approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

sampled data consensus of linear multi agent systems with packet losses in this paper the consensus problem is studied for a class of multi agent systems with sampled data and packet losses where random and deterministic packet losses are considered respectively for random packet losses a bernoulli distributed white sequence is used to describe packet dropouts among agents in a stochastic way for deterministic packet losses a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way the purpose of this paper is to derive consensus criteria such that linear multi agent systems with sampled data and packet losses can reach consensus by means of the lyapunov function approach and the decomposition method the design problem of a distributed controller is solved in terms of convex optimization the interplay among the allowable bound of the sampling interval the probability of random packet losses and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions the obtained criteria are closely related to the maximum eigenvalue of the laplacian matrix versus the second minimum eigenvalue of the laplacian matrix which reveals the intrinsic effect of communication topologies on consensus performance finally simulations are given to show the effectiveness of the proposed results

a fast semi discrete kansa method to solve the two dimensional spatiotemporal fractional diffusion equation abstract fractional order diffusion equations fdes extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media real world diffusion can be multi dimensional requiring efficient numerical solvers that can handle long term memory embedded in mass transport to address this challenge a semi discrete kansa method is developed to approximate the two dimensional spatiotemporal fde where the kansa approach first discretizes the fde then the gauss jacobi quadrature rule solves the corresponding matrix and finally the mittag leffler function provides an analytical solution for the resultant time fractional ordinary differential equation numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes applications further show that the numerical method can efficiently solve two dimensional spatiotemporal fde models with either a continuous or discrete mixing measure hence this study provides an efficient and fast computational method for modeling super diffusive sub diffusive and mixed diffusive processes in large two dimensional domains with irregular shapes

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

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

event driven multi consensus of multi agent networks with repulsive links due to diversification in environment a network of agents may have several emergent behaviors this paper introduces an event driven paradigm for realizing multi consensus which as a generic version of group cluster consensus depends both on the initial underlying topology and initial states for the network feature we resort to a distributed event driven paradigm since event driven control has advantages on communication reduction and control energy saving moreover to relax assumption on network topology repulsive links are exploited at event instants to facilitate multiple coordination to realize multi consensus we design a distributed event driven controller based on coupled intra subgroup and extra subgroup information in this context with different control strengths the finial value of man would be varying this phenomenon matches the principle of multi consensus in the current study based on lasalle s invariance principle we prove that under the proposed event driven configuration the joint cooperation and competition contributes to multi consensus provided that the sampling period is no larger than a positive threshold it is shown there is no further requirement on network topologies in pursuit of multiple coordination this paper studies multiple coordination of multi agent networks under an event driven paradigm for an undirected connected graph a node clustering scheme is first adopted to ensure a relatively strong degree of connectivity within each potential subgroup and some repulsive effect is used to deal with the extra subgroup links to reduce unnecessary communication a distributed event driven controller is designed via coupled intra subgroup and extra subgroup information based on the lasalle s invariance principle it is shown that under the proposed event driven control configuration multi agent networks can realize multi consensus without any balanced requirement on the underlying topologies simulation work is presented to validate the theoretical results

flocking of multi agent systems with multiplicative and independent measurement noises abstract we investigate the effect of noise and group density on the emergence of flocking in a stochastic cucker smale type system we consider the independent and multiplicative measurement noise and derive sufficient conditions for flocking and non flocking in terms of the noise intensity and group density we report that in noise environments it is rather easy for the group with high density to form and maintain ordered group motion moreover we find that under the perturbation of multiplicative noise systems with weak coupling can tolerate relatively strong perturbation of noise

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

on prediction of individual sequences sequential randomized prediction of an arbitrary binary sequence isinvestigated no assumption is made on the mechanism of generating the bit sequence the goal of the predictor is to minimize its relative loss i e to make almost as few mistakes as the best expert in a fixed possibly infinite set of experts we point out a surprising connection between this prediction problem and empirical process theory first in the special case of static memoryless experts we completely characterize the minimax relative loss in terms of the maximum of an associated rademacher process then we show general upper and lower bounds on the minimaxrelative loss in terms of the geometry of the class of experts as main examples we determine the exact order of magnitude of the minimax relative loss for the class of autoregressive linear predictors and for the class of markov experts

