stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

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

a rbf based differential quadrature method for solving two dimensional variable order time fractional advection diffusion equation abstract numerical simulation technique of two dimensional variable order time fractional advection diffusion equation is developed in this paper using radial basis function based differential quadrature method rbf dq to the best of the authors knowledge this is the first application of this method to variable order time fractional advection diffusion equations for the general case of irregular geometries the meshless local form of rbf dq is used and the multiquadric type of radial basis functions is selected for the computations this approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the dirichlet and neumann boundary conditions on the irregular boundaries the method is validated by the well documented test examples involving variable order fractional modeling of air pollution the numerical results demonstrate that the proposed method provides accurate solutions for two dimensional variable order time fractional advection diffusion equations

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

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

dual mode predictive control algorithm for constrained hammerstein systems in the process industry there exist many systems which can be approximated by a hammerstein model moreover these systems are usually subjected to input magnitude constraints in this paper a multi channel identification algorithm mcia is proposed in which the coefficient parameters are identified by least squares estimation lse together with a singular value decomposition svd technique compared with traditional single channel identification algorithms the present method can enhance the approximation accuracy remarkably and provide consistent estimates even in the presence of coloured output noises under relatively weak assumptions on the persistent excitation pe condition of the inputs then to facilitate the following controller design this mcia is converted into a two stage single channel identification algorithm ts scia which preserves most of the advantages of mcia with this ts scia as the inner model a dual mode non linear model predictive control nmpc algorithm is developed in detail over a finite horizon an optimal input profile found by solving a open loop optimal control problem drives the non linear system state into the terminal invariant set afterwards a linear output feedback controller steers the state to the origin asymptotically in contrast to the traditional algorithms the present method has a maximal stable region a better steady state performance and a lower computational complexity finally simulation results on a heat exchanger are presented to show the efficiency of both the identification and the control algorithms

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

adding one power integrator a tool for global stabilization of high order lower triangular systems a new feedback design tool called adding a power integrator is introduced and used to solve the problem of global robust stabilization for a significant class of uncertain nonlinear systems that are of a lower triangular form but neither necessarily feedback linearizable fully or partially nor affine in the control input and therefore cannot be dealt with via conventional approaches sufficient conditions are presented under which a globally stabilizing smooth state feedback control law can be explicitly constructed using repeatedly the adding one power integrator technique this novel design technique permits removal of the feedback linearizability and linearity of the control input which have been two common assumptions so far in the literature of nonlinear control

comments on adaptive global stabilization of nonholonomic systems with strong nonlinear drifts systems control letters 46 2002 195 205 in the paper to be commented there is a technical problem inherent in the input based switching and it is highlighted here to avoid possible failure when the scheme is used in practice

functional series modeling approach to identification of nonstationary stochastic systems the problem of identification of a nonstationary stochastic system is considered and an estimation method based on the functional series modeling fsm of the system parameter trajectory is proposed for its solution it is shown that the parameter matching properties of fsm estimators can be described in terms of the appropriately defined time varying impulse and frequency responses it is suggested and verified by means of computer simulation that the averaged frequency characteristics associated with fsm estimators can yield useful information about their parameter matching abilities

an improved spectral meshless radial point interpolation for a class of time dependent fractional integral equations 2d fractional evolution equation abstract in this paper a kind of spectral meshless radial point interpolation smrpi technique is applied to the fractional evolution equation in two dimensional for arbitrary fractional order the applied approach is based on erudite combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of smrpi in the current work the thin plate splines tps are used as the basis functions and also the discretization technique employed is patterned after an idea of ch lubich it is proved the scheme is unconditional stable with respect to the time variable in l 2 norm and convergent by the order of convergence o t 1 0 1 the results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme three numerical examples show that the smrpi has reliable accuracy in irregular domains

continuous global stabilisation of high order time delay nonlinear systems this paper studies the global stabilisation problem for a class of high order nonlinear systems with multiple time delays a distinct property of the systems to be investigated is that powers on the upper bound restrictions of nonlinearities are allowed to take values on a continuous interval by introducing the sign functions and using the generalised method of adding a power integrator this paper successfully designs a continuous global state feedback controller independent of time delays moreover a novel lyapunov krasovskii functional is constructed to prove the globally asymptotic stability of the resulting closed loop system

