continuous time random walks on networks with vertex and time dependent forcing we have investigated the transport of particles moving as random walks on the vertices of a network subject to vertex and time dependent forcing we have derived the generalized master equations for this transport using continuous time random walks characterized by jump and waiting time densities as the underlying stochastic process the forcing is incorporated through a vertex and time dependent bias in the jump densities governing the random walking particles as a particular case we consider particle forcing proportional to the concentration of particles on adjacent vertices analogous to self chemotactic attraction in a spatial continuum our algebraic and numerical studies of this system reveal an interesting pair aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices the steady states do not exhibit this pair aggregation if the transport is random on the vertices i e without forcing the manifestation of pair aggregation on a transport network may thus be a signature of self chemotactic like forcing

beyond the poisson renewal process a tutorial survey after sketching the basic principles of renewal theory and recalling the classical poisson process we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of mittag leffler and of wright type we compare these three processes with each other

anomalous rotational relaxation a fractional fokker planck equation approach in this study we have analytically obtained the relaxation function in terms of rotational correlation functions based on brownian motion for complex disordered systems in a stochastic framework we found out that the rotational relaxation function has a fractional form for complex disordered systems which indicates that relaxation has nonexponential character and obeys the kohlrausch william watts law following the mittag leffler decay

lie symmetry classification analysis for nonlinear coupled diffusion the group properties of the 1 1 dimensional matrix diffusion equation of the form with respect to point symmetries are given it is shown that in particular cases the group properties of this equation are similar to those of the corresponding scalar equation namely the lie algebra is extended for power and exponential functions with additional extensions when powers of and 2 exist however a specific form of is shown to exist for cases admitting both infinite and finite groups the result obtained is used to construct new group invariant solutions for impulsive boundary inputs using two similarity variables a reduction is applied to the coupled diffusion of temperature and volumetric moisture content in porous media under periodic boundary conditions a perturbation method is employed to obtain an explicit solution for the case when can be expressed exponentially in terms of the similarity variables

particle tracking for fractional diffusion with two time scales abstract previous work y zhang m m meerschaert b baeumer particle tracking for time fractional diffusion phys rev e 78 2008 036705 showed how to solve time fractional diffusion equations by particle tracking this paper extends the method to the case where the order of the fractional time derivative is greater than one a subordination approach treats the fractional time derivative as a random time change of the corresponding cauchy problem with a first derivative in time one novel feature of the time fractional case of order greater than one is the appearance of clustering in the operational time subordinator which is non markovian solutions to the time fractional equation are probability densities of the underlying stochastic process the process models movement of individual particles the evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations or langevin equations monte carlo simulation yields particle location and the ensemble density approximates the solution to the variable coefficient time fractional diffusion equation in one or several spatial dimensions the particle tracking code is validated against inverse transform solutions in the simplest cases further applications solve model equations for fracture flow and upscaling flow in complex heterogeneous porous media these variable coefficient time fractional partial differential equations in several dimensions are not amenable to solution by any alternative method so that the grid free particle tracking approach presented here is uniquely appropriate

a nystrom based finite element method on polygonal elements abstract we consider families of finite elements on polygonal meshes that are defined implicitly on each mesh cell as solutions of local poisson problems with polynomial data functions in the local space on each mesh cell are evaluated via nystrom discretizations of associated integral equations allowing for curvilinear polygons and non polynomial boundary data several experiments demonstrate the approximation quality of interpolated functions in these spaces

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

a phase based interior penalty discontinuous galerkin method for the helmholtz equation with spatially varying wavenumber this paper is concerned with an interior penalty discontinuous galerkin ipdg method based on a flexible type of non polynomial local approximation space for the helmholtz equation with varying wavenumber the local approximation space consists of multiple polynomial modulated phase functions which can be chosen according to the phase information of the solution we obtain some approximation properties for this space and a priori l2l2 error estimates for the h convergence of the ipdg method using duality argument we also provide ample numerical examples to show that building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces

a cauchy type problem involving a weighted sequential derivative in the space of integrable functions a cauchy type nonlinear problem for a class of fractional differential equations involving sequential derivatives is considered some properties and composition identities are derived the equivalence with the associated integral equation is established the existence and uniqueness of global solutions in the space of lebesgue integrable functions are proved

