linear and nonlinear algorithms for identification in h sub infinity with error bounds a linear algorithm and a nonlinear algorithm for the problem of system identification in h sub infinity posed by helmicki et al 1990 for discrete time systems are presented the authors derive some error bounds for the linear algorithm which indicate that it is not robustly convergent however the worst case identification error is shown to grow as log n where n is the model order a robustly convergent nonlinear algorithm is derived and bounds on the worst case identification error in the h sub infinity norm are obtained

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

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

application of subival in solving initial value problems with fractional derivatives abstract the application of a numerical method for the approximation of the fractional derivative in riemann liouville and caputo definitions in initial value problems is discussed the method previously known as the subinterval based method is now referred to by its acronym subival for simpler future references it is dependent on subinterval partitions performed according to a proposed algorithm interpolations using selected time axis nodes i e nodes where solutions have been computed and analytical monomial integrodifferentiation formulae two exemplary circuit problems have been introduced as a test for the method these problems have analytical solutions available in literature the evaluations of these solutions have been compared with results obtained through an adaptive step size predictor corrector scheme where the core computations relied on the proposed numerical method subival has been implemented into an activex dynamic link library dll the paper contains instructions on how the predictor corrector algorithm can be implemented into a computer algebra system where the subival library is applied for the fractional derivative approximation examples of this are given in the form of scripts in matlab and mathematica these scripts generally allow to solve systems of fractional order state equations to which the introduced circuit examples can be brought to

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

chebycheff approximation in system identification and model reduction the aim of this paper is to show how chebycheff approximation may be used in frequency weighted h system identification and model reduction we show how these techniques may be combined with known results from hankel norm model reduction theory to reduce the error between the plant and the low order approximation in identification we also apply an l norm variant of levy s method levy 1959

eigenvalue clustering of coefficient matrices in the iterative stride reductions for linear systems solvers for linear systems with tridiagonal coefficient matrices sometimes employ direct methods such as the gauss elimination method or the cyclic reduction method in each step of the cyclic reduction method nonzero offdiagonal entries in the coefficient matrix move incrementally away from diagonal entries and eventually vanish the steps of the cyclic reduction method are generalized as forms of the stride reduction method for example the 2 stride reduction method coincides with the 1st step of the cyclic reduction method which transforms tridiagonal linear systems into pentadiagonal systems in this paper we explain arbitrary stride reduction for linear systems with coefficient matrices with three nonzero bands we then show that arbitrary stride reduction is equivalent to a combination of 2 stride reduction and row and column permutations we thus clarify eigenvalue clustering of coefficient matrices in the step by step process of the stride reduction method we also provide two examples verifying this property

chaos control of a fractional order modified coupled dynamos system abstract this paper analyzes some routh hurwitz stability conditions generalized to the fractional order case and discusses the stability region of the fractional order system we analyze the chaotic behavior of the fractional order modified coupled dynamos system concretely and provide the conditions suppressing chaos to unstable equilibrium points then use the feedback control method to control chaos in the fractional order modified coupled dynamos system numerical simulations show the effectiveness of the 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

worst case identification of continuous time systems via interpolation we consider a worst case robust control oriented identification problem recently studied by several authors this problem is one of h identification in the continuous time setting we give a more general formulation of this problem the available a priori information in this paper consists of a lower bound on the relative stability of the plant a frequency dependent upper bound on a certain gain associated with the plant and an upper bound on the noise level the available experimental information consists of a finite number of noisy plant point frequency response samples the objective is to identify from the given a priori and experimental information an uncertain model that includes a stable nominal plant model and a bound on the modeling error measured in h norm our main contributions include both a new identification algorithm and several new explicit lower and upper bounds on the identification error the proposed algorithm belongs to the class of interpolatory algorithms which are known to possess a desirable optimality property under a certain criterion the error bounds presented improve upon the previously available ones in the aspects of both providing a more accurate estimate of the identification error as well as establishing a faster convergence rate for the proposed algorithm

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

on zero order hold equivalents of distributed parameter systems various connections are established between linear time invariant distributed parameter continuous time systems and their zero order hold discrete time equivalents these connections are established in both the time and frequency domains the time domain connections relate various growth constants and norm bounds of the continuous time systems considered to those of their zero order hold discrete time equivalents the frequency domain connection provides an upper bound on the difference between the frequency response of a continuous time system and that of its zero order hold discrete time equivalent

