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

finite element method for nonlinear riesz space fractional diffusion equations on irregular domains in this paper we consider two dimensional riesz space fractional diffusion equations with nonlinear source term on convex domains applying galerkin finite element method in space and backward difference method in time we present a fully discrete scheme to solve riesz space fractional diffusion equations our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes which can help us to deal with space fractional terms on any convex domain the stability and convergence of the scheme are also discussed numerical examples are given to verify accuracy and stability of our scheme a finite element method is developed to solve 2d space fractional diffusion equations with nonlinear source term we develop a finite element method for fpdes on irregular domain we obtain explicit expressions for fractional derivatives of shape functions we give the details on how to compute fractional stiffness matrix fractional fitzhugh nagumo model is solved on circular domain

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

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

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

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

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

numerical methods and analysis for a class of fractional advection dispersion models in this paper a class of fractional advection dispersion models fadms is considered these models include five fractional advection dispersion models i e the time fadm the mobile immobile time fadm with a time caputo fractional derivative 0 c 1 the space fadm with two sides riemann liouville derivatives the time space fadm and the time fractional advection diffusion wave model with damping with index 1 c 2 these equations can be used to simulate the regional scale anomalous dispersion with heavy tails we propose computationally effective implicit numerical methods for these fadms the stability and convergence of the implicit numerical methods are analysed and compared systematically finally some results are given to demonstrate the effectiveness of theoretical analysis

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

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

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

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

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

variations of hodge structure legendre submanifolds and accessibility variations of hodge structure of weight two are integral manifolds for a distribution in the tangent bundle of a period domain this distribution has dimension h2 0h 1 and is nonintegrable for h2 0 1 in this case it is known that the dimension of an integral manifold does not exceed 1h2oh 2 here we give a new proof based on an analogy between griffiths horizontal differential system of algebraic geometry and the contact system of classical mechanics we show also that any two points in such a domain can be joined by a horizontal curve which is piecewise holomorphic

numerical solution of distributed order fractional differential equations in this paper a method for the numerical solution of distributed order fdes fractional differential equations of a general form is presented the method applies to both linear and nonlinear equations the caputo type fractional derivative is employed the distributed order fde is approximated with a multi term fde which is then solved by adjusting appropriately the numerical method developed for multi term fdes by katsikadelis several example equations are solved and the response of mechanical systems described by such equations is studied the convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results

the euler lagrange and legendre equations for functionals involving distributed order fractional derivatives abstract in this paper we extend some fractional calculus of variations results by considering functionals depending on distributed order fractional derivatives using variational techniques we deduce optimal necessary conditions of euler lagrange and legendre type we also study the case where integral and holonomic constraints are imposed finally a numerical procedure is given to solve some proposed problems

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

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

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

a galerkin finite element method for the modified distributed order anomalous sub diffusion equation abstract in this paper we consider a modified time distributed order anomalous sub diffusion equation for the description of processes becoming less anomalous over the course of time first we discretize the integral term using a mid point quadrature rule and transform the time distributed order fractional diffusion equation into a multi term equation then applying the backward difference method in time and galerkin finite element method in space we present a fully discrete scheme to solve the time distributed order fractional diffusion equation moreover the stability and convergence of the proposed method are also discussed numerical examples are given to verify the accuracy and stability of our scheme

flow heat and mass transfer of three dimensional fractional maxwell fluid over a bidirectional stretching plate with fractional fourier s law and fractional fick s law abstract this paper investigates the flow heat and mass transfer of fractional maxwell fluid over a bidirectional stretching sheet and research is carried out for the three dimensional case by analogy with the constitutive equation of fractional maxwell fluid fractional derivative is introduced into fourier s law and fick s law meanwhile the magnetic field and chemical reaction are considered furthermore the stretching speeds are not only power law dependent on time but also power law dependent on the distance of each space direction combining with l1 algorithm a newly finite difference method is developed to solve the governing equations and convergence of the method is verified by constructing a numerical example the influences of various physical parameters on velocity temperature and concentration are analyzed through three dimensional graphs the velocity fractional parameter presents an interesting effect on the velocity when the powers of each space direction coincide the smaller the velocity fractional parameter is the thinner the velocity boundary layer is on the contrary larger velocity fractional parameter results in decreasing velocity at first and then increasing for different powers of each space direction furthermore fractional fourier s law leads to more obvious heat transfer phenomena which is similar with the effect of fractional fick s law on the mass transfer

