a fast galerkin finite element method for a space time fractional allen cahn equation abstract in this paper a space time fractional allen cahn equation is investigated to account for the memory effect of certain materials or the anomalous diffusion processes in heterogeneous distributed media for the mixture of two immiscible phases due to the nonlocal features of fractional derivatives conventional solvers such as the gauss elimination method require o m 2 m n memory units and o m n 2 m 3 n operations to solve the space time fractional allen cahn equation where m is the number of spatial unknowns and n is the number of time steps by exploring the special structure of the stiffness matrix and utilizing the invariant energy quadratization approach we developed an efficient linearized divide and conquer galerkin finite element method without resorting to any lossy compression this new method is efficient as it significantly reduces the memory requirement to o m n and the operation count to o m n log 2 n log m we have developed a fast galerkin finite element method with local refinement that allows accurate capture of the coarsening dynamics at the interface furthermore we have investigated the modeling capacities and energy dissipation properties of the space time fractional allen cahn equation numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space time fractional allen cahn equation

the high order compact numerical algorithms for the two dimensional fractional sub diffusion equation in this paper performing the average operators on the space variables a numerical scheme with third order temporal convergence for the two dimensional fractional sub diffusion equation is considered for which the unconditional stability and convergence in l1 l norm are strictly analyzed for 0 0 9569347 by using the discrete energy method therewith adding small perturbation terms we construct a compact alternating direction implicit difference scheme for the two dimensional case finally some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all 0 1

finite time stability of quantum systems governed by schrodinger s equation in this paper we investigate the finite time stability fts and finite time boundedness ftb of quantum systems which are governed by schrodinger s equation we first study the fts and fts at one target state and propose new results for the mentioned quantum systems in the form of sufficient conditions then we present sufficient conditions for the ftb and ftb at one target state in terms of matrix inequalities one numerical simulation is presented to illustrate the effectiveness of the obtained theoretical result

approximations of fractional integrals and caputo fractional derivatives abstract in this paper we propose two algorithms for numerical fractional integration and caputo fractional differentiation we present a modification of trapezoidal rule that is used to approximate finite integrals the new modification extends the application of the rule to approximate integrals of arbitrary order 0 we then using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order 0 where the fractional derivative based on caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points the study is conducted through illustrative examples and error analysis

stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks this paper is concerned with the stability of n dimensional stochastic differential delay systems with nonlinear impulsive effects first the equivalent relation between the solution of the n dimensional stochastic differential delay system with nonlinear impulsive effects and that of a corresponding n dimensional stochastic differential delay system without impulsive effects is established then some stability criteria for the n dimensional stochastic differential delay systems with nonlinear impulsive effects are obtained finally the stability criteria are applied to uncertain impulsive stochastic neural networks with time varying delay the results show that this convenient and efficient method will provide a new approach to study the stability of impulsive stochastic neural networks some examples are also discussed to illustrate the effectiveness of our theoretical results

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

computing eigenelements of boundary value problems with fractional derivatives variational iteration method has been successfully implemented to handle linear and nonlinear differential equations the main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper first a general framework of the variational iteration method is presented for analytic treatment of differential equations of fractional order where the fractional derivatives are described in caputo sense second the new framework is used to compute approximate eigenvalues and the corresponding eigenfunctions for boundary value problems with fractional derivatives numerical examples are tested to show the pertinent features of this method this approach provides a new way to investigate eigenvalue problems with fractional order derivatives

fox h functions in fractional diffusion the h functions introduced by fox in 1961 are special functions of a very general nature which allow one to treat several phenomena including anomalous diffusion in a unified and elegant framework in this paper we express the fundamental solutions of the cauchy problem for the space time fractional diffusion equation in terms of proper fox h functions based on their mellin barnes integral representations we pay attention to the particular cases of space fractional time fractional and neutral fractional diffusion

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

finite time stability conditions for non autonomous continuous systems finite time stability is defined for continuous non autonomous systems starting with a result from haimo 1986 we then extend this result to n dimensional non autonomous systems through the use of smooth and non smooth lyapunov functions as in perruquetti and drakunov 2000 one obtains two different sufficient conditions and a necessary one for finite time stability of continuous non autonomous systems