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

worst case system identification in l 1 ne equation 1 optimal algorithms and error bounds abstract this paper analyses worst case identification of linear shift invariant systems with an l 1 error criterion the assumed apriori information on the plant are bounds on the system gain and decay rate of the impulse response of the unknown system and the experimental data is assumed to be a finite set of corrupted impulse or step response samples of the system

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

a class of algorithms for identification in h sub infinity continuous time case the problem of system identification in h sub infinity for the continuous time case is investigated it is shown that the class of systems with a lower bound on the relative stability an upper bound on the steady state gain and an upper bound on the roll off rate is admissible this allows one to develop a class of robustly convergent nonlinear algorithms the algorithms in this class have a two stage structure and are characterized by the use of window functions explicit worst case error bounds in h sub infinity norm between the identified model and the unknown system are given for a particular algorithm an example is provided to illustrate the application of the results obtained

crank nicolson difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative in this paper the crank nicolson cn difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative is studied the existence of this difference solution is proved by the brouwer fixed point theorem the stability and convergence of the cn scheme are discussed in the l 2 norm when the fractional order is two all those results are in accord with the difference scheme developed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also presented

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

novel numerical methods for solving the time space fractional diffusion equation in two dimensions in this paper a time space fractional diffusion equation in two dimensions tsfde 2d with homogeneous dirichlet boundary conditions is considered the tsfde 2d is obtained from the standard diffusion equation by replacing the first order time derivative with the caputo fractional derivative t d gamma gamma in 0 1 and the second order space derivatives with the fractional laplacian delta alpha 2 alpha in 1 2 using the matrix transfer technique proposed by ilic et al m ilic f liu i turner and v anh fract calc appl anal 9 2006 pp 333 349 the tsfde 2d is transformed into a time fractional differential system as t d gamma mathbf u k alpha mathbf a alpha 2 mathbf u where mathbf a is the approximate matrix representation of delta traditional approximation of mathbf a alpha 2 requires diagonalization of mathbf a which is very time consuming for large sparse matrices the novelty of our proposed numerical schemes is that using either the finite difference method or the laplace transform to handle the caputo time fractional derivative the solution of the tsfde 2d is written in terms of a matrix function vector product f mathbf a mathbf b at each time step where mathbf b is a suitably defined vector depending on the method used to generate the matrix mathbf a the product f mathbf a mathbf b can be approximated using either the preconditioned lanczos method when mathbf a is symmetric or the mathbf m lanzcos method when mathbf a is nonsymmetric which are powerful techniques for solving large linear systems we give error bounds for the new methods and illustrate their roles in solving the tsfde 2d we also derive the analytical solution of the tsfde 2d in terms of the mittag leffler function finally numerical results are presented to verify the proposed numerical solution strategies

worst case deterministic identification in h sub infinity the continuous time case results obtained by the authors 1991 worst case deterministic h sub infinity identification of discrete time plants are extended to continuous time plants the problem involves identification of the transfer function of a stable strictly proper continuous time plant from a finite number of noisy point samples of the plant frequency response the assumed information consists of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant an upper bound on the roll off rate of the plant and an upper bound on the noise level concrete plans of identification algorithms are provided for this problem explicit worst case deterministic error bounds for each algorithm establish that they are robustly convergent and essentially asymptotically optimal additionally these bounds provide an a priori computable h sub infinity uncertainty specification corresponding to the resulting identified plant transfer function as an explicit function of the plant and noise prior information and the data cardinality

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

numerical approximations for fractional diffusion equations via splines a one dimensional fractional diffusion model is considered where the usual second order derivative gives place to a fractional derivative of order a with 1 email protected 2 we consider the caputo derivative as the space derivative which is a form of representing the fractional derivative by an integral operator an implicit numerical method is derived which uses a spline approximation for the caputo derivative the consistency and stability of the method are examined and numerical results are presented

time and space fractional partial differential equations the fundamental solution for time and space fractional partial differential operator dt a2 2 0 is given in terms of the fox s h function here the time fractional derivative in the sense of generalized functions distributions dt is defined by the convolution dt f t t f t where t t 1 and f t 0 as t 0 and the fractional n dimensional laplace operator 2 is defined by its fourier transform with respect to spatial variable f 2g x f g x the solutions for initial value problems for time and space fractional partial differential equation in the sense of caputo and riemann liouville time fractional derivatives respectively are obtained by the fundamental solution