the entropy production paradox for fractional master equations abstract time fractional evolution equations for probability distributions provide a means to describe an important class of stochastic processes their solutions show features which are essential in modeling a variety of phenomena in real world applications one aspect which has been observed in time fractional diffusion equations shows a surprising and unexpected behavior of the entropy production rate induced by these equations the entropy production rate increases as one moves away from the fully irreversible case corresponding to classical diffusion this rate is analyzed for a new class of systems with state spaces that are finite and denumerable we find that the entropy production paradox reemerges nonetheless but in a new and unexpected form

operator adapted spectral element methods i harmonic and generalized harmonic polynomials the paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted systems of functions compared with standard polynomials these operator adapted systems have superior local approximation properties first the case of laplace s equation and harmonic polynomials as operator adapted functions is analyzed and rates of convergence in a sobolev space setting are given for the approximation with harmonic polynomials special attention is paid to the approximation of singular functions that arise typically in corners these results for harmonic polynomials are extended to general elliptic equations with analytic coefficients by means of the theory of bergman and vekua the approximation results for laplace s equation hold true verbatim if harmonic polynomials are replaced with generalized harmonic polynomials the partition of unity method is used in a numerical example to construct an operator adapted spectral method for laplace s equation that is based on approximating with harmonic polynomials locally

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

brief paper a constructive solution for stabilization via immersion and invariance the cart and pendulum system immersion and invariance ii ii the construction of an invariant manifold such that the restriction of the system dynamics to this manifold coincides with the target dynamics iii the design of a control law that renders the manifold attractive and ensures that all signals are bounded the second step requires the solution of a partial differential equation pde that may be difficult to obtain in this short note we use the classical cart and pendulum system to show that by interlacing the first and second steps and invoking physical considerations it is possible to obviate the solution of the pde to underscore the generality of the proposed variation of i i we show that it is also applicable to a class of n dimensional systems that contain as a particular case the cart and pendulum system

random walk access times on partially disordered complex networks an effective medium theory an analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices considering transition rates f that are different for steps across lattice bonds from the rates f across network shortcuts the theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter access times for random walks on these partially disordered structures are compared to those on small world networks which on average appear to provide the most effective means of decreasing access times uniformly across the network

integrator forwarding a new recursive nonlinear robust design we consider the global stabilization of nonlinear systems in strict feedforward form we show that these systems while not feedback linearizable can be globally transformed by feedback and diffeomorphism into lower triangular form the transformation is explicit and recursive we employ this transformation to design a globally stabilizing controller with optimality properties and an input to state stability property with respect to matched uncertainties c 1997 elsevier science ltd

the general time fractional wave equation for a vibrating string the solution of a general time fractional wave equation for a vibrating string is obtained in terms of the mittag leffler type functions and complete set of eigenfunctions of the sturm liouville problem the time fractional derivative used is taken in the caputo sense and the method of separation of variables and the laplace transform method are used to solve the equation some results for special cases of the initial and boundary conditions are obtained and it is shown that the corresponding solutions of the integer order equations are special cases of those of time fractional equations the proposed general equation may be used for modeling different processes in complex or viscoelastic media disordered materials etc

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

immersion and invariance a new tool for stabilization and adaptive control of nonlinear systems a new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented the method relies upon the notions of system immersion and manifold invariance and in principle does not require the knowledge of a control lyapunov function the construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the invariant output zeroing manifold and its friend the method is well suited in situations where we know a stabilizing controller of a nominal reduced order model which we would like to robustify with respect to higher order dynamics this is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one we also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective our construction does not invoke certainty equivalence nor requires a linear parameterization furthermore viewed from a lyapunov perspective it provides a procedure to add cross terms between the parameter estimates and the plant states finally it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form yielding new stabilizing control laws we illustrate the method with several academic and practical examples including a mechanical system with flexibility modes an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application

subordinated stochastic processes with aged operational time in this paper subordinated stochastic processes are considered where the renewal process acting as the operational time it is assumed that the observation of the process begins at a certain time after the start of the renewal process a recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process two point correlation functions for certain subordinated stochastic processes particularly for the generalized ornstein uhlenbeck process were calculated a model of relaxation in a disordered medium with traps and obstacles is proposed