second order numerical methods for multi term fractional differential equations smooth and non smooth solutions abstract starting with the asymptotic expansion of the error equation of the shifted grunwald letnikov formula we derive a new modified weighted shifted grunwald letnikov wsgl formula by introducing appropriate correction terms we then apply one special case of the modified wsgl formula to solve multi term fractional ordinary and partial differential equations and we prove the linear stability and second order convergence for both smooth and non smooth solutions when the regularity of the solutions is known we show theoretically and numerically that numerical solutions with good accuracy can be obtained with only a few correction terms moreover the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions numerical simulations verify the theoretical results and demonstrate that the new wsgl formula leads to better performance compared to other known numerical approximations with similar resolution

weighted least mean square design of 2 d fir digital filters this paper is concerned with weighted least mean square design of two dimensional 2 d zero phase fir filters with quadrantally symmetric and antisymmetric frequency responses the optimal solutions are first characterized by certain integral equations and the existence and uniqueness of the weighted least mean square solution for 2 d fir filter design are then established using contraction mapping and fixed point theorem explicit solution in closed form for discretized weighted least mean square error is obtained for 2 d zero phase fir filters which can be considered as an extension of the methods of ahmed and wang 1989 and tufts and francis 1970 further two numerical algorithms are proposed to iteratively solve the weighted least mean square solution convergence of the iterative algorithms is established and estimates of the convergence rate are also derived examples are used to illustrate the effectiveness of the proposed design algorithms

distributed order fractional diffusions on bounded domains fractional derivatives can be used to model time delays in a diffusion process when the order of the fractional derivative is distributed over the unit interval it is useful for modeling a mixture of delay sources this paper provides explicit strong solutions and stochastic analogues for distributed order time fractional diffusion equations on bounded domains with dirichlet boundary conditions

a weighted least squares algorithm for quasi equiripple fir and iir digital filter design it has been demonstrated by several authors that if a suitable frequency response weighting function is used in the design of a finite impulse response fir filter the weighted least squares solution is equiripple the crux of the problem lies in the determination of the necessary least squares frequency response weighting function a novel iterative algorithm for deriving the least squares frequency response weighting function which will produce a quasi equiripple design is presented the algorithm converges very rapidly it typically produces a design which is only about 1 db away from the minimax optimum solution in two iterations and converges to within 0 1 db in six iterations convergence speed is independent of the order of the filter it can be used to design filters with arbitrarily prescribed phase and amplitude response

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

coupled contact systems and rigidity of maximal dimensional variations of hodge structure in this article we prove that locally griffiths horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system as a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of hlodge structure for example we prove that if the weight is odd greater than one then there is a unique germ of maximal dimensional variation of hodge structure through every point of the period domain similar results hold if the weight is even with the exception of one case

fractional langevin equations of distributed order distributed order fractional langevin like equations are introduced and applied to describe anomalous diffusion without unique diffusion or scaling exponent it is shown that these fractional langevin equations of distributed order can be used to model the kinetics of retarding subdiffusion whose scaling exponent decreases with time and the strongly anomalous ultraslow diffusion with mean square displacement which varies asymptotically as a power of logarithm of time

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

a novel finite volume method for the riesz space distributed order diffusion equation abstract in recent years considerable attention has been devoted to distributed order differential equations mainly because they appear to be more effective for modelling complex processes which obey a mixture of power laws or flexible variations in space in this paper we propose a novel finite volume method fvm for a distributed order space fractional diffusion equation fde firstly we use the mid point quadrature rule to transform the space distributed order diffusion equation into a multi term fractional equation secondly the transformed multi term fractional equation is solved by discretising in space using the finite volume method and then in time using the crank nicolson scheme thirdly we prove that the crank nicolson scheme with fvm is unconditionally stable and convergent with second order accuracy in both time and space finally two numerical examples are presented to show the effectiveness of the numerical method these methods and techniques can also be used to solve other types of fractional partial differential equations

a novel finite element method for the distributed order time fractional cable equation in two dimensions abstract in this paper the unstructured mesh galerkin finite element method with a weighted and shifted grunwald difference approximation and composite trapezoid formula is presented to solve the nonhomogeneous two dimensional distributed order time fractional cable equation on irregular convex domains the crank nicolson type discretization of the finite element scheme is implemented to obtain the numerical solution the stability and convergence of the numerical scheme are discussed and derived finally some numerical examples on irregular convex domains are given to confirm our theoretical results

