a numerical method for an integro differential equation with memory in banach spaces qualitative properties a first order method is considered for the discretization in time of an integro differential equation which can be written as d alpha u t a u t f t 1 alpha 2 where a d a subset x to x is a sectorial operator in a banach space x qualitative properties of the numerical solution such as contractivity and positivity are studied a numerical illustration is provided

adaptive discretization of fractional order viscoelasticity using sparse time history an efficient numerical method to integrate the constitutive response of fractional order viscoelasticity is developed the method can handle variable time steps to overcome the problem of the growing amount of data that has to be stored and used in each time step we introduce sparse quadrature we use an internal variable formulation of the viscoelastic equations where the internal variable is of stress type the rate equation that governs the evolution of the internal variable involves a fractional integral and can be identified as a volterra integral equation of the second kind with a weakly singular kernel for the numerical integration of the rate equation we adopt the finite element method in time in particular the discontinuous galerkin method with piecewise constant basis functions is used a priori and a posteriori error estimates are proved an adaptive strategy based on the a posteriori error estimate is developed finally the precision and effectiveness of the method are demonstrated by comparing the numerical solutions with analytical solutions

spectral methods for the time fractional diffusion wave equation in a semi infinite channel in this paper we consider the numerical approximation of the time fractional diffusion wave equation in a semi infinite channel the time fractional derivative is described in caputo sense with order 1 2 a fully discrete spectral scheme based on a finite difference method in the time direction and a laguerre legendre spectral method in the space direction is proposed we also propose an alternating direction implicit adi spectral scheme in order to reduce the amount of computation the stability and convergence of both schemes are rigorously established numerical results are presented to support our theoretical analysis

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

numerical analysis of volterra integro differential equations for viscoelastic rods and membranes abstract we consider the initial boundary value problems for a homogeneous volterra integro differential equations for viscoelastic rods and membranes in a bounded smooth domain the memory kernel of the equation is made up in a complicated way from the distinct moduli of stress relaxation for compression and shear which is challenging to approximate the literature reported on the numerical solution of this model is extremely sparse in this paper we will study the second order continuous time galerkin approximation for its space discretization and propose a fully discrete scheme employing the crank nicolson method for the time discretization then we derive the uniform long time error estimates in the norm l t 1 0 l x 2 for the finite element solutions some numerical results are presented to illustrate our theoretical error bounds

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

spectral and discontinuous spectral element methods for fractional delay equations we first develop a spectrally accurate petrov galerkin spectral method for fractional delay differential equations fddes this scheme is developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently presented in m zayernouri and g e karniadakis j comput phys 252 2013 pp 495 517 specifically we obtain solutions to fddes in terms of new nonpolynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of the first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of the second kind fslp ii we prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the pg spectral method in contrast to standard nondelay fractional differential equations the delay character of fddes might induce solutions which are either nonsmooth or piecewise smooth in order to effectively trea

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

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

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

spectral and pseudospectral approximations for the time fractional diffusion equation on an unbounded domain in this paper we consider the numerical approximation of the time fractional diffusion equation with variable coefficients on a semi infinite spatial domain a fully discrete scheme based on finite difference method in time and spectral approximation using laguerre functions in space is proposed stability and convergence of the proposed scheme are rigorously established the scheme is unconditionally stable and convergent with order o 2 n 1 m 2 where n and m are the time step size polynomial degree and regularity in the space variable of the exact solution respectively a pseudospectral scheme is also proposed and analysed some numerical examples are presented to demonstrate the theoretical results

a spectral order method for inverting sectorial laplace transforms laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered a spectral order parallelizable method for their numerical inversion is proposed the method takes into account the available information about the errors arising in the evaluations several numerical illustrations are provided

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

high order compact alternating direction implicit method for the generalized sine gordon equation high order compact alternating direction implicit scheme is given for solving the generalized sine gordon equation in a two dimensional rectangular domain we apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives and we give a linearized three time level algorithm for solving the original nonlinear equation error estimate is given by the energy method numerical results are provided to verify the accuracy and efficiency of this algorithm

time discretization and global solution for a doubly nonlinear volterra equation this note addresses the analysis of an abstract doubly nonlinear volterra equation with a nonsmooth kernel and possibly unbounded and degenerate operators by exploiting a suitable implicit time discretization technique we obtain the existence of a global strong solution as a by product the discrete scheme is proved to be conditionally stable and convergent