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

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

analysis of the spectral meshless radial point interpolation for solving fractional reaction subdiffusion equation abstract the present paper is devoted to the development of spectral meshless radial point interpolation smrpi technique for solving fractional reaction subdiffusion equation in one and two dimensional cases the time fractional derivative is described in the riemann liouville sense the applied approach is based on a combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of smrpi it is proved that the scheme is unconditionally stable with respect to the time variable in h 1 and we show convergence order of the time discrete scheme is o t 0 1 in the current work the thin plate splines tps are used to construct the basis functions the results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme

optimal choice of time scaling factor for linear system approximations using laguerre models clowes 1965 illustrated how to select the optimal time scaling factor for approximations of rational transfer functions using a series expansion of laguerre functions the present paper extends this result to approximations for general l sub 2 stable linear systems as a direct application of this extended result we are able to obtain approximations for irrational transfer functions which minimize the frequency domain error in the l sub 2 sense also we give empirical solutions for the optimal time scaling factor for approximations of first order plus delay systems which can then be applied to a larger class of overdamped systems

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

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

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

a compact difference scheme for a two dimensional nonlinear fractional klein gordon equation in polar coordinates abstract in this paper a high order finite difference scheme for a two dimensional nonlinear fractional klein gordon equation in polar coordinates is proposed the difficulty caused by the variable coefficient and the nonlinear term in the equation is carefully handled stability and convergence are analyzed using the matrix form of the scheme numerical examples are provided to demonstrate the theoretical results

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

nonparametric identification of wiener systems by orthogonal series a wiener system i e a system comprising a linear dynamic and a nonlinear memoryless subsystems connected in a cascade is identified both the input signal and disturbance are random white and gaussian the unknown nonlinear characteristic is strictly monotonous and differentiable and therefore the problem of its recovering from input output observations of the whole system is nonparametric it is shown that the inverse of the characteristic is a regression function and a class of orthogonal series nonparametric estimates recovering the regression is proposed and analyzed the estimates apply the trigonometric legendre and hermite orthogonal functions pointwise consistency of all the algorithms is shown under some additional smoothness restrictions the rates of their convergence are examined and compared an algorithm to identify the impulse response of the linear subsystem is proposed

numerical solution of a class of fractional optimal control problems via the legendre orthonormal basis combined with the operational matrix and the gauss quadrature rule abstract a numerical direct method for solving a general class of fractional optimal control problems focps is presented in the discussed focp the fractional derivative in the dynamical system is considered in the caputo sense to solve the problem first the focp is transformed into an equivalent variational problem then using the legendre orthonormal basis the problem is reduced to the problem of solving a system of algebraic equations with the aid of an operational matrix of riemann liouville fractional integration gauss quadrature formula and newton s iterative method for solving a system of algebraic equations the problem is solved approximately approximations achieved by this method satisfy all the initial conditions of the problem which is an important property the convergence of the method is extensively discussed and finally some illustrative examples are included to demonstrate the applicability of the new technique

a fourth order compact solution of the two dimensional modified anomalous fractional sub diffusion equation with a nonlinear source term this work is concerned to the study of high order difference scheme for the solution of a two dimensional modified anomalous sub diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses the space fractional derivatives are described in the riemann liouville sense in the proposed scheme we discretize the space derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme we prove the stability and convergence of proposed scheme using the fourier analysis the convergence order of the proposed method is o t h x 4 h y 4 comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme

a fourier method and an extrapolation technique for stokes first problem for a heated generalized second grade fluid with fractional derivative in this article we consider stokes first problem for a heated generalized second grade fluid with fractional derivative sfp hgsgf implicit and explicit numerical approximation schemes for the sfp hgsgf are presented the stability and convergence of the numerical schemes are discussed using a fourier method in addition the solvability of the implicit numerical approximation scheme is also analyzed a richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed finally a numerical test is given the numerical results demonstrate the good performance of our theoretical analysis