algorithm for some anomalously diffusive hyperbolic systems in molecular dynamics theoretical analysis and pattern formation abstract departing from a two dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions we investigate a general form of that model using a finite difference approach the model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms the presence of two dimensional diffusive terms consisting of fractional operators of the riesz type is considered here using spatial differentiation orders in the set 0 1 1 2 we impose initial conditions on a closed and bounded rectangle and a finite difference methodology based on the use of fractional centered differences is proposed among the most important results of this work we prove the existence and the uniqueness of the solutions of the numerical method and establish analytically the second order consistency of our scheme moreover the discrete energy method is employed to prove the stability and the quadratic convergence of the technique some numerical simulations obtained through our method show the presence of turing patterns and wave instabilities in agreement with some reports found in the literature on superdiffusive hyperbolic activator inhibitor systems we show numerically that the presence of turing patterns is independent of the size of the spatial domain as a new application we show that turing patterns are also present in subdiffusive scenarios

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

an unstructured mesh control volume method for two dimensional space fractional diffusion equations with variable coefficients on convex domains in this paper we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains which to the best of our knowledge is a new contribution to the literature firstly we present the finite volume scheme for the two dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements secondly we explore the property of the stiffness matrix generated by the integral of the space fractional derivative we find that the stiffness matrix is sparse and not regular therefore we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system which is more efficient than using the gaussian elimination method finally we present several examples to verify our method in which we make a comparison of our method with the finite element method for solving a riesz space fractional diffusion equation on a circular domain the numerical results demonstrate that our method can reduce cpu time significantly while retaining the same accuracy and approximation property as the finite element method the numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains

time distributed order diffusion wave equation ii applications of laplace and fourier transformations a cauchy problem for a time distributed order multi dimensional diffusion wave equation containing a forcing term is reinterpreted in the space of tempered distributions and a distributional diffusion wave equation is obtained the distributional equation is solved in the general case of weight function or distribution solutions are given in terms of solution kernels green s functions which are studied separately for two cases the first case is when the order of the fractional derivative is in the interval 0 1 while in the second case the order of the fractional derivative is in the interval 0 2 solutions of fractional diffusionwave and fractional telegraph equations are obtained as special cases numerical experiments are also performed an analogue of the maximum principle is also presented

distributed order calculus and equations of ultraslow diffusion abstract we consider equations of the form d u t x u t x f t x t 0 x r n where d is a distributed order derivative that is d t 0 1 d t d d is the caputo dzhrbashyan fractional derivative of order is a positive weight function the above equation is used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement in this work we develop a mathematical theory of such equations study the derivatives and integrals of distributed order

modified methods for solving two classes of distributed order linear fractional differential equations abstract this paper introduces two methods for the numerical solution of distributed order linear fractional differential equations the first method focuses on initial value problems ivps and based on the th caputo fractional definition with the shifted chebyshev operational matrix of fractional integration by applying this method the ivps are converted into simple linear differential equations which can be easily handled the other method focuses on boundary value problems bvps based on picard s method frame this method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods also the response of mechanical system described by such equations is studied

a conservative fourth order stable finite difference scheme for the generalized rosenau kdv equation in both 1d and 2d abstract in the present work a conservative fourth order stable finite difference scheme is proposed to solve the generalized rosenau kdv equation in both 1d and 2d the existence uniqueness and mass and energy conservations of the numerical solution are proved using the discrete energy method the new scheme is convergent with o 2 h 4 and unconditionally stable numerical experiments are carried out to show that the scheme is efficient and reliable

a crank nicolson adi galerkin legendre spectral method for the two dimensional riesz space distributed order advection diffusion equation abstract in the paper a crank nicolson alternating direction implicit adi galerkin legendre spectral scheme is presented for the two dimensional riesz space distributed order advection diffusion equation the gauss quadrature has a higher computational accuracy than the mid point quadrature rule which is proposed to approximate the distributed order riesz space derivative so that the considered equation is transformed into a multi term fractional equation moreover the transformed equation is solved by discretizing in space by the adi galerkin legendre spectral scheme and in time using the crank nicolson difference method stability and convergence analysis are verified for the numerical approximation a lot of numerical results are demonstrated to justify the theoretical analysis

