a finite element approximation for a class of caputo time fractional diffusion equations abstract we develop a fully discrete scheme for time fractional diffusion equations by using a finite difference method in time and a finite element method in space the fractional derivatives are used in caputo sense stability and error estimates are derived the accuracy and efficiency of the presented method is shown by conducting two numerical examples

variable order fractional differential operators in anomalous diffusion modeling abstract the purpose of this paper is to offer a unified discussion of variable order differential operators in anomalous diffusion modeling the characteristics of the new models in contrast to constant order fractional diffusion models change with time space concentration or other independent quantities we introduced a classification of variable order fractional diffusion models based on the possible physical origins which prompt the variable order some potential applications of the variable order fractional diffusion models are also discussed

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

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

contribution to the determination of vanishing points using hough transform we propose a method to locate three vanishing points on an image corresponding to three orthogonal directions of the scene this method is based on two cascaded hough transforms we show that even in the case of synthetic images of high quality a naive approach may fail essentially because of the errors due to the limitation of the image size we take into account these errors as well as errors due to detection inaccuracy of the image segments and provide a method efficient even in the case of real complex scenes

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

particle occupation factors without large number approximations abstract new derivations of particle occupation factors that are based on mean values and do not require large number approximations lna are provided for fermions bosons and boltzmann particles the derivations are closely related to traditional combinatorial approaches so the physical content of the latter is preserved without recourse to most probable values or lna the approach is based on the use of eigenvalues of the density matrix as probabilities entering into the shannon entropy the degeneracies of the eigenvalues of the density matrix can be identified with boltzmann enumeration factors problems associated with steepest descent procedures limit theorem techniques and the boltzmann thermodynamic probability are avoided in addition since the approach is based on the eigenvalues of the density matrix the concept of ensembles is not required

maximum principle and numerical method for the multi term time space riesz caputo fractional differential equations the maximum principle for the space and time space fractional partial differential equations is still an open problem in this paper we consider a multi term time space riesz caputo fractional differential equations over an open bounded domain a maximum principle for the equation is proved the uniqueness and continuous dependence of the solution are derived using a fractional predictor corrector method combining the l1 and l2 discrete schemes we present a numerical method for the specified equation two examples are given to illustrate the obtained results

error estimates of crank nicolson type difference schemes for the subdiffusion equation a crank nicolson type difference scheme is proposed for solving the subdiffusion equation with fractional derivative and the truncation error is analyzed in detail at each temporal level only a tridiagonal linear system needs to be solved and the thomas algorithm may be used the solvability unconditional stability and h 1 norm convergence are proved the convergence order is rm min 2 gamma 2 1 gamma in the temporal direction and two in the spatial direction by the sobolev embedding inequality we obtain the maximum norm error estimate a spatial compact scheme based on the crank nicolson type difference scheme is also presented and similar results are given the convergence order is mathcal o tau rm min 2 gamma 2 1 gamma h 4 numerical experiments are included to support the theoretical results and comparisons with the related works are presented to show the effectiveness of our 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

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

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

anisotropic linear triangle finite element approximation for multi term time fractional mixed diffusion and diffusion wave equations with variable coefficient on 2d bounded domain abstract a fully discrete approximate scheme is established for the 2d multi term time fractional mixed diffusion and diffusion wave equations with spatial variable coefficient by using linear triangle finite element method in space and classical l 1 time stepping method combined with crank nicolson scheme in time then the unconditionally stable analysis of the fully discrete scheme is presented by employing some important lemmas at the same time both the spatial superclose property in h 1 norm and convergence result in l 2 norm are derived by skillfully dealing with numerical errors without any restrictions of time step and mesh size h as a necessary way for obtaining the aimed numerical analysis the relationship between the nonstandard projection operator r h and the interpolation operator i h of linear triangle finite element is introduced moreover the global superconvergence is deduced by adopting interpolation postprocessing technique finally numerical tests are provided to illustrate the efficiency and correctness of the theoretical results