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

fractional diffusion probability distributions and random walk models abstract we present a variety of models of random walk discrete in space and time suitable for simulating random variables whose probability density obeys a space time fractional diffusion equation

on impulsive control of a periodically forced chaotic pendulum system we consider impulsive control of a periodically forced pendulum system which has rich chaos and bifurcation phenomena a new impulsive control method for chaos suppression of this pendulum system is developed some simple sufficient conditions for driving the chaotic state to its zero equilibrium are presented and some criteria for eventually exponentially asymptotical stability are established this work provides a rigorous theoretical analysis to support some early experimental observations on controlling chaos in the periodically forced pendulum system

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

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

numerical methods 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

distributed impulsive group consensus in second order multi agent systems under directed topology this paper investigates the impulsive group consensus problems of second order multi agent systems under directed network topology with acyclic partition two distributed impulsive sampled data protocols are presented to realise group consensus for both cases with leaders and leaderless and then some simple yet generic criteria on convergence for such algorithms are established it is shown that for the case with leaderless a feature of the present protocols is to give an explicit expression of group consensus states in terms of the initial values of the agents which allows us to develop a simple approach yielding the desired group or cluster consensus while for the case of virtual leaders a pinning like tracking technique is effectively designed to regulate all the agents access to the virtual leaders respectively for the purpose of practical control strategy finally simulation results are presented to illustrate the effectiveness of the theoretical analysis

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

an efficient fourier spectral exponential time differencing method for the space fractional nonlinear schrodinger equations abstract a fourth order time discretization scheme based on the exponential time differencing approach with fourier spectral method in space is proposed for the space fractional nonlinear schrodinger equations the stability and convergence of the numerical scheme are discussed it is shown that the proposed numerical scheme is fourth order convergent in time and spectral convergent in space numerical experiments are performed on one two and three dimensional fractional nonlinear schrodinger equations and systems of two and three dimensional equations in addition a realistic two dimensional example with the solution of a singularity occurring in finite time is included the results demonstrate accuracy efficiency and reliability of the scheme computational results arising from the experiments are compared with relevant known schemes such as the fourth order split step fourier method the fourth order explicit runge kutta method and a mass conservative spectral method

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

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

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

wave field simulation for heterogeneous porous media with singular memory drag force the objective of this paper is to use biot s theory and the johnson koplik dashen dynamic permeability model in wave field simulation of a heterogeneous porous medium the johnson koplik dashen dynamic permeability model was originally formulated in the frequency domain in this paper the time domain drag force expression of the model is expressed in terms of the shifted fractional derivative of the relative fluid velocity in contrast to the exponential type viscous relaxation models the convolution operator in the johnson koplik dashen dynamic permeability model cannot be replaced by memory variables satisfying first order relaxation differential equations a new method for calculating the shifted fractional derivative without storing and integrating the entire velocity histories is developed using the new method to calculate the fractional derivative the governing equations for the two dimensional porous medium are reduced to a system of first order differential equations for velocities stresses pore pressure and the quadrature variables associated with the drag forces spatial derivatives involved in the first order differential equations system are calculated by fourier pseudospectral method while the time derivative of the system is discretized by a predictor corrector method for the demonstration of our method some numerical results are presented

numerical study of interacting particles approximation for integro differential equations the paper develops a numerical method based on the interacting particles approximation propagation of chaos for the solution of a large class of evolution problems involving the fractional laplacian operator and a non local quadratic type non linearity coupled stochastic differential equations driven by levy symmetric a stable processes are integrated numerically using euler s method and the solutions of the governing equations are obtained from their statistics the method is tested on several one and two dimensional examples and established analytical properties of the solutions are verified for the numerical approximates when they are available for initial conditions that are either integrable or monotone bounded functions it is shown that these methods represent viable tools for constructing the solution to the cauchy problem