error estimate for the numerical solution of fractional reaction subdiffusion process based on a meshless method in this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction subdiffusion equation firstly we obtain a time discrete scheme based on a finite difference scheme then we use the meshless galerkin method to approximate the spatial derivatives and obtain a full discrete scheme in the proposed scheme some integrals appear over the boundary and the domain of problem which will be approximated using gauss legendre quadrature rule then we prove that the time discrete scheme is unconditionally stable and convergent using the energy method we show convergence order of the time discrete scheme is o the aim of this paper is to obtain an error estimate and to show convergence for the meshless galerkin method based on the radial basis functions numerical examples confirm the efficiency and accuracy of the proposed scheme

numerical approximation of higher order time fractional telegraph equation by using a combination of a geometric approach and method of line we propose a simple and accurate numerical scheme for solving the time fractional telegraph tft equation within caputo type fractional derivative a fictitious coordinate is imposed onto the problem in order to transform the dependent variable u x t into a new variable with an extra dimension in the new space with the added fictitious dimension a combination of method of line and group preserving scheme gps is proposed to find the approximate solutions this method preserves the geometric structure of the problem power and accuracy of this method has been illustrated through some examples of tft equation

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

numerical scheme with high order accuracy for solving a modified fractional diffusion equation abstract in recent years some researchers have developed various numerical schemes to solve the modified fractional diffusion equation for the numerical solutions of the modified fractional diffusion equation there are already some important progresses in this paper a numerical scheme with second order temporal accuracy and fourth order spatial accuracy is developed to solve a modified fractional diffusion equation the convergence stability and solvability of the numerical scheme are analyzed by fourier analysis the theoretical results extremely consistent with the numerical experiment

numerical methods with fourth order spatial accuracy for variable order nonlinear stokes first problem for a heated generalized second grade fluid stokes first problem has in recent years received much attention in this paper we focus on the variable order nonlinear stokes first problem for a heated generalized second grade fluid a numerical scheme with fourth order spatial accuracy is developed to solve the problem the stability solvability and convergence of the numerical scheme are discussed via fourier analysis an improved numerical scheme is also developed in addition a numerical example is given and the numerical results support the effectiveness of our theoretical analysis results

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

some new operational matrices and its application to fractional order poisson equations with integral type boundary constrains abstract enormous application of fractional order partial differential equations fpdes subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world the prime objective of this article is to find approximate solution of a general fpdes subject to nonlocal integral type boundary conditions on both ends of the domain the proposed method is based on spectral method we construct some new operational matrices which have the ability to handle integral type non local boundary constrains these operational matrices can be effectively applied to convert the fpdes together with its integral types boundary conditions to easily solvable matrix equation the accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems the proposed method has the ability to solve non local fpdes with high accuracy and low computational cost different aspects of presented approach are compared with two other recently developed methods haar wavelets collocation method and a family of collocation methods which are based on radial base functions it is observed that the proposed method is highly accurate robust efficient and stable as compared to these methods

stability analysis of systems with impulse effects the authors establish conditions for the uniform stability and the uniform asymptotic stability of equilibria of systems with impulse effects described by systems of nonlinear time varying ordinary differential equations under mild additional assumptions these sufficient conditions turn out to be necessary as well for the case of time invariant systems with impulse effects described by nonlinear ordinary differential equations the above results are used to establish conditions under which the uniform asymptotic stability of equilibria can be deduced from the linearizations of such systems

laguerre series approximation of infinite dimensional systems laguerre fourier approximations of stable systems are shown to exhibit many desirable properties for various classes of infinite dimensional systems specifically time domain supremum and l1 norm convergence results and frequency domain h norm convergence results are given for laguerre fourier approximations it is also shown that the theory of laguerre polynomials solves explicitly the problem of determining laguerre fourier approximations for a large class of delay systems furthermore it is believed that these results are important for the study of orthonormal series identification as a general technique for identification of infinite dimensional systems