a novel approach of high accuracy analysis of anisotropic bilinear finite element for time fractional diffusion equations with variable coefficient abstract in this paper by using bilinear finite element and l 1 approximation a fully discrete scheme is established for time fractional diffusion equation with variable coefficient on anisotropic meshes unconditionally stable analysis of the proposed scheme are presented in both l 2 norm and h 1 norm moreover convergence superclose and superconvergence results are derived by combining interpolation with projection which is the key technique for the numerical analysis specifically by defining a novel projection operator the error estimate between the projection and the exact solution is obtained on anisotropic meshes furthermore high accuracy analysis on interpolation of bilinear finite element and projection is gained by means of some known results about the interpolation and mean value technique based on the related results about projection and interpolation the optimal error estimate in l 2 norm and superclose of interpolation in h 1 norm are deduced by skillfully dealing with fractional derivative at the same time the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

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

initial boundary value problems for multi term time fractional diffusion equations with positive constant coefficients in this paper we investigate the well posedness and the long time asymptotic behavior for initial boundary value problems for multi term time fractional diffusion equations the governing equation under consideration includes a linear combination of caputo derivatives in time with decreasing orders in 0 1 and positive constant coefficients by exploiting several important properties of multinomial mittag leffler functions various estimates follow from the explicit solutions in form of these special functions then we prove the uniqueness and continuous dependency on initial values and source terms from which we further verify the lipschitz continuous dependency of solutions with respect to coefficients and orders of fractional derivatives finally by a laplace transform argument it turns out that the decay rate of the solution as t is given by the minimum order of the time fractional derivatives

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

numerical approximation of higher order time fractional telegraph equation by using a combination of a geometric approach and method of line we propose a simple and accurate numerical scheme for solving the time fractional telegraph tft equation within caputo type fractional derivative a fictitious coordinate is imposed onto the problem in order to transform the dependent variable u x t into a new variable with an extra dimension in the new space with the added fictitious dimension a combination of method of line and group preserving scheme gps is proposed to find the approximate solutions this method preserves the geometric structure of the problem power and accuracy of this method has been illustrated through some examples of tft equation

subordination approach to multi term time fractional diffusion wave equations abstract this paper is concerned with the fractional evolution equation with a discrete distribution of caputo time derivatives such that the largest and the smallest orders and m satisfy the conditions 1 2 and m 1 first based on a study of the related propagation function the nonnegativity of the fundamental solutions to the spatially one dimensional cauchy and signaling problems is proven and propagation speed of a disturbance is discussed next we study the equation with a general linear spatial differential operator defined in a banach space and suppose it generates a cosine family a subordination principle is established which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function explicit representation of the probability density function is derived the subordination principle is applied for obtaining regularity results the analytical findings are supported by numerical work

analytical time domain green s functions for power law media frequency dependent loss and dispersion are typically modeled with a power law attenuation coefficient where the power law exponent ranges from 0 to 2 to facilitate analytical solution a fractional partial differential equation is derived that exactly describes power law attenuation and the szabo wave equation time domain wave equations for lossy media obeying a frequency power law j acoust soc am 96 491 500 1994 is an approximation to this equation this paper derives analytical time domain green s functions in power law media for exponents in this range to construct solutions stable law probability distributions are utilized for exponents equal to 0 1 3 1 2 2 3 3 2 and 2 the green s function is expressed in terms of dirac delta exponential airy hypergeometric and gaussian functions for exponents strictly less than 1 the green s functions are expressed as fox functions and are causal for exponents greater than or equal than 1 the green s functions are expressed as fox and

a diffusion wave equation with two fractional derivatives of different order we analyse a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains thus our model represents a generalized telegraph equation solutions to signalling and cauchy problems in terms of a series and integral representation are given classical wave and heat conduction equations are obtained as limiting cases