a compact difference scheme for a two dimensional fractional klein gordon equation with neumann boundary conditions in this paper a high order finite difference scheme for a two dimensional fractional klein gordon equation subject to neumann boundary conditions is proposed the difficulty induced by the nonlinear term and the neumann conditions is carefully handled in the proposed scheme the stability and convergence of the finite difference scheme are analyzed using the matrix form of the scheme numerical examples are provided to demonstrate the theoretical results

spectral iterative method and convergence analysis for solving nonlinear fractional differential equation abstract in this study a new spectral iterative method sim based on fractional interpolation is presented for solving nonlinear fractional differential equations fdes involving caputo derivative this method is equipped with a pre algorithm to find the singularity index of solution of the problem this pre algorithm gives us a real parameter as the index of the fractional interpolation basis for which the sim achieves the highest order of convergence in comparison with some recent results about the error estimates for fractional approximations a more accurate convergence rate has been attained we have also proposed the order of convergence for fractional interpolation error under the l 2 norm finally general error analysis of sim has been considered the numerical results clearly demonstrate the capability of the proposed method

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

fast tensor product solvers for optimization problems with fractional differential equations as constraints fractional differential equations have recently received much attention within computational mathematics and applied science and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms an optimization problem with constraints given by fractional differential equations is considered which in its discretized form leads to a high dimensional tensor equation to reduce the computation time and storage the solution is sought in the tensor train format we compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned krylov solvers or tailored alternating schemes the competitiveness of these approaches is presented using several examples with constant and variable coefficients

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions this paper provides a rigorous and delicate analysis for exponential decay of jacobi polynomial expansions of analytic functions associated with the bernstein ellipse using an argument that can recover the best estimate for the chebyshev expansion we derive various new and sharp bounds of the expansion coefficients which are featured with explicit dependence of all related parameters and valid for degree n ge 1 we demonstrate the sharpness of the estimates by comparing with existing ones in particular the very recent results in siam j numer anal 50 2012 pp 1240 1263 we also extend this argument to estimate the gegenbauer gauss quadrature remainder of analytic functions which leads to some new tight bounds for quadrature errors

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

modeling and numerical analysis of fractional order bloch equations this paper deals with the bloch equations which are a set of macroscopic equations that are used for modeling of nuclear magnetization as a function of time these equations were introduced by felix bloch in 1946 and they are used for a description of the nuclear magnetic resonance nmr this physical phenomenon is used in medicine chemistry physics and engineering to study complex material fractional order generalization of the bloch equations was presented by richard magin et al in 2008 as an opportunity to extend their use to describe a wider range of experimental situations involving heterogeneous porous or composite materials in this paper we describe numerical and simulation models created for matlab simulink of the classical and the fractional order bloch equations the behaviour and stability analysis of the bloch equations are presented as well

fractional bloch equation with delay in this paper we investigate a fractional generalization of the bloch equation that includes both fractional derivatives and time delays the appearance of the fractional derivative on the left side of the bloch equation encodes a degree of system memory in the dynamic model for magnetization the introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation the analysis of this system shows different stability behavior for the t 1 and the t 2 relaxation processes the t 1 decay is stable for the range of delays tested 1 100 ms while the t 2 relaxation in this model exhibited a critical delay typically 6 ms above which the system was unstable delays are expected to appear in nmr systems in both the system model and in the signal excitation and detection processes therefore by including both the fractional derivative and finite time delays in the bloch equation we believe that we have established a more complete and more realistic model for nmr resonance and relaxation

the bem for numerical solution of partial fractional differential equations a numerical method is presented for the solution of partial fractional differential equations fdes arising in engineering applications and in general in mathematical physics the solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape the method is based on the concept of the analog equation which in conjunction with the boundary element method bem enables the spatial discretization and converts a partial fde into a system of coupled ordinary multi term fdes then this system is solved using the numerical method for the solution of such equations developed recently by katsikadelis the method is illustrated by solving second order partial fdes and its efficiency and accuracy is validated