adaptive disturbance attenuation for generalized high order uncertain nonlinear systems abstract this paper investigates the problem of adaptive disturbance attenuation for a class of generalized high order uncertain nonlinear systems the control strategy is on the basis of continuous domination and delicate adaptive technique and it can cope with serious coexistence among uncertainties including time varying control coefficients which have unknown upper and lower bounds nonlinear parameters and external disturbances adaptive state feedback controller is one dimensional irrespective of the number of unknown parameters and its performance is evaluated in terms of l 2 l 2 p gain

adaptive stabilisation for a large class of high order uncertain non linear systems the problem of global adaptive stabilisation by state feedback is investigated for a class of high order non linear systems with uncertain control coefficients and zero dynamics first some appropriate unknown parameters are introduced to obtain the updating laws when adapting control design then by the flexible way of combining adding a power integrator with adaptive technique and the idea of changing supply functions the requirement on the uncertain control coefficients is relaxed and a recursive design procedure is successfully developed to achieve a continuous adaptive stabilising controller finally an example is provided to illustrate the correctness of the theoretical results

equivalence of laguerre based model order reduction and moment matching the equivalence between the classical moment matching and the laguerre based reduction approaches is shown in addition this equivalence is generalized to include a family of coefficients known as generalized markov parameters

technical communique decentralized adaptive control of linear interconnected systems based on laguerre series representation this paper investigates the problem of completely decentralized adaptive control of large scale systems when the subsystems are stable a new approach which eliminates the assumptions about the interconnection terms the time delay the order etc is proposed each subsystem is modeled by an orthonormal laguerre network put in state space form a simple decentralized predictive control law is proposed the potential of such a scheme is illustrated by simulation results

global stabilisation of high order nonlinear systems with multiple time delays this paper focuses on the stabilisation for a class of high order nonlinear systems with multiple time delays growth restriction on system nonlinearities is further relaxed design procedures of a continuous controller are provided by the method of adding a power integrator and the stability of the resulting closed loop system is rigorously proven with the help of the elegant choice of a lyapunov krasovskii functional finally a simulation example is provided to demonstrate the validness of the proposed approach

numerical simulation with the second order compact approximation of first order derivative for the modified fractional diffusion equation abstract in this paper based on the second order compact approximation formula of first order derivative the numerical simulation method with second order temporal accuracy and fourth order spatial accuracy is developed to simulate the modified fractional diffusion equation the convergence stability and solvability of the numerical simulation method are analyzed by fourier analysis and algebraic theory respectively the theoretical results extremely consistent with the numerical experiment

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

deterministic adaptive control based on laguerre series representation the behaviour of adaptive controllers in the presence of unmodelled dynamics and the need for reduced a priori information have led us to abandon the usual arma transfer function representation for a representation by an orthonormal series the appeal of our new approach is that it eliminates the need for assumptions about the plant order and the time delay the plant is modelled by an orthonormal laguerre network put in state space form a simple predictive control law is proposed an explicit deterministic adaptive controller is then designed simulations show that it is easy to use able to handle non minimum phase plants and more robust than the conventional model based approach although we chose laguerre functions other orthonormal functions may be used we have already tested some with success

decentralized adaptive control of interconnected systems the adaptive control of a class of large scale systems formed of an arbitrary interconnection of subsystems with unknown parameters nonlinearities and bounded disturbances is investigated it is first shown that no matter how weak the interconnections are a decentralized adaptive control scheme can become unstable approaches are then developed for stabilization and tracking using new decentralized adaptive controllers in the case where the relative degree n of the transfer function of each decoupled subsystem is less than or equal to two sufficient conditions are established which guarantee boundedness and exponential convergence of the state and parameter errors to bounded residual sets in the absence of disturbances and interconnections the decentralized adaptive control schemes guarantee exact convergence of the tracking errors to zero the effectiveness of the proposed adaptive schemes is demonstrated using a simple example