on a general link between anomalous diffusion and nonextensivity stochastic processes of systems with composable entropy measures are studied the processes are described by fokker planck equations which are nonlinear with respect to their probability densities and whose diffusion coefficients are derived from the entropy measures in questions in nonextensive extensive systems anomalous normal diffusion is found

convergence of the grunwald letnikov scheme for time fractional diffusion using bivariate generating functions we prove convergence of the grunwald letnikov difference scheme for the fractional diffusion equation in one space dimension with and without central linear drift in the fourier laplace domain as the space and time steps tend to zero in a well scaled way this implies convergence in distribution weak convergence of the discrete solution towards the probability of sojourn of a diffusing particle the difference schemes allow also interpretation as discrete random walks for fractional diffusion with central linear drift we show that in the fourier laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation

effective medium approximation for lattice random walks with long range jumps we consider the random walk on a lattice with random transition rates and arbitrarily long range jumps we employ bruggeman s effective medium approximation ema to find the disorder averaged coarse grained dynamics the ema procedure replaces the disordered system with a cleverly guessed reference system in a self consistent manner we give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion in the case of a power law scaling between transition rates and distance lattice variants of l evy flights emerge as the effective medium and the problem is solved analytically bearing the effective anomalous diffusivity finally we discuss several example distributions and demonstrate very good agreement with numerical simulations

barrier lyapunov functions for the output tracking control of constrained nonlinear switched systems abstract in this paper we investigate the output tracking control problem of constrained nonlinear switched systems in lower triangular form first when all the states are subjected to constraints we employ a barrier lyapunov function blf which grows to infinity whenever its arguments approach some finite limits to prevent the states from violating the constraints based on the simultaneous domination assumption we design a continuous feedback controller for the switched system which guarantees that asymptotic output tracking is achieved without transgression of the constraints and all closed loop signals remain bounded provided that the initial states are feasible then we further consider the case of asymmetric time varying output constraints by constructing an appropriate blf finally the effectiveness of the proposed results is demonstrated with a numerical example

dynamics of reaction diffusion systems in a subdiffusive regime in this paper we examine the dynamics of reaction diffusion systems with fractional time derivatives it is shown that in these conditions diffusion is anomalous in the sense that the mean square displacement r 2 ensuremath sim t ensuremath gamma where ensuremath gamma l1 a situation known as subdiffusion we study the conditions for the appearance of a diffusion driven instability and show that the restrictive conditions for a turing instability are relaxed this implies that systems whose kinetics are not of the activator inhibitor kind can have a turing instability and a modulated final state we demonstrate our results with numerical calculations in two dimensions using a generic turing model

extensions of effective medium theory of transport in disordered systems the effective medium theory of transport in disordered systems whose basis is the replacement of spatial disorder by temporal memory is extended in several practical directions restricting attention to a one dimensional system with bond disorder for specificity a transformation procedure is developed to deduce explicit expressions for the memory functions from given distribution functions characterizing the system disorder it is shown how to use the memory functions in the laplace domain forms in which they first appear and in the time domain forms which are obtained via numerical inversion algorithms to address time evolution of the system beyond the asymptotic domain of large times normally treated an analytic but approximate procedure is provided to obtain the memories in addition to the inversion algorithm good agreement of effective medium theory predictions with numerically computed exact results is found for all time ranges for the distributions used except near the percolation limit as expected the use of ensemble averages is studied for normal as well as correlation observables the effect of size on effective medium theory is explored and it is shown that even in the asymptotic limit finite size corrections develop to the well known harmonic mean prescription for finding the effective rate a percolation threshold is shown to arise even in one dimension for finite but not infinite systems at a concentration of broken bonds related to the system size spatially long range transfer rates are shown to emerge naturally as a consequence of the replacement of spatial disorder by temporal memories in spite of the fact that the original rates possess nearest neighbor character pausing time distributions in continuous time random walks corresponding to the effective medium memories are calculated

non markovian diffusion equations and processes analysis and simulations abstract in this paper we introduce and analyze a class of diffusion type equations related to certain non markovian stochastic processes we start from the forward drift equation which is made non local in time by the introduction of a suitable chosen memory kernel k t the resulting non markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l t we then consider the subordinated process y t x l t where x t is a markovian diffusion the corresponding time evolution of the marginal density function of y t is governed by a non markovian fokker planck equation which involves the memory kernel k t we develop several applications and derive the exact solutions we consider different stochastic models for the given equations providing path simulations