persistent identification of time varying systems identification of time varying systems especially slowly time varying systems is of importance in the development of a comprehensive theory of adaptation the persistent identification measures employed in this paper capture a main characterization in such identification problems namely one input signal must be used for identification of all possible observation windows this paper establishes several essential features in persistent identification problems which highlight their potential utility in adaptation 1 they have computable upper and lower bounds for typical classes of prior uncertainty sets 2 any full rank n periodic signals are optimal and the simple least squares estimates are optimal identification algorithms 3 optimal probing inputs can be approximately generated in a closed loop configuration when the plant and the controller are slowly time varying and 4 n periodic signals are asymptotically optimal for slowly time varying systems the main results of this paper have been successfully combined with a certain slow h sup spl infin design to derive an adaptive stabilization scheme

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

ill posed distributed parameter systems a control viewpoint the use of linear time invariant distributed parameter systems ltidpss as models of physical processes is considered from a control viewpoint specifically classes of ltidpss exhibiting properties which potentially limit their usefulness in feedback design are defined and termed ill posed the structure of these classes is investigated within a general mathematical framework which encompasses many distributed parameter systems of engineering interest within this framework necessary and sufficient conditions are derived which completely characterize these classes and establish their equivalence

heat conduction modeling by using fractional order derivatives the article deals with the heat conduction modeling a brief historical overview of the authors who have dealt with the heat conduction and overview of solving methods is listed in the introduction of article in the next section a mathematical model of one dimensional heat conduction with using derivatives of integer and fractional order is described the methods of solving models of heat conduction are described namely analytical and numerical methods in the case of numerical methods regards the finite difference method by using grunwald letnikov definition for the fractional time derivative implementation of these individual methods was realized in matlab the two libraries of m functions for the heat conduction model have been created namely heat conduction toolbox and fractional heat conduction toolbox at the conclusion of the article the simulations examples with using toolboxes are listed

model order reduction for meshfree solution of poisson singularity problems summary model order reduction mor techniques for enriched reproducing kernel meshfree methods are proposed for analysis of poisson problems with mild and strong singularities the employment of an integrated singular basis function method isbfm in conjunction with the selection of harmonic near tip asymptotic basis functions leads to a galerkin formulation in which the non smooth near tip basis functions appear only on the boundaries away from the singularity point this approach avoids the need of integrating the derivatives of non smooth functions near the singularity point and yields a discrete system that allows effective mor procedures under this framework a decomposed reduction method equipped with two distinct projections for smooth and non smooth parts of the finite dimensional space is proposed compared with the uniform reduction approach using a single projection operator the decomposed projection on isbfm discrete system preserves the singularity behavior of the fine scale solution in its lower dimensional approximation analytical error estimation and stability analysis show that isbfm with the decomposed projection can achieve better accuracy with only a slight increase of condition number compared with the uniform reduction approach in the reduced order solution of singularity problems the numerical tests validate the effectiveness of the proposed methods copyright 2014 john wiley sons ltd

study on fractional order gradient methods abstract in this paper convergence capability of the conventional fractional order gradient methods fogms is analyzed and a new fogm with guaranteed and faster convergence ability is proposed in particular we further consider the case that the derivative order varies between 1 and 2 and give some discussion on how to determine the derivative order in the algorithm settings simulation results from a number of numerical examples are provided to illustrate the approaches

identification of rational approximate models in h sup spl infin using generalized orthonormal basis robust identification with fir models fails to be successful when the number of coefficients to be estimated becomes large caused by lightly damped modes of the plant or poles very close to the unit circle the paper presents a two stage algorithm to obtain a low order approximate model in frequency domain in a generalized orthonormal basis with guaranteed h sup spl infin error bound for deterministic linear time invariant stable systems the first stage being an l sup spl infin rational approximation and the second nonlinear step being an h sup spl infin rational approximation

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

stability and convergence of a new explicit finite difference approximation for the variable order nonlinear fractional diffusion equation in this paper we consider the variable order nonlinear fractional diffusion equation u x t t b x t xr a x t u x t f u x t where xr a x t is a generalized riesz fractional derivative of variable order a x t 1 a x t 2 and the nonlinear reaction term f u x t satisfies the lipschitz condition f u 1 x t f u 2 x t