a novel high order approximate scheme for two dimensional time fractional diffusion equations with variable coefficient abstract based on the spatial quasi wilson nonconforming finite element method and temporal l 2 1 formula a fully discrete approximate scheme is proposed for a two dimensional time fractional diffusion equations with variable coefficient on anisotropic meshes in order to demonstrate the stable analysis and error estimates several lemmas are provided which focus on high accuracy about projection and superclose estimate between the interpolation and projection unconditionally stable analysis are derived in l 2 norm and broken h 1 norm moreover convergence result of accuracy o h 2 2 and superclose property of accuracy o h 2 2 are deduced by combining interpolation with projection where h and are the step sizes in space and time respectively and then the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

post processing discontinuous galerkin solutions to volterra integro differential equations analysis and simulations this paper presents a superconvergence extraction technique for volterra integro differential equations with smooth and non smooth kernels specifically extracting superconvergence is done via a post processed discontinuous galerkin dg method obtained from interpolating the dg solution using lagrange polynomials at the nodal points a global superconvergence error bound in the l norm is established for a non smooth kernel a family of non uniform time meshes is used to compensate for the singular behaviour of the exact solution near t 0 the derived theoretical results are numerically validated in a sample of test problems demonstrating higher than expected convergence rates

an h 1 galerkin mixed finite element method for an evolution equation with a positive type memory term an h1 galerkin mixed finite element method is analyzed for a class of evolution equations with memory when a classical mixed method is applied to such problems it has not been possible to obtain any estimate for the flux however the proposed approach yields optimal order convergence without the lbb consistency condition and quasi uniformity of the finite element mesh compared to the results proved for one space variable the l2 estimate of the flux is not optimal for problems in two and three space dimensions therefore a modification of the method is proposed and analyzed a maximum norm estimate is also derived in one and two space variables a backward euler approximation of the modified method is also analyzed

quasi wavelet based numerical method for a class of partial integro differential equation abstract in this paper we study the numerical solution of initial boundary problem for a class of partial integro differential equations the quasi wavelet method is proposed to handle the spatial derivatives while the forward euler method is used to discretize the temporal derivatives detailed discrete schemes are given and some numerical experiments are included to demonstrate the effectiveness of the discrete technique the comparisons of the present numerical results with the exact analytical solutions show that the quasi wavelet based numerical method has distinctive local property and can achieve accurate results

a second order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel abstract a second order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel the graded meshes are employed to compensate for the singular behavior of the exact solution at t 0 for the time discretization the product integration rule is used to approximate the riemann liouville fractional integral a generalized crank nicolson time stepping is considered and shown that the error is of order k 2 where k denotes the maximum time step a fully discrete difference scheme is constructed with space discretization by compact difference method numerical experiment is carried out to support the theoretical results the comparison between the method on uniform grids and graded grids shows the efficiency of our method

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

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

developing finite element methods for maxwell s equations in a cole cole dispersive medium in this paper we consider the time dependent maxwell s equations when cole cole dispersive medium is involved the cole cole model contains a fractional time derivative term which couples with the standard maxwell s equations in free space and creates some challenges in developing and analyzing time domain finite element methods for solving this model as mentioned in our earlier work j li j sci comput 47 2001 pp 1 26 by adopting some techniques developed for the fractional diffusion equations v j ervin n heuer and j p roop siam j numer anal 45 2007 pp 572 591 y lin and c xu j comput phys 225 2007 pp 1533 1552 f liu p zhuang v anh i turner and k burrage appl math comput 191 2007 pp 12 20 we propose two fully discrete mixed finite element schemes for the cole cole model numerical stability and optimal error estimates are proved for both schemes the proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis

time stepping discontinuous galerkin methods for fractional diffusion problems time stepping hp hp versions discontinuous galerkin dg methods for the numerical solution of fractional subdiffusion problems of order alpha with 1 alpha 0 1 0 will be proposed and analyzed generic hp hp version error estimates are derived after proving the stability of the approximate solution for h h version dg approximations on appropriate graded meshes near t 0 t 0 we prove that the error is of order o k max 2 p frac alpha 2 o kmax 2 p 2 where k k is the maximum time step size and p ge 1 p 1 is the uniform degree of the dg solution for hp hp version dg approximations by employing geometrically refined time steps and linearly increasing approximation orders exponential rates of convergence in the number of temporal degrees of freedom are shown finally some numerical tests are given