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

high order central difference scheme for caputo fractional derivative abstract in this paper we propose a class of central difference schemes for resolving the caputo fractional derivative the accuracy may reach any selected integer order more precisely the caputo fractional derivative operator is decomposed into symmetric and antisymmetric components starting from difference schemes of lower order accuracy for each component we enhance the accuracy by a weighted average of shifted differences the weights are calculated by matching the symbols of the scheme and the operators we further illustrate the application of the proposed schemes to a fractional advection diffusion equation together with the crank nicolson algorithm it reaches designed accuracy order and is unconditionally stable numerical tests are presented to demonstrate the nice features

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

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 compact integrated rbf method for time fractional convection diffusion reaction equations abstract in this paper a local compact integrated radial basis function cirbf method is proposed to solve the time fractional convection diffusion reaction equations the proposed cirbf scheme derives high accuracy and fast convergence rate with grid refinement the spatial derivatives are discretized by combining with the integrated rbf approximation and compact approximation in a stencil the nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the rbf weight space the time discretization is replaced by the second order shifted grunwald scheme moreover the unconditional stability and convergence of the proposed method are shown finally some numerical examples on 2d and 3d domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations

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

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

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

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

on finite time stability of nonautonomous nonlinear systems this paper is concerned with the finite time stability fts of nonautonomous nonlinear systems firstly a new definition of fts for nonautonomous nonlinear systems is introduced then some suffic

a high order schema for the numerical solution of the fractional ordinary differential equations in this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations fodes this technique is based on the so called block by block approach which is a common method for the integral equations in our approach the classical block by block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps while preserving the good stability property of the block by block schemes by using this new approach we are able to construct a high order schema for fodes of the order a a 0 the stability and convergence of the schema is rigorously established we prove that the numerical solution converges to the exact solution with order 3 a for 0 1 a series of numerical examples are provided to support the theoretical claims

controllability of quantum systems with switching control this article focuses on wavefunction controllability of a class of quantum systems with switching control first we present a specific quantum system which is not wavefunction controllable under the action of any one of two controllers with different fixed structures however we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers then we present a sufficient condition for wavefunction controllability of a class of quantum systems with switching control and briefly discuss the problem of control design for such quantum control systems finally two illustrative examples are presented to demonstrate a possible control design process for the proposed method

finite time stabilization of linear systems via impulsive control this paper introduces a new concept of finite time stability of linear time invariant impulsive systems the sufficient condition of finite time stability of the system via impulsive control at fixed times and variable times is obtained respectively such sufficient conditions of finite time stabilization are given in terms of matrix inequalities a numerical example is presented to illustrate the efficiency of the proposed results

optimal control of multi level quantum system with energy cost functional in this paper we have synthesized optimal control of multi level quantum system by representing unitary operator in terms of the projection operators of the hamiltonian of the system we have solved the steering problem for multi level quantum systems minimizing energy cost functional of the control the optimal control problem of the time evolution of quantum spin of pauli two level system subjected to an external field with the minimum energy function is also illustrated and formulated in terms of the quantum spin up and spin down states of the pauli two level system

three point combined compact difference schemes for time fractional advection diffusion equations with smooth solutions a ccd method with the l1 formula is proposed for solving time fractional advection diffusion equations the method is globally at least fifth order accurate in space and 2 th order accurate in time the unconditional stability is proved by means of fourier analysis numerical experiments are conducted to demonstrate the efficiency the present method preserves the higher order accuracy for advection dominated problems with the smooth solutions in this paper we study the numerical solution for the time fractional advection diffusion equation tfade a high order accurate three point combined compact difference scheme with the l1 formula is proposed to solve a class of tfades the method is globally 2 th order accurate in time and at least fifth order accurate in space for the constant coefficient tfades subject to periodic boundary conditions where is the order of time fractional derivative in the governing equation the unconditional stability and the high order convergence are proved using the fourier analysis method the proposed method can be extended to treat the variable coefficient tfades subject to other local boundary conditions several numerical examples are computed to validate the numerical accuracy effectiveness and robustness of the present method for tfades with smooth solutions and different boundary conditions moreover the proposed algorithm preserves the higher order accuracy for advection dominated problems with the smooth solutions