adaptive global stabilization of nonholonomic systems with strong nonlinear drifts a new input to state scaling scheme is first introduced to transform a class of nonholonomic systems in a chained form with strong nonlinear drifts and unknown constant parameters into a strict feedback form the backstepping technique is then applied to design a global adaptive stabilization controller a switching strategy based on the control input magnitude rather than the time is derived to get around the phenomenon of uncontrollability simulation examples validate the effectiveness of the proposed controller

output feedback adaptive stabilization control design for non holonomic systems with strong non linear drifts this paper investigates the problem of output feedback adaptive stabilization control design for non holonomic chained systems with strong non linear drifts including modelled non linear dynamics unmodelled dynamics and those modelled but with unknown parameters an observer and an estimator are introduced for state and parameter estimates respectively by using the integrator backstepping approach and based on the observer and parameter estimator a constructive design procedure for output feedback adaptive stabilization control is given it is shown that under some conditions the control design ensures the closed loop system is globally asymptotically stable when there is no non linear drift in the first subsystem and semiglobally asymptotically stable otherwise an example is given to show the effectiveness of the theory

system identification using laguerre models the traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear in the parameters predictor models leads to approximations of very high order in cases of rapid sampling and or dispersion in time constants by using prior information about the time constants of the system more appropriate expansions related to laguerre networks are introduced and analyzed it is shown that the model order can be reduced compared to arx fir ar modeling by using laguerre models furthermore the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the laguerre parameter consistency error bounds persistence of excitation conditions and asymptotic statistical properties are investigated this analysis is based on the result that the covariance matrix of the regression vector of a laguerre model has a toeplitz structure

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

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

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

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

strang type preconditioners for solving fractional diffusion equations by boundary value methods the finite difference scheme with the shifted grunwald formula is employed to semi discrete the fractional diffusion equations this spatial discretization can reduce to the large system of ordinary differential equations odes with initial values recently the boundary value method bvm was developed as a popular algorithm for solving the large systems of odes this method requires the solutions of one or more nonsymmetric and large scale linear systems in this paper the gmres method with the block circulant preconditioner is proposed to solve relevant linear systems some conclusions about the convergence analysis and spectrum of the preconditioned matrices are also drawn if the diffusion coefficients are constant finally extensive numerical experiments are reported to show the performance of our method for solving the fractional diffusion equations

a compact lod method and its extrapolation for two dimensional modified anomalous fractional sub diffusion equations a crank nicolson type compact locally one dimensional lod finite difference method is proposed for a class of two dimensional modified anomalous fractional sub diffusion equations with two time riemann liouville fractional derivatives of orders 1 and 1 0 1 the resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one dimensional problems this property evidently enhances the simplicity of programming and makes the computations more easy the unconditional stability and convergence of the scheme are rigorously proved the error estimates in the standard h 1 and l 2 norms and the weighted l norm are obtained and show that the proposed compact lod method has the accuracy of the order 2 min in time and 4 in space a richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min 4 min if and min 1 4 if a comparison study of the compact lod method with the other existing methods is given to show its superiority numerical results confirm our theoretical analysis and demonstrate the accuracy and the effectiveness of the compact lod method and the extrapolation algorithm

adaptive control of nonlinearly parameterized systems the smooth feedback case studies global adaptive control of nonlinearly parameterized systems with uncontrollable linearization using a parameter separation technique and the tool of adding a power integrator we develop a feedback domination design approach for the explicit construction of a smooth adaptive controller that solves the problem of global state regulation in contrast to the existing results in the literature a key feature of our adaptive regulator is its minimum order property namely no matter how big the number of unknown parameters is the order of the dynamic compensator is identical to one and is therefore minimal as a consequence global state regulation of feedback linearizable systems with nonlinear parameterization is achieved by one dimensional adaptive controllers without imposing any extra e g convex concave conditions on the unknown parameters