quasi synchronization of heterogeneous dynamic networks via distributed impulsive control this paper studies leader follower synchronization in heterogeneous dynamic networks via distributed impulsive control the goal is to drive the followers to approximately synchronize with the leader within a nonzero error bound referred to as quasi synchronization some fundamental and yet challenging problems are addressed 1 how to obtain a tight quasi synchronization error bound 2 how many and which nodes should be controlled 3 how to design the coupling strength to select the pinned nodes and to determine the impulse intervals to optimize the error bound or to achieve quasi synchronization within a prescribed error bound first some general quasi synchronization conditions are derived with explicit expressions of the error bound second the pinning strategy and the coupling strength in achieving quasi synchronization are explored and expressed in term of impulse intervals it is interesting to find that a large coupling strength will destroy synchrony in contrast to continuous pinning control furthermore the common requirement that the leader has a directed path to every node is needed only in the case that the impulse intervals can be arbitrarily small a new concept referred to as impulse pinning controllability is introduced to the case that impulses cannot take place too frequently third an optimization of the error bound is discussed and formulated under a prescribed error bound the design problem of the coupling strength pinned nodes and impulse intervals is solved finally three examples are given to verify the theoretical results

sampled data discrete time coordination algorithms for double integrator dynamics under dynamic directed interaction in this article we study two sampled data based discrete time coordination algorithms for multi vehicle systems with double integrator dynamics under dynamic directed interaction for both algorithms we derive sufficient conditions on the interaction graph the damping gain and the sampling period to guarantee coordination by using the property of infinity products of stochastic matrices when the conditions on the damping gain and the sampling period are satisfied the first algorithm guarantees coordination on positions with a zero final velocity if the interaction graph has a directed spanning tree jointly while the second algorithm guarantees coordination on positions with a constant final velocity if the interaction graph has a directed spanning tree at each time interval simulation results are presented to show the effectiveness of the theoretical results

least squares spectral method for the solution of a fractional advection dispersion equation fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields this article describes a least squares spectral method for solving advection dispersion equations using caputo or riemann liouville fractional derivatives a gauss lobatto jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition exponential convergence rate of the operator is verified when increasing the order of the approximation solutions are calculated for fractional time and fractional space differential equations comparisons with finite difference schemes are included a significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate

convergence analysis for three parareal solvers we analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem mathbf u a mathbf u g the coarse propagator mathcal g is fixed to the backward euler method and three time integrators are chosen for the mathcal f propagator the trapezoidal rule the third order diagonal implicit runge kutta rk dirk method and the fourth order gauss rk method it is well known that the parareal euler algorithm using the backward euler method for mathcal f and mathcal g converges rapidly but less is known when one uses for mathcal f the trapezoidal rule or the fourth order gauss rk method especially when the mesh ratio j delta t delta t is small we show that for a specified lambda max the maximal eigenvalue of a or its upper bound there exists some critical j rm cri such that the parareal solvers derived from these three choices of mathcal f converge as fast as parareal euler provided j geq j rm cri

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

stable evaluations of fractional derivative of the muntz legendre polynomials and application to fractional differential equations abstract the aim of this paper is to present efficient and stable methods to compute caputo fractional derivative cfd of the muntz legendre polynomials based on three term recurrence relations and gauss jacobi quadrature rules this approach with collocation method at chebyshev gauss lobatto points has been applied for solving linear and nonlinear fractional multi order differential equations fdes described in caputo sense the main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations in this work for the first time we present the new rates of convergence for projection error which are more accurate than the rate presented by shen and wang in shen and wang 2016 moreover we present convergence rate for spectral collocation method for linear fdes with initial value on a finite interval and endpoint singularities also we propose an error analysis for jacobi gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper finally the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation

coordination and consensus of networked agents with noisy measurements stochastic algorithms and asymptotic behavior this paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors states for consensus seeking we propose stochastic approximation type algorithms with a decreasing step size and introduce the notions of mean square and strong consensus although the decreasing step size reduces the detrimental effect of the noise it also reduces the ability of the algorithm to drive the individual states towards each other the key technique is to ensure a trade off for the decreasing rate of the step size by following this strategy we first develop a stochastic double array analysis in a two agent model which leads to both mean square and strong consensus and extend the analysis to a class of well studied symmetric models subsequently we consider a general network topology and introduce stochastic lyapunov functions together with the so called direction of invariance to establish mean square consensus finally we apply the stochastic lyapunov analysis to a leader following scenario

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

generalized fractional diffusion equations for accelerating subdiffusion and truncated levy flights fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time for processes lacking such scaling the corresponding description may be given by diffusion equations with fractional derivatives of distributed order such equations were introduced in a v chechkin r gorenflo and i sokolov phys rev e 66 046129 2002 for the description of the processes getting more anomalous in the course of time decelerating subdiffusion and accelerating superdiffusion here we discuss the properties of diffusion equations with fractional derivatives of the distributed order for the description of anomalous relaxation and diffusion phenomena getting less anomalous in the course of time which we call respectively accelerating subdiffusion and decelerating superdiffusion for the former process by taking a relatively simple particular example with two fixed anomalous diffusion exponents we show that the proposed equation effectively describes the subdiffusion phenomenon with diffusion exponent varying in time for the latter process we demonstrate by a particular example how the power law truncated levy stable distribution evolves in time to the distribution with power law asymptotics and gaussian shape in the central part the special case of two different orders is characteristic for the general situation in which the extreme orders dominate the asymptotics

on hybrid impulsive and switching systems and application to nonlinear control in this note a new class of hybrid impulsive and switching models is introduced and their asymptotic stability properties are investigated using switched lyapunov functions some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established in addition a new hybrid impulsive and switching control strategy for nonlinear systems is developed a typical example the unified chaotic system is given to illustrate the theoretical results

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

the grunwald letnikov method for fractional differential equations this paper is devoted to the numerical treatment of fractional differential equations based on the grunwald letnikov definition of fractional derivatives finite difference schemes for the approximation of the solution are discussed the main properties of these explicit and implicit methods concerning the stability the convergence and the error behavior are studied related to linear test equations the asymptotic stability and the absolute stability of these methods are proved error representations and estimates for the truncation propagation and global error are derived numerical experiments are given

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

impulsive consensus algorithms for second order multi agent networks with sampled information in this paper the distributed consensus problem for second order continuous time multi agent networks with sampled data communication is investigated motivated by impulsive control strategy two kinds of impulsive distributed consensus algorithm are proposed these algorithms only utilize the sampled information and are implemented at sampled times by using the stability theory of impulsive systems and properties of the laplacian matrix necessary and sufficient conditions are obtained to ensure the consensus of the controlled networks it is shown that the control gains the sampling period and the eigenvalues of the laplacian matrix of the communication graph play key roles in achieving the consensus a numerical example is given to illustrate the results

an mhss like iteration method for two by two linear systems with application to fde optimization problems abstract in this paper we exploit the numerical solvers for a fractional differential equations fde optimization problem with constraints given by fractional elliptic state equations the discretized linear system can be rewritten as a two by two linear system by making use of the strategy of modified hermitian and skew hermitian splitting mhss iteration method proposed by bai we propose an mhss like iteration method comparing with the mhss iteration method the mhss like method only requires one to solve the two by two linear systems by a sparse cholesky factorization and a fast fourier transform hence the mhss like method has less workload than the mhss method the convergence properties and the quasi optimal parameters are analyzed in detail numerical examples are used to testify the efficiency of the new method

discrete random walk models for symmetric levy feller diffusion processes we propose a variety of models of random walk discrete in space and time suitable for simulating stable random variables of arbitrary index 0 2 in the symmetric case we show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous markovian stochastic processes which we refer to as levy feller diffusion processes