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

analysis of time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise abstract in this paper we establish the existence and uniqueness of the mild solution and the holder continuity for the time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise a full discrete scheme with spectral galerkin method in space and exponential method in time is proposed it is rigorously proved that the numerical scheme admits an unique solution finally we present the optimal error estimates of the proposed scheme based on the regularity analysis

an improved pc scheme for nonlinear fractional differential equations a novel computationally effective fractional predictorcorrector pc scheme is proposed to solve fractional differential equations involving caputo derivative the properties of the caputo derivative are used to reduce the fractional differential equation into a volterra integral equation to design high order numerical solution of fdes the simpsons 3 8 rule is applied to the volterra type integral equation the scheme is capable of handling both linear and nonlinear fractional differential equations a detailed error analysis and stability analysis of the numerical scheme are rigorously established the proposed scheme is compared with the pc schemes of literature for illustrating the effectiveness of the algorithm

a fourth order extrapolated compact difference method for time fractional convection reaction diffusion equations with spatially variable coefficients abstract this paper is concerned with numerical methods for a class of time fractional convection reaction diffusion equations the convection and reaction coefficients of the equation may be spatially variable based on the weighted and shifted grunwald letnikov formula for the time fractional derivative and a compact finite difference approximation for the spatial derivative we establish an unconditionally stable compact difference method the local truncation error and the solvability of the resulting scheme are discussed in detail the stability of the method and its convergence of third order in time and fourth order in space are rigorously proved by the discrete energy method combining this method with a richardson extrapolation we present an extrapolated compact difference method which is fourth order accurate in both time and space a rigorous proof for the convergence of the extrapolation method is given numerical results confirm our theoretical analysis and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method

time splitting schemes for fractional differential equations i smooth solutions we propose three time splitting schemes for nonlinear time fractional differential equations with smooth solutions where the order of the fractional derivative is 0 alpha 1 while one of the schemes is of order alpha the other two schemes are of order 1 alpha and 2 alpha and thus they can be combined to provide flexible numerical methods with convergence order no less than 3 2 we prove the convergence and stability of the proposed schemes numerical examples illustrate the flexibility and the efficiency of these time splitting schemes and show that they work for multirate and stiff time fractional differential systems effectively

a new high order adi numerical difference formula for time fractional convection diffusion equation abstract based on exponential transformation quadratic interpolation polynomial and pade approximation a new high order finite difference scheme is proposed for solving the two dimensional 2d time fractional convection dominated diffusion equation of order 0 1 the resulting scheme is of 3 order accuracy in time and fourth order accuracy in space in order to reduce the amount of computation the alternating direction implicit adi scheme is further developed numerical experiments are given to demonstrate the high accuracy and robustness of our new scheme

an efficient numerical solver for anisotropic subdiffusion problems abstract this paper presents an efficient and robust numerical solver for anisotropic subdiffusion problems which are important but not addressed directly in the literature the chebyshev spectral collocation method is utilized for discretization of the spatial laplacian whereas a linear interpolation is used for discretizing the fractional order caputo temporal derivative this solver is stable and catches the main features of subdiffusion numerical experiments are presented to demonstrate the accuracy and efficiency of this new solver

numerical method for solving diffusion wave phenomena we find solutions for the diffusion wave problem in 1d with n term time fractional derivatives whose orders belong to the intervals 0 1 1 2 and 0 2 respectively using the method of the approximation of the convolution by laguerre polynomials in the space of tempered distributions this method transfers the diffusion wave problem into the corresponding infinite system of linear algebraic equations through the coefficients which are uniquely solvable under some relations between the coefficients with index zero the method is applicable for nonlinear problems too