the implicit midpoint method for the modified anomalous sub diffusion equation with a nonlinear source term in this paper the implicit midpoint method is used to solve the semi discrete modified anomalous sub diffusion equation with a nonlinear source term and the weighted and shifted grunwald letnikov difference operator and the compact difference operator are applied to approximate the riemann liouville fractional derivative and space partial derivative respectively then the new numerical scheme is constructed the stability and the convergence of this method are analyzed numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results

numerical computation of decentralized fixed modes abstract in this paper we use an algebraic characterization of fixed modes of a decentralized linear multivariable system to show that the fixed modes are related to the blocking zeros of certain subsystems derived from the given decentralized system a numerical algorithm is then presented which enables us to compute the fixed modes in a reliable manner examples are provided to illustrate the main results of the paper

the monte carlo markov chain method for solving the modified anomalous fractional sub diffusion equation abstract in this paper the monte carlo markov chain method for solving the modified anomalous fractional sub diffusion equation is studied most of the previous methods are low in temporal and spatial accuracy order based on the idea of monte carlo markov chain method and compact finite difference schemes a probability model for solving the modified anomalous fractional sub diffusion equation is established numerical examples are given to show the feasibility of the proposed scheme compared with the compact finite difference method the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order and it is also applied to solve partial differential equation in irregular domains

a continuous feedback approach to global strong stabilization of nonlinear systems we present a continuous feedback approach to the problem of global strong stabilization for genuinely nonlinear systems that may not be stabilized even locally by any smooth feedback we describe conditions under which it is possible to prove while no smooth controllers exist the existence of continuous state feedback control laws that achieve global strong stability gss in the sense of kurzweil 1956 the proof is constructive and carried out by developing a machinery which combines the theory of homogeneous systems with the idea of adding a power integrator for the explicit construction of globally stabilizing continuous controllers we then illustrate by means of examples how this machinery can be used to overcome the topological obstruction caused by smooth feedback and hence resulting in new solutions to a variety of open control problems including global stabilization of an underactuated unstable two degree of freedom mechanical system

finite volume element method and its stability analysis for analyzing the behavior of sub diffusion problems in this paper we analyze the spatially semi discrete piecewise linear finite volume element method for the time fractional sub diffusion problem in two dimensions and give an approximate solution of this problem at first we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods furthermore we use the standard finite element ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method finally some numerical examples are included to illustrate the effectiveness of the new technique

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

structurally fixed modes a notion of structurally fixed modes is introduced in the framework of decentralized control systems to identify the modes that cannot be shifted by decentralized feedback regardless of the numerical values of system parameters in this way a qualitative characterization of decentralized fixed modes is provided which is suitable for delineating decentrally stabilizable systems the characterization leads naturally to the notion of fixed modes with respect to arbitrary feedback structure constraints

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

a new method based on legendre polynomials for solutions of the fractional two dimensional heat conduction equation abstract in this paper we develop a new scheme for numerical solutions of the fractional two dimensional heat conduction equation on a rectangular plane our main aim is to generalize the legendre operational matrices of derivatives and integrals to the three dimensional case by the use of these operational matrices we reduce the corresponding fractional order partial differential equations to a system of easily solvable algebraic equations the method is applied to solve several problems the results we obtain are compared with the exact solutions and we find that the error is negligible

finite difference methods and a fourier analysis for the fractional reaction subdiffusion equation various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations fpde for example systems biology chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments however effective numerical methods and numerical analysis for fpde are still in their infancy in this paper we consider a fractional reaction subdiffusion equation fr subde in which both the motion and the reaction terms are affected by the subdiffusive character of the process using the relationship between the riemann liouville and grunwald letnikov definitions of fractional derivatives an implicit and an explicit difference methods for the fr subde are presented the stability and the convergence of the two numerical methods are investigated by a fourier analysis the solvability of the implicit finite difference method is also proved the high accuracy algorithm is structured using richardson extrapolation finally a comparison between the exact solution and the two numerical solutions is given the numerical results are in excellent agreement with our theoretical analysis