local discontinuous galerkin method for a nonlinear time fractional fourth order partial differential equation abstract in this article a fully discrete local discontinuous galerkin ldg method with high order temporal convergence rate is presented and developed to look for the numerical solution of nonlinear time fractional fourth order partial differential equation pde in the temporal direction for approximating the fractional derivative with order 0 1 the weighted and shifted grunwald difference wsgd scheme with second order convergence rate is introduced and for approximating the integer time derivative two step backward euler method with second order convergence rate is used for the spatial direction the ldg method is used for the numerical theories the stability is derived and a priori error results are proved further some error results and convergence rates are calculated by numerical procedure to illustrate the effectiveness of proposed method

orientation diffusions diffusions are useful for image processing and computer vision because they provide a convenient way of smoothing noisy data analyzing images at multiple scales and enhancing discontinuities a number of diffusions of image brightness have been defined and studied so far they may be applied to scalar and vector valued quantities that are naturally associated with intervals of either the real line or other flat manifolds some quantities of interest in computer vision and other areas of engineering that deal with images are defined on curved manifolds typical examples are orientation and hue that are defined on the circle generalizing brightness diffusions to orientation is not straightforward especially in the case where a discrete implementation is sought an example of what may go wrong is presented a method is proposed to define diffusions of orientation like quantities first a definition in the continuum is discussed then a discrete orientation diffusion is proposed the behavior of such diffusions is explored both analytically and experimentally it is shown how such orientation diffusions contain a nonlinearity that is reminiscent of edge process and anisotropic diffusion a number of open questions are proposed

error estimates for a semidiscrete finite element method for fractional order parabolic equations we consider the initial boundary value problem for a homogeneous time fractional diffusion equation with an initial condition v x and a homogeneous dirichlet boundary condition in a bounded convex polygonal domain omega we study two semidiscrete approximation schemes i e the galerkin finite element method fem and lumped mass galerkin fem using piecewise linear functions we establish almost optimal with respect to the data regularity error estimates including the cases of smooth and nonsmooth initial data i e v in h 2 omega cap h 1 0 omega and v in l 2 omega for the lumped mass method the optimal l 2 norm error estimate is valid only under an additional assumption on the mesh which in two dimensions is known to be satisfied for symmetric meshes finally we present some numerical results that give insight into the reliability of the theoretical study

dequantizing image orientation we address the problem of computing a local orientation map in a digital image we show that standard image gray level quantization causes a strong bias in the repartition of orientations hindering any accurate geometric analysis of the image in continuation a simple dequantization algorithm is proposed which maintains all of the image information and transforms the quantization noise in a nearby gaussian white noise we actually prove that only gaussian noise can maintain isotropy of orientations mathematical arguments are used to show that this results in the restoration of a high quality image isotropy in contrast with other classical methods it turns out that this property can be obtained without smoothing the image or increasing the signal to noise ratio snr as an application it is shown in the experimental section that thanks to this dequantization of orientations such geometric algorithms as the detection of nonlocal alignments can be performed efficiently we also point out similar improvements of orientation quality when our dequantization method is applied to aliased images

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

comparative study of three numerical schemes for fractional integro differential equations this paper presents a comparative study of three numerical schemes such as linear quadratic and quadratic linear scheme for the fractional integro differential equations defined in terms of the caputo fractional derivatives the error estimates of the respective approximations are also established numerical tests of the discussed schemes show that all schemes work well and when the number of terms approximating the solution is increased the desired solution is achieved the accuracy of the numerical schemes with respect to the step size h is analysed and illustrated through various tables finally comparative performances of the schemes are discussed comparative study of three schemes such as linear quadratic and quadratic linear schemes is presented for the fractional integro differential equations the convergences of the schemes are also established the performances of the schemes are compared through the test examples

convergence and superconvergence of a fully discrete scheme for multi term time fractional diffusion equations using finite element method in spatial direction and classical l1 approximation in temporal direction a fully discrete scheme is established for a class of two dimensional multi term time fractional diffusion equations with caputo fractional derivatives the stability analysis of the approximate scheme is proposed the spatial global superconvergence and temporal convergence of order o h2 2 for the original variable in h1 norm is presented by means of properties of bilinear element and interpolation postprocessing technique where h and are the step sizes in space and time respectively finally several numerical examples are implemented to evaluate the efficiency of the theoretical results