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

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

hypersurface variations are maximal ii the variation of hodge structure defined by the natural family of hypersurfaces of degreed and dimensionn is maximal if the cohomology has hodge level 1 there is a small list of hypersurfaces of level one which give non maximal variations plane curves of degreed 5 cubics of dimension 3 and 5 and quartic threefolds

a semi alternating direction method for a 2 d fractional fitzhugh nagumo monodomain model on an approximate irregular domain a fitzhugh nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue in this paper we consider a two dimensional fractional fitzhugh nagumo monodomain model on an irregular domain the model consists of a coupled riesz space fractional nonlinear reaction diffusion model and an ordinary differential equation describing the ionic fluxes as a function of the membrane potential second we use a decoupling technique and focus on solving the riesz space fractional nonlinear reaction diffusion model a novel spatially second order accurate semi implicit alternating direction method siadm for this model on an approximate irregular domain is proposed third stability and convergence of the siadm are proved finally some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two dimensional fractional fitzhugh nagumo model on both an approximate circular and an approximate irregular domain

unstructured mesh galerkin finite element method for the two dimensional multi term time space fractional bloch torrey equations on irregular convex domains abstract models based on partial differential equations containing time space fractional derivatives have attracted considerable interest in the past decade because of their ability to model anomalous transport phenomena these phenomena are strongly connected to the interactions within complex and non homogeneous media exhibiting spatial heterogeneity the class of equations with multi term time space derivatives of fractional orders has been found to be very useful in the description of such interactions this motivates the extension of the classical bloch torrey equation through the application of the operators of fractional calculus to new multi term time space fractional bloch torrey equations with riesz fractional operators in this paper we firstly propose an unstructured mesh galerkin finite element method for the two dimensional multi term time space fractional diffusion equation with riesz fractional operators on irregular convex domains secondly we rigorously establish the stability and convergence of the numerical scheme thirdly we extend the computational model to solve a system of coupled two dimensional multi term time space fractional bloch torrey equations finally some numerical results are given to demonstrate the versatility and application of the models

fast simulation of multi dimensional wave problems by the sparse scheme of the method of fundamental solutions in this work a meshless scheme is presented for the fast simulation of multi dimensional wave problems the present method is rather simple and straightforward the houbolt method is used to eliminate the time dependence of spatial variables then the original wave problem is converted into equivalent systems of modified helmholtz equations the sparse scheme of the method of fundamental solutions in combination with the localized method of approximate particular solutions is employed for efficient implementation of spatial variables to demonstrate the effectiveness and simplicity of this new approach three numerical examples have been assessed with excellent performance

generic integral manifolds for weight two period domains we define the notion of a generic integral element for the griffiths distribution on a weight two period domain draw the analogy with the classical contact distribution and then show how to explicitly construct an infinite dimensional family of integral manifolds tangent to a given element

coupling of the improved singular boundary method and dual reciprocity method for multi term time fractional mixed diffusion wave equations abstract the main purpose of this work is to present an impressive numerical scheme to solve two dimensional multi term time fractional mixed diffusion wave differential equations tfmdwe the proposed method is based on the compact dual reciprocity method and the meshless improved singular boundary method isbm the most significant privilege of the proposed method is mathematically simple efficient quite accurate which requires relatively less computational cost singular boundary method is one of truly meshless methods and it does not need any element or mesh for both field interpolation and background integration for the considered problems time derivatives are approximated via the time stepping method several numerical results are provided to show the accuracy of the present method for multi term time fractional mixed diffusion wave equations

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

matrix based algorithms for constrained least squares and minimax designs of 2 d linear phase fir filters the impulse response coefficients of a two dimensional 2 d finite impulse response fir filter are in a matrix form in nature conventional optimal design algorithms rearrange the filter s coefficient matrix into a vector and then solve for the coefficient vector using design algorithms for one dimensional 1 d fir filters some recent design algorithms have exploited the matrix nature of the 2 d filter s coefficients but not incorporated with any constraints and thus are not applicable to the design of 2 d filters with explicit magnitude constraints in this paper we develop some efficient algorithms exploiting the coefficients matrix nature for the constrained least squares cls and minimax designs of quadrantally symmetric 2 d linear phase fir filters both of which can be formulated as an optimization problem or converted into a sequence of subproblems with a positive definite quadratic cost and a finite number of linear constraints expressed in terms of the filter s coefficient matrix design examples and comparisons with several existing algorithms demonstrate the effectiveness and efficiency of the proposed algorithms