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

several effective algorithms for nonlinear time fractional models abstract in this paper several effective algorithms are proposed to simulate the time fractional models a complete analysis of the algorithms for nonlinear problems is presented while the previous investigations mainly focus on construction and analysis of l1 type method for the linear time fractional problems after that by using the sum of exponentials approximation we develop the corresponding accelerated methods we obtain that computational cost is reduced from o n 2 to o log n or o log 2 n where n denotes total number of time steps finally several numerical experiments on several real world models are proposed to confirm the effectiveness of the algorithms

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

global finite time stabilisation by output feedback for a class of uncertain nonlinear systems this article considers the problem of global finite time stabilisation by output feedback for a class of nonlinear systems comprised of a chain of power integrators perturbed by an uncertain vector field to solve the problem we first construct a homogeneous observer and controller in a recursive way for the nominal system without the perturbing nonlinearities then using the homogeneous domination approach we scale the homogeneous observer and controller with an appropriate choice of gain to render the uncertain nonlinear system globally finite time stable due to the use of a reduced order observer the proposed output feedback controller is applicable to those systems with unknown gains associated with the power integrators

stability and convergence of finite difference schemes for a class of time fractional sub diffusion equations based on certain superconvergence this paper is devoted to the construction and analysis of finite difference methods for solving a class of time fractional subdiffusion equations based on the certain superconvergence at some particular points of the fractional derivative by the traditional first order grunwald letnikov formula some effective finite difference schemes are derived the obtained schemes can achieve the global second order numerical accuracy in time which is independent of the values of anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme respectively are established for the one dimensional problem along with the strict analysis on the unconditional stability and convergence of these schemes by the discrete energy method furthermore the extension to the two dimensional case is also considered numerical experiments support the correctness of the theoretical analysis and effectiveness of the new developed difference schemes the superconvergent points of the traditional first order grunwald letnikov formula to approximate the riemann liouville fractional derivatives are used to construct the several finite difference schemes for solving a class of time fractional sub diffusion equations the obtained schemes can achieve the global second order accuracy in time which is independent of the anomalous diffusion exponent 0 1 in the governing equation the spatial second order scheme and the spatial fourth order compact scheme are presented respectively the strict stability and convergence analysis on the resulting difference schemes are given by the discrete energy method the extension to the two dimensional case is also discussed the compact difference scheme with the convergence of second order in time and fourth order in space is derived along with the strict stability and convergence analysis several numerical experiments are implemented to illustrate the computational efficiency numerical accuracy and robustness of the proposed schemes

finite time stability of linear time varying singular systems with impulsive effects in this paper we introduce a new concept of finite time stability of linear time varying singular systems with impulses at fixed times and present new results for the above mentioned class of system in the form of sufficient conditions a sufficient condition is given in terms of matrix inequalities which gives the opportunity of fitting the finite time control problem in the general framework via impulsive control a numerical example is presented to illustrate the efficiency of the proposed result

sequential amplitude estimation in multiuser communications considers the problem of multiuser amplitude estimation i e the problem of estimating the amplitudes of several digital communications signals superimposed in the same channel this problem is of importance in communications environments such as spread spectrum radio networks in which nonorthogonal multiplexing is used multiuser amplitude estimation is a critical prerequisite to the optimum demodulation of such signals using for example verdu s algorithm in the present paper a sequential detection estimation approach is applied to this problem and several estimation paradigms including the method of moments and likelihood based estimators are considered the consistency asymptotic variance and complexity of these estimators are examined a new method of constructing a recursive consistent and asymptotically efficient estimation algorithm out of a consistent estimator sequence is also suggested and is applied to the current setup it is seen that detector estimators that use these estimators in verdu s algorithm result asymptotically in known amplitude optimum error probabilities with little relative increase in complexity per demodulated bit