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

a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation in this paper a new alternating direction implicit galerkin legendre spectral method for the two dimensional riesz space fractional nonlinear reaction diffusion equation is developed the temporal component is discretized by the crank nicolson method the detailed implementation of the method is presented the stability and convergence analysis is strictly proven which shows that the derived method is stable and convergent of order 2 in time an optimal error estimate in space is also obtained by introducing a new orthogonal projector the present method is extended to solve the fractional fitzhugh nagumo model numerical results are provided to verify the theoretical analysis

numerical reconstruction of heat fluxes this paper studies the reconstruction of heat fluxes on an inner boundary of a heat conductive system when the measurement of temperature in a small subregion near the outer boundary of the physical domain is available we will first consider two different regularization formulations for this severely ill posed inverse problem and justify their well posedness then we will propose two fully discrete finite element methods to approximate the resultant nonlinear minimization problems the existence and uniqueness of the discrete minimizers and convergence of the finite element solution are rigorously demonstrated a conjugate gradient method is formulated to solve the nonlinear finite element optimization problems numerical experiments are given to demonstrate the stability and effectiveness of the proposed reconstruction methods

finite difference methods and a fourier analysis for the fractional reaction subdiffusion equation various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations fpde for example systems biology chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments however effective numerical methods and numerical analysis for fpde are still in their infancy in this paper we consider a fractional reaction subdiffusion equation fr subde in which both the motion and the reaction terms are affected by the subdiffusive character of the process using the relationship between the riemann liouville and grunwald letnikov definitions of fractional derivatives an implicit and an explicit difference methods for the fr subde are presented the stability and the convergence of the two numerical methods are investigated by a fourier analysis the solvability of the implicit finite difference method is also proved the high accuracy algorithm is structured using richardson extrapolation finally a comparison between the exact solution and the two numerical solutions is given the numerical results are in excellent agreement with our theoretical analysis

strong maximum principle for multi term time fractional diffusion equations and its application to an inverse source problem abstract in this paper we establish a strong maximum principle for fractional diffusion equations with multiple caputo derivatives in time and investigate a related inverse problem of practical importance exploiting the solution properties and the involved multinomial mittag leffler functions we improve the weak maximum principle for the multi term time fractional diffusion equation to a stronger one which is parallel to that for its single term counterpart as expected as a direct application we prove the uniqueness for determining the temporal component of the source term with the help of the fractional duhamel s principle for the multi term case

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

numerical simulations of 2d fractional subdiffusion problems the growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes currently most algorithms are designed for 1d problems due to the memory effect in fractional derivatives in this work the 2d fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach the proposed algorithm is also used to simulate a subdiffusion convection equation

numerical identification of a robin coefficient in parabolic problems this paper studies a regularization approach for an inverse problem of estimating a spatially and temporally dependent robin coefficient arising in the analysis of convective heat transfer the parameter to state map is analyzed especially a differentiability result is established a regularization approach is proposed and the properties e g existence and optimality system of the functional are investigated a finite element method is adopted for discretizing the continuous optimization problem and the convergence of the finite element approximations as the mesh size and temporal step size tend to zero is established numerical results by the conjugate gradient method for oneand two dimensional problems are presented

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

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

a compact exponential difference method for multi term time fractional convection reaction diffusion problems with non smooth solutions abstract this paper is concerned with a numerical method for a class of one dimensional multi term time fractional convection reaction diffusion problems where the differential equation contains a sum of the caputo time fractional derivatives of different orders between 0 and 1 in general the solutions of such problems typically exhibit a weak singularity at the initial time a compact exponential finite difference method using the well known l1 formula for each time fractional derivative and a fourth order compact exponential difference approximation for the spatial discretization is proposed on a mesh that is generally nonuniform in time and uniform in space taking into account the initial weak singularity of the solution the stability and convergence of the method is proved and the optimal error estimate in the discrete l2 norm is obtained by developing a discrete energy analysis technique which enables us to overcome the difficulties caused by the nonsymmetric discretization matrices the error estimate shows that the method has the spatial fourth order convergence and reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence the extension of the method to two dimensional problems is also discussed numerical results confirm the theoretical convergence result and show the applicability of the method to convection dominated problems