a kronecker product splitting preconditioner for two dimensional space fractional diffusion equations abstract we study preconditioned iterative methods for the linear system arising in the numerical discretization of a two dimensional space fractional diffusion equation our approach is based on a formulation of the discrete problem that is shown to be the sum of two kronecker products by making use of an alternating kronecker product splitting iteration technique we establish a class of fixed point iteration methods theoretical analysis shows that the new method converges to the unique solution of the linear system moreover the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real the basic iteration is accelerated by a krylov subspace method like gmres the corresponding preconditioner is in a form of a kronecker product structure and requires at each iteration the solution of a set of discrete one dimensional fractional diffusion equations we use structure preserving approximations to the discrete one dimensional fractional diffusion operators in the action of the preconditioning matrix numerical examples are presented to illustrate the effectiveness of this approach

brief paper second order consensus in multi agent dynamical systems with sampled position data this paper studies second order consensus in multi agent dynamical systems with sampled position data a distributed linear consensus protocol with second order dynamics is designed where both the current and some sampled past position data are utilized it is found that second order consensus in such a multi agent system cannot be reached without any sampled position data under the given protocol while it can be achieved by appropriately choosing the sampling period a necessary and sufficient condition for reaching consensus of the system in this setting is established based on which consensus regions are then characterized it is shown that if all the eigenvalues of the laplacian matrix are real then second order consensus in the multi agent system can be reached for any sampling period except at some critical points depending on the spectrum of the laplacian matrix however if there exists at least one eigenvalue of the laplacian matrix with a nonzero imaginary part second order consensus cannot be reached for sufficiently small or sufficiently large sampling periods in such cases one nevertheless may be able to find some disconnected stable consensus regions determined by choosing appropriate sampling periods finally simulation examples are given to verify and illustrate the theoretical analysis

a meshless method based on point interpolation method pim for the space fractional diffusion equation this paper aims to develop a meshless approach based on the point interpolation method pim for numerical simulation of a space fractional diffusion equation two fully discrete schemes for the one dimensional space fractional diffusion equation are obtained by using the pim and the strong forms of the space diffusion equation numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach

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

spectral approximation methods and error estimates for caputo fractional derivative with applications to initial value problems abstract in this paper we revisit two spectral approximations including truncated approximation and interpolation for caputo fractional derivative the two approaches have been studied to approximate riemann liouville r l fractional derivative by chen et al and zayernouri et al respectively in their most recent work for truncated approximation the reconsideration partly arises from the difference between fractional derivative in r l sense and caputo sense caputo fractional derivative requires higher regularity of the unknown than r l version another reason for the reconsideration is that we distinguish the differential order of the unknown with the index of jacobi polynomials which is not presented in the previous work also we provide a way to choose the index when facing multi order problems by using generalized hardy s inequality the gap between the weighted sobolev space involving caputo fractional derivative and the classical weighted space is bridged then the optimal projection error is derived in the non uniformly jacobi weighted sobolev space and the maximum absolute error is presented as well for the interpolation analysis of interpolation error was not given in their work in this paper we build the interpolation error in non uniformly jacobi weighted sobolev space by constructing fractional inverse inequality with combining collocation method the approximation technique is applied to solve fractional initial value problems fivps numerical examples are also provided to illustrate the effectiveness of this algorithm

brief paper some necessary and sufficient conditions for second order consensus in multi agent dynamical systems this paper studies some necessary and sufficient conditions for second order consensus in multi agent dynamical systems first basic theoretical analysis is carried out for the case where for each agent the second order dynamics are governed by the position and velocity terms and the asymptotic velocity is constant a necessary and sufficient condition is given to ensure second order consensus and it is found that both the real and imaginary parts of the eigenvalues of the laplacian matrix of the corresponding network play key roles in reaching consensus based on this result a second order consensus algorithm is derived for the multi agent system facing communication delays a necessary and sufficient condition is provided which shows that consensus can be achieved in a multi agent system whose network topology contains a directed spanning tree if and only if the time delay is less than a critical value finally simulation examples are given to verify the theoretical analysis