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

a higher order numerical method for time fractional partial differential equations with nonsmooth data abstract gao et al 11 2014 introduced a numerical scheme to approximate the caputo fractional derivative with the convergence rate o k 3 0 1 by directly approximating the integer order derivative with some finite difference quotients in the definition of the caputo fractional derivative see also lv and xu 20 2016 where k is the time step size under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth lv and xu 20 2016 proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate o k 3 0 1 uniformly with respect to the time variable t however in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate o k 3 0 1 uniformly with respect to the time variable t in this paper we first obtain a similar approximation scheme to the riemann liouville fractional derivative with the convergence rate o k 3 0 1 as in gao et al 11 2014 by approximating the hadamard finite part integral with the piecewise quadratic interpolation polynomials based on this scheme we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using laplace transform methods that the time discretization scheme has the convergence rate o k 3 0 1 for any fixed t n 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases numerical examples are given to show that the theoretical results are consistent with the numerical results

new explicit and accelerated techniques for solving fractional order differential equations abstract this paper proposes new direct and acceleration numerical methods for solving fractional order differential equations fdes for the caputo differential operator with fractional order 0 o h 2 and o h 3 respectively in order to overcome a strong singular issue as 0 we propose the explicit predictor corrector scheme with perturbation technique in case of 1 the truncation errors are reduced by o h and o h 2 in order to accelerate the convergence rate we decompose 2 2 convert the fde into the system of fdes and apply the predictor corrector scheme numerical tests for linear nonlinear variable order and time fractional sub diffusion problems demonstrate that the proposed methods give a prominent performance we also compare the numerical results with other high order explicit and implicit schema

efficient numerical solution of the time fractional diffusion equation by mapping from its brownian counterpart the solution of a caputo time fractional diffusion equation of order 0 1 is expressed in terms of the solution of a corresponding integer order diffusion equation we demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem in the context of an n point finite difference time discretisation the mapping allows for an improvement in time computational complexity from o n 2 to o n given a precomputation of o n 1 ln n the mapping is applied successfully to the least squares fitting of a fractional advection diffusion model for the current in a time of flight experiment resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes a mapping between fractional diffusion and normal diffusion is explored numerically an efficient solution method for fractional diffusion equations is proposed a benchmark fractional order model fit for scher montroll transport is presented

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

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

high order approximation to caputo derivatives and caputo type advection diffusion equations iii in this paper a series of new high order numerical approximations to th 0 1 order caputo derivative is constructed by using r th degree interpolation approximation for the integral function where r 4 is a positive integer as a result the new formulas can be viewed as the extensions of the existing jobs cao et al 2015 li et al 2014 the convergence orders are o r 1 where is the time stepsize two test examples are given to demonstrate the efficiency of these schemes then we adopt the derived schemes to solve the caputo type advection diffusion equation with dirichlet boundary conditions the local truncation error of the derived difference scheme is o r 1 h 2 where is the time stepsize and h the space one the stability and convergence of the proposed schemes for r 4 are also considered without loss of generality we only display the numerical examples for r 4 5 which support the numerical algorithms

continuous stabilization controllers for singular bilinear systems the state feedback case global asymptotic stabilization for a class of singular bilinear systems is first studied in this paper new approaches are developed by means of the lasalle invariant principle for nonlinear systems a new set of sufficient condition is first derived via the continuous static state feedback the feedback not only guarantees the existence of solution but also the global asymptotical stabilization for the closed loop system

analysis of lyapunov method for control of quantum states the natural trajectory tracking problem is studied for generic quantum states represented by density operators a control design based on the hilbert schmidt distance as a lyapunov function is considered the control dynamics is redefined on an extended space where the lasalle invariance principle can be correctly applied even for non stationary target states lasalle s invariance principle is used to derive a general characterization of the invariant set which is shown to always contain the critical points of the lyapunov function critical point analysis of the latter is used to show that for generic states it is a morse function with n isolated critical points including one global minimum one global maximum and n 2 saddles it is also shown however that the actual dynamics of the system is not a gradient flow and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points this analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular and in fact convergence to the target state trajectory in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space on the other hand if either of these sufficient conditions is not satisfied the target state ceases to be asymptotically stable a center manifold emerges around the target state and the control design ceases to be effective

lyapunov control methods of closed quantum systems according to special geometric or physical meanings the paper summarizes three lyapunov functions in controlling closed quantum systems and their controller designing processes specially for the average value based method the paper gives the generalized condition of the largest invariant set in the original reference and develops the construction method of the imaginary mechanical quantity for the error based method this paper gives its strict mathematical proof train of thought on the asymptotic stability and the corresponding physical meaning also we study the relations among the three lyapunov functions and give a unified form of these lyapunov functions finally we compare the control effects of three lyapunov methods by doing some simulation experiments