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

the galerkin finite element method for a multi term time fractional diffusion equation we consider the initial boundary value problem for a diffusion equation involving multiple time fractional derivatives on a bounded convex polyhedral domain we analyze a space semidiscrete scheme based on the standard galerkin finite element method using continuous piecewise linear functions nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived which cover both smooth and nonsmooth data further we develop a fully discrete scheme based on a finite difference discretization of the time fractional derivatives and discuss its stability and error estimate extensive numerical experiments for one and two dimensional problems confirm the theoretical convergence rates

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

vanishing point detection without any a priori information even though vanishing points in digital images result from parallel lines in the 3d scene most of the proposed detection algorithms are forced to rely heavily either on additional properties like orthogonality or coplanarity and equal distance of the underlying 3d lines or on knowledge of the camera calibration parameters in order to avoid spurious responses in this work we develop a new detection algorithm that relies on the helmoltz principle recently proposed for computer vision by desolneux et al 2001 2003 both at the line detection and line grouping stages this leads to a vanishing point detector with a low false alarms rate and a high precision level which does not rely on any a priori information on the image or calibration parameters and does not require any parameter tuning

gradient schemes for the signorini and the obstacle problems and application to hybrid mimetic mixed methods gradient schemes is a framework which enables the unified convergence analysis of many different methods such as finite elements conforming non conforming and mixed and finite volumes methods for 2nd order diffusion equations we show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed dirichlet neumann and signorini boundary conditions this extension allows us to provide error estimates for numerical approximations of such models recovering known convergence rates for some methods and establishing new convergence rates for schemes not previously studied for variational inequalities the general framework we develop also enables us to design a new numerical method for the obstacle and signorini problems based on hybrid mimetic mixed schemes we provide numerical results that demonstrate the accuracy of these schemes and confirm our theoretical rates of convergence

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

a new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion wave equations abstract we consider as discretization in space the gdm gradient discretization method developed recently in droniou et al 2018 to approximate multidimensional time fractional diffusion and diffusion wave equations where the fractional order of the time derivative is respectively satisfying 0 1 and 1 2 the time fractional derivative is given in the caputo sense the time discretization is performed using a uniform mesh for the time fractional diffusion equation we derive an implicit scheme in addition to an l l 2 a priori estimate which can be derived from a reasoning in bradji 2018 we present and prove a new l 2 h 0 1 a priori estimate for the discrete problem under the assumption that the exact solution is sufficiently smooth these a priori estimates allow to prove error estimates in discrete norms of l 2 h 0 1 and l l 2 the convergence order in these estimates is optimal in the sense of two points of view the first one is that the order in space is the same one proved in droniou et al 2018 for elliptic equation whereas the second point of view is that the particular case of this order when 1 is the same one obtained in droniou et al 2018 for the standard heat equation for the time fractional diffusion wave equation we derive two implicit schemes a full convergence analysis is carried out for both schemes in particular we develop new a priori estimates which yield error estimates in several discrete norms for each scheme the convergence is proved to be unconditional and optimal the convergence results are obtained thanks to a comparison with appropriately chosen auxiliary gradient schemes and to the stated a priori estimates these results improve the ones of bradji 2017 in which a conditional convergence is proved for a sushi scheme using stabilization and hybrid interfaces approximating a time fractional diffusion wave equation for both cases of time fractional diffusion and diffusion wave equations we show the well posedness in the sense that the discrete solutions depend continuously on the data of the considered problems the stated results can be extended to multi term time fractional diffusion and diffusion wave equations one of the main features when using gdm is that their results hold for all the schemes within the framework of gdm conforming and nonconforming finite element mixed finite element hybrid mixed mimetic family some multi point flux approximation finite volume schemes and some discontinuous galerkin schemes some examples of schemes recovered by the gss gradient schemes presented in this work are sketched among these examples we quote the one presented in jin et al 2015 to approximate time fractional diffusion equations this work extends and improves some results presented in brief or stated without proof in the notes bradji 2018 we present some numerical tests using sushi introduced in eymard et al 2010 to support the theoretical results

