d time fractional convection diffusion equation of order 1 2 the original equation is first transformed to a fractional diffusion wave equation then using fourth order pade approximation for spatial derivatives and the center difference method for time derivative respectively a fully discrete implicit compact scheme is obtained furthermore based on different splitting terms three unconditionally stable adi compact schemes with optimal convergence order are developed respectively the resulting schemes in each adi solution step corresponding to a strictly diagonally dominant matrix equation can be solved using the 1d tridiagonal thomas algorithm with a considerable saving in computing time numerical experiments show that these schemes can significantly improve the time accuracy

direct meshless local petrov galerkin dmlpg method for time fractional fourth order reaction diffusion problem on complex domains abstract a new numerical scheme has been developed based on the fast and efficient meshless local weak form i e direct meshless local petrov galerkin dmlpg method for solving the fractional fourth order partial differential equation on computational domains with complex shape the fractional derivative is the riemann liouville fractional derivative at first a finite difference scheme with the second order accuracy has been employed to discrete the time variable then the dmlpg technique is employed to achieve a full discrete scheme the time discrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the l 2 space also some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and non regular grid points

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

compact exponential scheme for the time fractional convection diffusion reaction equation with variable coefficients high order compact exponential finite difference scheme for solving the time fractional convection diffusion reaction equation with variable coefficients is considered in this paper the convection diffusion and reaction coefficients can depend on both the spatial and temporal variables we begin with the one dimensional problem and after transforming the original equation to one with diffusion coefficient unity the new equation is discretized by a compact exponential finite difference scheme with a high order approximation for the caputo time derivative we prove the solvability of this fully discrete implicit scheme and analyze its local truncation error for the fractional equation with constant coefficients we use fourier method to prove the stability and utilize matrix analysis as a tool for the error estimate then we discuss the two dimensional problem give the compact adi scheme with the restriction that besides the time variable the convection coefficients can only depend on the corresponding spatial variables respectively numerical results are provided to verify the accuracy and efficiency of the proposed algorithm the time fractional convection diffusion reaction equation is numerically solved compact exponential scheme for the variable coefficients equation is given fourier method and matrix analysis are used for the analysis the compact exponential adi scheme is given for the two dimensional problem high accuracy and good efficiency of the proposed algorithm is demonstrated

superconvergence of an h1 galerkin nonconforming mixed finite element method for a parabolic equation by choosing a suitable pair of approximating spaces an h 1 galerkin nonconforming mixed finite element method fem is proposed for a class of parabolic equations under semi discrete backward euler and crank nicolson fully discrete schemes in which the famous eq 1 r o t element and zero order raviart thomas element are used to approximate the primitive solution u and the flux p email protected u respectively based on special characters of the elements considered the corresponding optimal order error estimates for u in broken h 1 norm and p in h div norm are obtained for the above schemes furthermore the global superconvergence results are derived through the postprocessing technique the numerical results show the validity of the theoretical analysis