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

two alternating direction implicit difference schemes with the extrapolation method for the two dimensional distributed order differential equations the grunwald formula is used to solve the two dimensional distributed order differential equations two alternating direction implicit adi difference schemes are derived it is proved that the schemes are unconditionally stable and convergent with the convergence orders o h 1 2 h 2 2 2 and o h 1 4 h 2 4 4 in h 1 norm respectively where h 1 h 2 and are the step sizes in time space in x and y direction and distributed order the extrapolation method is applied to improve the approximate accuracy to the orders o 2 ln 2 h 1 2 h 2 2 2 and o 2 ln 2 h 1 4 h 2 4 4 respectively several numerical examples are given to confirm the theoretical results

numerical analysis for distributed order differential equations in this paper we present and analyse a numerical method for the solution of a distributed order differential equation of the general form 0 ma r d ru t dr f t where m is a positive real number and where the derivative d r is taken to be a fractional derivative of caputo type of order r we give a convergence theory for our method and conclude with some numerical examples

fully discrete spectral methods for solving time fractional nonlinear sine gordon equation with smooth and non smooth solutions abstract we consider the initial boundary value problem of the time fractional nonlinear sine gordon equation and the fractional derivative is described in caputo sense with the order 1

the variations of hodge structure of maximal dimension with associated hodge numbers ℎ 2 0 2 and ℎ 1 1 2 1 do not arise from geometry the specified variations are proved to be covered by a bounded contractible domain q2 after classifying the analytic boundary components of q with respect to a fixed realization the group of the biholomorphic automorphisms aut q2 and the aut q orbit structure of q2 are found explicitly then n is shown to admit no quasiprojective arithmetic quotients whereas the lack of geometrically arising variations covered by q2 the weight 1 period domain sp n r u arises from geometry in as much as it classifies the first cohomologies of the n dimensional abelian varieties with fixed polarization the geometric locus on the weight 2 period domain is shown to satisfy the griffiths horizontality cf 9

weighted least mean square design of 2 d fir digital filters the general case this paper solves the weighted least mean square wlms design of two dimensional 2 d finite impulse response fir filters with general half plane symmetric frequency responses and nonnegative weighting functions the optimal solution is characterized by a pair of coupled integral equations and the existence and uniqueness of the wlms solution for 2 d fir filter design are established two efficient numerical algorithms using a 2 d fast fourier transform fft are proposed to solve the wlms solution one is based on the contraction mapping and fix point theorem characterizing the coupled integral equation the other uses conjugate gradient techniques which guarantees finite convergence the associated computational complexity is analyzed and compared with existing algorithms examples are used to illustrate the effectiveness of the proposed design algorithms the selection of weighting functions to improve the minimax performance of the filter is also discussed

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

multi dimensional spectral tau methods for distributed order fractional diffusion equations abstract the distributed order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power law scaling over the whole time domain an important application of distributed order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate to broaden the range of applicability of distributed order fractional diffusion models efficient numerical methods are needed to solve the model equation in this work we develop spectral tau schemes to discretize the fractional diffusion equation with distributed order fractional derivative in time and dirichlet boundary conditions the model solution is expanded in multi dimensions in terms of legendre polynomials and the discrete equations are obtained with the tau method numerical examples are provided to highlight the convergence rate and the flexibility of this approach the proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth our results confirm that nonlocal numerical methods are best suited to discretize distributed order fractional differential equations as they naturally take the global behavior of the solution into account

some high order difference schemes for the distributed order differential equations two difference schemes are derived for both one dimensional and two dimensional distributed order differential equations it is proved that the schemes are unconditionally stable and convergent in an l 1 l norm with the convergence orders o 2 h 2 2 and o 2 h 4 4 respectively where h and are the step sizes in time space and distributed order variables several numerical examples are given to confirm the theoretical results

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

a novel legendre operational matrix for distributed order fractional differential equations abstract in this paper for the first time the shifted legendre operational matrix of distributed order fractional derivative has been derived also this new operational matrix is used together with tau method for approximation of solutions of linear distributed order fractional differential equations and diffusion equations with distributed order in time moreover eight numerical examples are implemented in order to show the validity and reliability of the suggested methods

bounds on the dimension of variations of hodge structure we derive upper bounds on the dimension of a variation of hodge structure of weight two and show that these bounds are sharp using them we exhibit maximal geometric variations of hodge structure analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis and variations coming from hypersurfaces are shown to be nondegenerate maximal geometric variations of higher weight are also constructed

compact difference scheme for distributed order time fractional diffusion wave equation on bounded domains in this paper we derive and analyse a compact difference scheme for a distributed order time fractional diffusion wave equation this equation is approximated by a multi term fractional diffusion wave equation which is then solved by a compact difference scheme the unique solvability of the difference solution is discussed using the discrete energy method we prove the compact difference scheme is unconditionally stable and convergent finally numerical results are presented to support our theoretical analysis