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

a high order l2 compact difference method for caputo type time fractional sub diffusion equations with variable coefficients abstract a high order compact finite difference method is proposed for solving a class of time fractional sub diffusion equations the diffusion coefficient of the equation may be spatially variable and the time fractional derivative is in the caputo sense with the order 0 1 the caputo time fractional derivative is discretized by a 3 th order numerical formula called the l2 formula here which is constructed by piecewise quadratic interpolating polynomials but does not require any sub stepping scheme for the approximation at the first time level the variable coefficient spatial differential operator is approximated by a fourth order compact finite difference operator by developing a technique of discrete energy analysis a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all 0 1 the optimal error estimate is obtained in the l2 norm and shows that the proposed method has the temporal 3 th order accuracy and the spatial fourth order accuracy further approximations are also considered for enlarging the applicability of the method while preserving its high order accuracy applications are given to three model problems and numerical results are presented to demonstrate the theoretical analysis results

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

numerical analysis and physical simulations for the time fractional radial diffusion equation we do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field based on rewriting the equation in a new form we first present two kinds of implicit finite difference schemes for numerically solving the equation then we strictly establish the stability and convergence results we prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes finally we do the physical simulations some interesting physical phenomena are revealed we verify that the long time asymptotic survival probability kt a but independent of the dimension where a is the anomalous diffusion exponent

lyapunov control of bilinear schrodinger equations a lyapunov based approach for trajectory tracking of the schrodinger equation is proposed in the finite dimensional case convergence is precisely analyzed connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied when the linearized system is controllable such a feedback ensures almost global asymptotic convergence when the linearized system is not controllable the stability of the closed loop system is not asymptotic to overcome such lack of convergence we propose when the reference trajectory is an eigenstate a modification based on adiabatic invariance simulations illustrate the simplicity and also the interest for trajectory generation

a center box method for radially symmetric solution of fractional subdiffusion equation in this paper a center box difference method is considered for the radially symmetric solution of fractional subdiffusion equation by method of order reduction the derivative boundary condition is transformed into dirichlet boundary condition and thus the geometrical singularity is successfully removed from the original problem as a matter of course a natural discretization scheme is obtained to investigate the stability and convergence of the method we define a new norm with a weight r d 1 thus the usual sobolev inequality is not suitable to the new norm therefore we prove three new sobolev like embedding inequalities which can also be applied to the other problems in polar coordinates then the scheme is proved to be unconditionally stable and convergent in maximum norm with the help of the new sobolev like embedding inequalities some illustrative examples are provided to demonstrate the theoretical results by some comparisons it can be seen that the natural discretization scheme is accurate and effective in physical simulations and it can be used to both long time and short time computation

some high order difference schemes for the space and time fractional bloch torrey equations in this paper several difference schemes are proposed for both one dimensional and two dimensional space and time fractional bloch torrey equations the spatial second order scheme and the spatial fourth order compact scheme are established respectively the obtained schemes can achieve the global second order numerical accuracy in time the unique solvability unconditional stability and convergence of the proposed schemes are proved by the energy method two adi schemes are also discussed for the two dimensional problem numerical examples are given to verify the numerical accuracy and efficiency of the difference schemes

galerkin finite element method for two dimensional space and time fractional bloch torrey equation abstract in this paper we consider the galerkin finite element method for the two dimensional space and time fractional bloch torrey equation 2d stfbte utilizing the l 2 1 formula to discretize temporal caputo derivative we obtain the local truncation error with o 3 in the temporal direction where is the order of time fractional derivative furthermore the semi discrete form for the problem is given and the stability and convergence of the semi discrete variational formulation are rigorously proved in l 2 norm and fractional norm respectively then we derive a fully discrete scheme for the 2d stfbte and investigate its convergence theoretically finally extensive numerical results based on linear piecewise polynomials verify the theoretical results and show the effectiveness of the proposed scheme

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