implicit sampling for hierarchical bayesian inversion and applications in fractional multiscale diffusion models abstract this paper presents an implicit sampling method for hierarchical bayesian inverse problems a widely used approach for sampling posterior distribution is markov chain monte carlo mcmc however the samples generated by mcmc are usually strongly correlated which may lead to a small size of effective samples the implicit sampling method proposed in chorin and tu 2009 can generate independent samples and capture inherent non gaussian features of the posterior in implicit sampling method the posterior samples are generated by a map and distribution around the maximum a posterior map point for practical applications some parameters in prior density are often unknown and the hierarchical bayesian formulation is used to estimate the map point effectively it is applied to the bayesian inverse problems of multi term time fractional diffusion models in heterogeneous media to effectively capture the heterogeneity effect we present a mixed generalized multiscale finite element method mixed gmsfem to substantially speed up the bayesian inversion finally we carry out a few numerical examples to effectively identify various unknown inputs

finite volume element method and its stability analysis for analyzing the behavior of sub diffusion problems in this paper we analyze the spatially semi discrete piecewise linear finite volume element method for the time fractional sub diffusion problem in two dimensions and give an approximate solution of this problem at first we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods furthermore we use the standard finite element ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method finally some numerical examples are included to illustrate the effectiveness of the new technique

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

a gradient discretization method to analyze numerical schemes for nonlinear variational inequalities application to the seepage problem using the gradient discretization method gdm we provide a complete and unified numerical analysis for nonlinear variational inequalities vis based on leray lions operators and subject to nonh

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

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

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

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

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

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

spike controls for elliptic and parabolic pdes abstract we analyze the use of measures of the minimal norm to control elliptic and parabolic equations we prove the sparsity of the optimal control in the parabolic case we prove that the solution of the optimization problem is a borel measure supported in a set of lebesgue measure zero in both cases the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations we also analyze the corresponding dual problem

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

boundary particle method for laplace transformed time fractional diffusion equations highlights error analysis and numerical experiments demonstrate the efficiency of our method for time fractional diffusion equations our method avoids costly convolution integral calculation and dilemmatic selection of time step size a truly boundary only meshless method is applied to laplace transformed inhomogeneous problem our method effectively simulates 3d long time history fractional diffusion systems this paper develops a novel boundary meshless approach laplace transformed boundary particle method ltbpm for numerical modeling of time fractional diffusion equations it implements laplace transform technique to obtain the corresponding time independent inhomogeneous equation in laplace space and then employs a truly boundary only meshless boundary particle method bpm to solve this laplace transformed problem unlike the other boundary discretization methods the bpm does not require any inner nodes since the recursive composite multiple reciprocity technique rc mrm is used to convert the inhomogeneous problem into the higher order homogeneous problem finally the stehfest numerical inverse laplace transform nilt is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding bpm solutions in comparison with finite difference discretization the ltbpm introduces laplace transform and stehfest nilt algorithm to deal with time fractional derivative term which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability consequently it can effectively simulate long time history fractional diffusion systems error analysis and numerical experiments demonstrate that the present ltbpm is highly accurate and computationally efficient for 2d and 3d time fractional diffusion equations

high order finite element methods for time fractional partial differential equations the aim of this paper is to develop high order methods for solving time fractional partial differential equations the proposed high order method is based on high order finite element method for space and finite difference method for time optimal convergence rate o dt 2 a n r is proved for the r 1 th order finite element method r 2

fractal von neumann entropy abstract we consider the fractal von neumann entropy associated with the fractal distribution function and we obtain for some universal classes h of fractons their entropies we obtain also for each of these classes a fractal deformed heisenberg algebra this one takes into account the braid group structure of these objects which live in two dimensional multiply connected space