a new derivative with normal distribution kernel theory methods and applications abstract new approach of fractional derivative with a new local kernel is suggested in this paper the kernel introduced in this work is the well known normal distribution that is a very common continuous probability distribution this distribution is very important in statistics and also highly used in natural science and social sciences to portray real valued random variables whose distributions are not known two definitions are suggested namely atangana gomez averaging in liouville caputo and riemann liouville sense we presented some relationship with existing integrals transform operators numerical approximations for first and second order approximation are derived in detail some applications of the new mathematical tools to describe some real world problems are presented in detail this is a new door opened the field of statistics natural and socials sciences

high order algorithms for riesz derivative and their applications ii in this paper we firstly develop two high order approximate formulas for the riesz fractional derivative secondly we propose a temporal second order numerical method for a fractional reaction dispersion equation where we discretize the riesz fractional derivative by using two numerical schemes we prove that the numerical methods for a spatial riesz fractional reaction dispersion equation are both unconditionally stable and convergent and the orders of convergence are o 2 h 6 and o 2 h 8 in which and h are spatial and temporal step sizes respectively finally we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis

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

state agreement for continuous time coupled nonlinear systems two related problems are treated in continuous time first the state agreement problem is studied for coupled nonlinear differential equations the vector fields can switch within a finite family associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems generalizing the work of moreau under the assumption that the vector fields satisfy a certain subtangentiality condition it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time the proof uses nonsmooth analysis second the rendezvous problem for kinematic point mass mobile robots is studied when the robots fields of view have a fixed radius the circumcenter control law of ando e ieee trans robotics automation 15 1999 pp 818 828 is shown to solve the problem the rendezvous problem is a kind of state agreement problem but the interaction structure is state dependent

fractional diffusion in inhomogeneous media starting from the continuous time random walk ctrw scheme with the space dependent waiting time probability density function pdf we obtain the time fractional diffusion equation with varying in space fractional order of time derivative as an example we study the evolution of a composite system consisting of two separate regions with different subdiffusion exponents and demonstrate the effects of non trivial drift and subdiffusion whose laws are changed in the course of time

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

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

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

consensus of second order multi agent systems via impulsive control using sampled hetero information in this paper the consensus problem for second order multi agent systems using impulsive control with sampled hetero information is investigated necessary and sufficient conditions for sampling interval to achieve consensus for the second order multi agent systems are obtained analysis for the upper bound of sampling interval convergence performance and communication cost of the proposed control protocols are also discussed some numerical examples are given to demonstrate the effectiveness of the proposed control protocols

robust consensus control with guaranteed rate of convergence using second order hurwitz polynomials abstractthis paper considers homogeneous networks of general linear time invariant second order systems we consider linear feedback controllers and require that the directed graph associated with the network contains a spanning tree and systems are stabilisable we show that consensus with a guaranteed rate of convergence can always be achieved using linear state feedback to achieve this we provide a new and simple derivation of the conditions for a second order polynomial with complex coefficients to be hurwitz we apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph laplacian matrix may have complex eigenvalues based on the conditions found methods to compute feedback gains are proposed we show that gains can be chosen such that consensus is achieved robustly over a variety of communication structures and system dynamics we also consider the use of static output feedback

impulsive synchronization of discrete time networked oscillators with partial input saturation abstract in this paper a novel impulsive control with partial input saturation is proposed to synchronize a class of discrete time delayed coupled networks the proposed impulsive control with partial input saturation means that the considered impulsive controllers are subject to input saturation and only a fraction of nodes are injected with controllers for this purpose the razumikhin type technique and mathematical induction are utilized to derive a sufficient criterion in addition the control gains are calculated by transforming the criterion into a convex optimization problem finally the result is verified through numerical simulation