discontinuous galerkin method for an evolution equation with a memory term of positive type we consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by c t s 1 where 0 1 for the time discretization we apply the discontinuous galerkin method using piecewise polynomials of degree at most q 1 for q 1 or 2 for the space discretization we use continuous piecewise linear finite elements the discrete solution satisfies an error bound of order k 9 h 2 l k where k and h are the mesh sizes in time and space respectively and l k max 1 log k 1 in the case q 2 we prove a higher convergence rate of order k 3 h 2 l k at the nodes of the time mesh typically the partial derivatives of the exact solution are singular at t 0 necessitating the use of non uniform time steps we compare our theoretical error bounds with the results of numerical computations

numerical methods for solving a two dimensional variable order modified diffusion equation in this paper a two dimensional variable order modified diffusion equation is considered we develop the numerical methods to solve the equation by fourier analysis we discuss the convergence stability and solvability of the numerical method the numerical method for improving temporal accuracy is also developed moreover our theoretical analysis results are demonstrated by the numerical example

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

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

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

alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive type memory term new numerical techniques are presented for the solution of a class of linear partial integro differential equations pides with a positive type memory term in the unit square in these methods orthogonal spline collocation osc is used for the spatial discretization and for the time stepping new alternating direction implicit adi methods based on the backward euler the crank nicolson and the second order bdf methods combined with judiciously chosen quadrature rules are considered the adi osc methods are proved to be of optimal accuracy in time and in the l 2 norm in space numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the l infty and h 1 norms and superconvergence phenomena

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

adaptive discretization of an integro differential equation with a weakly singular convolution kernel an integro differential equation involving a convolution integral with a weakly singular kernel is considered the kernel can be that of a fractional integral the integro differential equation is discretized using the discontinuous galerkin method with piecewise constant basis functions sparse quadrature is introduced for the convolution term to overcome the problem with the growing amount of data that has to be stored and used in each time step a priori and a posteriori error estimates are proved an adaptive strategy based on the a posteriori error estimate is developed finally the precision and effectiveness of the algorithm are demonstrated in the case that the convolution is a fractional integral this is done by comparing the numerical solutions with analytical solutions

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

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

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

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

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

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

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

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

discontinuous galerkin method for an integro differential equation modeling dynamic fractional order viscoelasticity an integro differential equation modeling dynamic fractional order viscoelasticity with a mittag leffler type convolution kernel is considered a discontinuous galerkin method based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem stability estimates of the discrete problem are proved that are used to prove optimal order a priori error estimates the theory is illustrated by a numerical example

derivation of stokes law from kirkwood s formula and the green kubo formula via large deviation theory we study the friction coefficient of a macroscopic sphere in a viscous fluid at low reynolds number first kirkwood s formula for the friction coefficient is reviewed on the basis of the hamiltonian description of particle systems according to this formula the friction coefficient is expressed in terms of the stress correlation on the surface of the macroscopic sphere then with the aid of large deviation theory we relate the surface stress correlation to the stress correlation in the bulk of the fluid where the latter is characterized by the viscosity in the green kubo formula by combining kirkwood s formula and the green kubo formula in large deviation theory we derive stokes law without explicitly employing the hydrodynamic equations

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

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

finite element approximation of diffusion equations with convolution terms approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered such problems arise from modeling of flows in fissured media convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented the advantages of implementation of the algorithm in a multiprocessing environment are discussed

time stepping error bounds for fractional diffusion problems with non smooth initial data we apply the piecewise constant discontinuous galerkin method to discretize a fractional diffusion equation with respect to time using laplace transform techniques we show that the method is first order accurate at the nth time level t n but the error bound includes a factor t n 1 if we assume no smoothness of the initial data we also show that for smoother initial data the growth in the error bound as t n decreases is milder and in some cases absent altogether our error bounds generalize known results for the classical heat equation and are illustrated for a model problem

superconvergence analysis of finite element method for time fractional thermistor problem abstract in this paper the superclose and superconvergence analysis of the nonlinear time fractional thermistor problem are investigated by bilinear finite element method fem for a fully discrete scheme in which the caputo derivative is approximated by the classical l 1 method by dealing with the error estimates in the spatial direction rigorously which are one order higher than the traditional fems the superclose estimates in h 1 norm are obtained for the corresponding variables based on the special properties of this element together with mean value technique subsequently the global superconvergence results are derived by employing the interpolation postprocessing approach finally a numerical experiment is carried out to confirm the theoretical analysis

lame equation and asymptotic higher order periodic solutions to nonlinear evolution equations based on an auxiliary lame equation and the perturbation method a direct method is proposed to construct asymptotic higher order periodic solutions to some nonlinear evolution equations it is shown that some asymptotic higher order periodic solutions to some nonlinear evolution equations in terms of jacobi elliptic functions are explicitly obtained with the aid of symbolic computation

nonconforming discontinuous galerkin methods for nonlocal variational problems we present a nonconforming discontinuous galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints the nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions the key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin it works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions a convergence theory is established for two types of finite dimensional spaces with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown

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

quasi rational function solutions of an elliptic equation and its application to solve some nonlinear evolution equations abstract using the differential transformation method and the homogeneous balance method some new solutions of an auxiliary elliptic equation are obtained these solutions possess the forms of rational functions in terms of trigonometric functions hyperbolic functions exponential functions power functions elliptic functions and their operation and composite functions and so on which are so called quasi rational function solutions based on these new quasi rational functions solutions a direct method is proposed to construct the exact solutions of some nonlinear evolution equations with the aid of symbolic computation the coupled kdv mkdv equation and broer kaup equations are chosen to illustrate the effectiveness and convenience of the suggested method for obtaining quasi rational function solutions of nonlinear evolution equations

linear response theory in stochastic many body systems revisited abstract the green kubo relation the einstein relation and the fluctuation response relation are representative universal relations among measurable quantities that are valid in the linear response regime we provide pedagogical proofs of these universal relations for stochastic many body systems through these simple proofs we characterize the three relations as follows the green kubo relation is a direct result of the local detailed balance condition the fluctuation response relation represents the dynamic extension of both the green kubo relation and the fluctuation relation in equilibrium statistical mechanics and the einstein relation can be understood by considering thermodynamics we also clarify the interrelationships among the universal relations

numerical schemes with high spatial accuracy for a variable order anomalous subdiffusion equation in this paper we consider a variable order anomalous subdiffusion equation a numerical scheme with first order temporal accuracy and fourth order spatial accuracy for the equation is proposed the convergence stability and solvability of the numerical scheme are discussed via the technique of fourier analysis another improved numerical scheme with second order temporal accuracy and fourth order spatial accuracy is also proposed some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

time discretization of fractional subdiffusion equations via fractional resolvent operators abstract in this work we study time discretization of subdiffusion equations that is fractional differential equations of order 0 1 assuming that a is the generator of a fractional resolvent family s t t 0 which allows to write the solution to the subdiffusion equation t u t a u t f t as a variation of constants formula we find an interesting connection between s t t 0 and a discrete resolvent family s n n n and then by using the properties of s t t 0 we study the existence of solutions to the discrete subdiffusion equation c u n a u n f n n n where based on the backward euler method for a 0 given c u n is an approximation of t u t at time t n n we study simultaneously the fractional derivative in the caputo and riemann liouville sense we also provide error estimates and some experiments to illustrate the results

finite difference spectral approximations for the distributed order time fractional reaction diffusion equation on an unbounded domain the numerical approximation of the distributed order time fractional reaction diffusion equation on a semi infinite spatial domain is discussed in this paper a fully discrete scheme based on finite difference method in time and spectral approximation using laguerre functions in space is proposed the scheme is unconditionally stable and convergent with order o 2 2 n 1 m 2 where n and m are the time step size step size in distributed order variable polynomial degree and regularity in the space variable of the exact solution respectively a pseudospectral scheme is also proposed and analyzed some numerical examples are presented to demonstrate the efficiency of the proposed scheme

hierarchical matrix approximations for space fractional diffusion equations abstract space fractional diffusion models generally lead to dense discrete matrix operators which lead to substantial computational challenges when the system size becomes large for a state of size n full representation of a fractional diffusion matrix would require o n 2 memory storage requirement with a similar estimate for matrix vector products in this work we present h 2 matrix representation and algorithms that are amenable to efficient implementation on gpus and that can reduce the cost of storing these operators to o n asymptotically matrix vector multiplications can be performed in asymptotically linear time as well performance of the algorithms is assessed in light of 2d simulations of space fractional diffusion equation with constant diffusivity attention is focused on smooth particle approximation of the governing equations which lead to discrete operators involving explicit radial kernels the algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain and then for the steady forced fractional diffusion equation in a bounded domain both matrix inverse and pseudo transient solution approaches are considered in the latter case our experiments show that the construction of the fractional diffusion matrix the matrix vector multiplication and the generation of an approximate inverse pre conditioner all perform very well on a single gpu on 2d problems with n in the range 1 0 5 1 0 6 in addition the tests also showed that for the entire range of parameters and fractional orders considered results obtained using the h 2 approximations were in close agreement with results obtained using dense operators and exhibited the same spatial order of convergence overall the present experiences showed that the h 2 matrix framework promises to provide practical means to handle large scale space fractional diffusion models in several space dimensions at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations

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

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid

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

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

discrete time orthogonal spline collocation method with application to two dimensional fractional cable equation discrete time orthogonal spline collocation osc methods are presented for the two dimensional fractional cable equation which governs the dynamics of membrane potential in thin and long cylinders such as axons or dendrites in neurons the proposed scheme is based on the osc method for space discretization and finite difference method for time which is proved to be unconditionally stable and convergent with the order o min 2 1 2 2 h r 1 in l 2 norm where h and r are the time step size space step size and polynomial degree respectively and 1 and 2 are two different exponents of fractional derivatives with 0 1 2 1 numerical experiments are presented to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the method described herein and super convergence phenomena at the partition nodes is also exhibited which is a characteristic of the osc methods namely the rates of convergence in the maximum norm at the partition nodes in u x and u y are approximately h r 1 in our numerical experiment

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

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

travelling wave solutions of reduced super kdv equation a perspective from lame equation abstract reduced supersymmetric korteweg de vries super kdv equation is investigated in this paper with the help of an auxiliary lame equation and jacobi elliptic function travelling wave solutions of reduced super kdv equation are obtained in the form of jacobi elliptic periodic function meanwhile trigonal solutions and hyperbolic solutions are derived

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

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

an analysis of the rayleigh stokes problem for a generalized second grade fluid we study the rayleigh stokes problem for a generalized second grade fluid which involves a riemann liouville fractional derivative in time and present an analysis of the problem in the continuous space semidiscrete and fully discrete formulations we establish the sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v v including v in l 2 omega v l2 a space semidiscrete galerkin scheme using continuous piecewise linear finite elements is developed and optimal with respect to initial data regularity error estimates for the finite element approximations are derived further two fully discrete schemes based on the backward euler method and second order backward difference method and the related convolution quadrature are developed and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data numerical results for one and two dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method and to verify the convergence theory

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

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

fast summation by interval clustering for an evolution equation with memory we solve a fractional diffusion equation using a piecewise constant discontinuous galerkin method in time combined with a continuous piecewise linear finite element method in space if there are n time levels and m spatial degrees of freedom then a direct implementation of this method requires o n 2m operations and o nm active memory locations owing to the presence of a memory term at each time step the discrete evolution equation involves a sum over all previous time levels we show how the computational cost can be reduced to o mn log n operations and o m log n active memory locations

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

numerical asymptotic stability for the integro differential equations with the multi term kernels we prove that a second order backward difference semi discretization approximation for the integro differential equations with the multi term kernels preserves the weighted asymptotic integrabilities of continuous solutions the method of proof extend and simulate numerically those introduced by hannsgen and wheeler relying on deep frequency domain techniques

a non uniform time stepping convex splitting scheme for the time fractional cahn hilliard equation abstract in this paper a non uniform time stepping convex splitting numerical algorithm for solving the widely used time fractional cahn hilliard equation is introduced the proposed numerical scheme employs the l 1 formula for discretizing the time fractional derivative and a second order convex splitting technique to deal with the non linear term semi implicitly then the pseudospectral method is utilized for spatial discretization as a result the fully discrete scheme has several advantages second order accurate in time spectrally accurate in space uniquely solvable mass preserving and unconditionally energy stable rigorous proofs are given along with several numerical results to verify the theoretical results and to show the accuracy and effectiveness of the proposed scheme also some interesting phase separation dynamics of the time fractional cahn hilliard equation has been investigated

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

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

an hp version discontinuous galerkin method for integro differential equations of parabolic type we study the numerical solution of a class of parabolic integro differential equations with weakly singular kernels we use an hp version discontinuous galerkin dg method for the discretization in time we derive optimal hp version error estimates and show that exponential rates of convergence can be achieved for solutions with singular temporal behavior near t 0 caused by the weakly singular kernel moreover we prove that by using nonuniformly refined time steps optimal algebraic convergence rates can be achieved for the h version dg method we then combine the dg time stepping method with a standard finite element discretization in space and present an optimal error analysis of the resulting fully discrete scheme our theoretical results are numerically validated in a series of test problems

hamiltonian derivation of a detailed fluctuation theorem we analyze the microscopic evolution of a system undergoing a far from equilibrium thermodynamic process explicitly accounting for the degrees of freedom of participating heat reservoirs we derive a hybrid result similar in form to both the fluctuation theorem and a statement of detailed balance we relate this result to the steady state fluctuation theorem and to a free energy relation valid far from equilibrium