an efficient galerkin spectral method for two dimensional fractional nonlinear reaction diffusion wave equation abstract the aim of this paper is to develop an efficient numerical treatment for the two dimensional fractional nonlinear reaction diffusion wave equation with the time fractional derivative of order 1 2 for this purpose we employ the alternating direction implicit adi method based on the crank nicolson scheme for the time stepping while we apply the legendre galerkin spectral method for the space discretization the stability and convergence analysis are rigorously set up in addition the proposed method is extended to solve the time fractional klein gordon and sine gordon models numerical experiments are included which verifies the theoretical predictions

further results on generalized multiple fractional part integrals for complex values in this paper the following multiple fractional part integrals i n 1 2 n 0 1 n j 1 n x j j s n 1 d x 1 d x n and j n 0 1 n s n s n 1 d x 1 d x n are studied for positive integer n and complex values of j j 1 2 n where u denotes the fractional part of u r s denotes the real part of s and s n x 1 x 2 x n it is proved that i 1 can be represented as a linear combination of the riemann zeta function the beta function and euler s constant as r 1 moreover i n 1 2 n can be expressed by i n 1 1 2 n 1 the beta function and the incomplete beta function for n 2 3 in addition the recurrence formula of j n n 2 3 is established and j n can be expressed by i 1 logarithmic function and some binomial coefficients

mixed spline function method for reaction subdiffusion equations in this paper we develop two classes of finite difference schemes for the reaction subdiffusion equations by using a mixed spline function in space direction forward and backward difference in time direction respectively it has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters by fourier method we prove that one class of difference scheme is unconditionally stable and convergent the other is conditionally stable and convergent finally some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

entanglement measure for the universal classes of fractons we introduce the notion of entanglement measure for the universal classes of fractons as an entanglement between occupation numbers of fractons in the lowest landau levels and the rest of the many body system of particles this definition came as an entropy of the probability distribution a la shannon fractons are charge flux systems classified in universal classes of particles or quasiparticles labelled by a fractal or hausdorff dimension defined within the interval 1 h 2 and associated with the fractal quantum curves of such objects they carry rational or irrational values of spin and the spin statistics connection occurs in this fractal approach to the fractional spin particles we take into account the fractal von neumann entropy associated with the fractal distribution function which each universal class of fractons satisfies we consider the fractional quantum hall effect fqhe given that fractons can model hall states according to our formulation entanglement between occupation numbers in this context increases with the universality classes of the quantum hall transitions considered as fractal sets of dual topological quantum numbers filling factors we verify that the hall states have stronger entanglement between occupation numbers and so we can consider this resource for fracton quantum computing

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

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

lsd a fast line segment detector with a false detection control we propose a linear time line segment detector that gives accurate results a controlled number of false detections and requires no parameter tuning this algorithm is tested and compared to state of the art algorithms on a wide set of natural images

und unite and divide method in fourier and radon domains for line segment detection in this paper we extend our previously proposed line detection method to line segmentation using a so called unite and divide und approach the methodology includes two phases namely the union of spectra in the frequency domain and the division of the sinogram in radon space in the union phase given an image its sinogram is obtained by parallel 2d multilayer fourier transforms cartesian to polar mapping and 1d inverse fourier transform in the division phase the edges of butterfly wings in the neighborhood of every sinogram peak are firstly specified with each neighborhood area corresponding to a window in image space by applying the separated sinogram of each such windowed image we can extract the line segments the division phase identifies the edges of butterfly wings in the neighborhood of every sinogram peak such that each neighborhood area corresponds to a window in image space line segments are extracted by applying the separated sinogram of each windowed image our experiments are conducted on benchmark images and the results reveal that the und method yields higher accuracy has lower computational cost and is more robust to noise compared to existing state of the art methods

spline collocation methods for linear multi term fractional differential equations abstract some regularity properties of the solution of linear multi term fractional differential equations are derived based on these properties the numerical solution of such equations by piecewise polynomial collocation methods is discussed the results obtained in this paper extend the results of pedas and tamme 2011 15 where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer in the present paper we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by some numerical examples

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