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

on fractional diffusion and continuous time random walks abstract a continuous time random walk model is presented with long tailed waiting time density that approaches a gaussian distribution in the continuum limit this example shows that continuous time random walks with long time tails and diffusion equations with a fractional time derivative are in general not asymptotically equivalent

an inverse problem for a generalized fractional diffusion we propose a method for determining the solution and source term of a generalized time fractional diffusion equation the method is based on selecting a bi orthogonal basis of l 2 space corresponding to a nonself adjoint boundary value problem uniqueness is proven and an existence result is obtained for smooth initial and final conditions the asymptotic behavior of the generalized mittag leffler function is used to relax the smoothness requirement on these conditions

on hilfer s objection to the fractional time diffusion equation abstract hilfer physica a 329 2003 35 claims to give an example of a continuous time random walk ctrw model with long tailed waiting time probability density that approaches a gaussian behavior in the continuum limit rigorous limit theorems derived previously show however that in the limit of long time such a ctrw converges to a non gaussian behavior we discuss two types of continuum limits for the ctrw model the fractional continuum limit and the one introduced by hilfer we show that the fractional limit yields the correct long time behavior of the ctrw while hilfer s continuum limit does not we discuss a general approach to find a continuum limit of the ctrw process

from the space time fractional integral of the continuous time random walk to the space time fractional diffusion equations a short proof and simulation abstract in this paper i prove the relation between the integral equation of the continuous time random walk ctrw and the space time fractional diffusion equations by using the definitions of the weyl fractional derivatives and integrals i generalize a transformation theorem between the independent variables of the solution of the space time fractional diffusion equation stfde and the solution of the space time fractional fokker planck equation stffpe i simulate the symmetric and the non symmetric random walks for the two mentioned models i use the asymptotic behavior of the mittag leffler function for generating the waiting time random variable and use the levy flight distribution function to generate the jump width random variable for the both models the simulation results are investigated for different values of the space and time fractional orders

anomalous subdiffusion with multispecies linear reaction dynamics we have introduced a set of coupled fractional reaction diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics the model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics the effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms one consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species the general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail we have carried out monte carlo simulations corresponding to these special cases and we find excellent agreement with theory

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

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

some recent advances in theory and simulation of fractional diffusion processes to offer an insight into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of current interest i the well scaled passage to the limit from continuous time random walk under power law assumptions to space time fractional diffusion ii the asymptotic universality of the mittag leffler waiting time law in time fractional processes iii our method of parametric subordination for generating particle trajectories

continuous time random walk galilei invariance and relation for the nth moment we consider a decoupled continuous time random walk model with a generic waiting time probability density function pdf for the force free case we derive an integro differential diffusion equation which is related to the galilei invariance for the probability density we also derive a general relation which connects the nth moment in the presence of any external force to the second moment without external force i e it is valid for any waiting time pdf this general relation includes the generalized second einstein relation which connects the first moment in the presence of any external force to the second moment without any external force these expressions for the first two moments are verified by using several kinds of the waiting time pdf moreover we present new anomalous diffusion behaviours for a waiting time pdf given by a product of power law and exponential function

traversal times for random walks on small world networks we study the mean traversal time for a class of random walks on newman watts small world networks in which steps around the edge of the network occur with a transition rate that is different from the rate for steps across small world connections when f f the mean time to traverse the network exhibits a transition associated with percolation of the random graph i e small world part of the network and a collapse of the data onto a universal curve this transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps we develop a simple self consistent effective medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition as is characteristic of most effective medium theories

adaptive tracking control for a class of uncertain switched nonlinear systems this paper is concerned with the problem of tracking control for a class of switched nonlinear systems in lower triangular form with unknown functions and arbitrary switchings two classes of state feedback controllers are constructed by adopting the adaptive backstepping technique and both of them are designed by using the common lyapunov function clf method the first controller is designed under multiple adaptive laws then the second one is designed based on constructing a maximum common adaptive parameter which can overcome the problem of over parameterization of the first controllers it is shown that the designed state feedback controllers can ensure that all the signals remain bounded and the tracking error converges to a small neighborhood of the origin finally simulation results are presented to show the effectiveness of the proposed approaches

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

brownian walkers within subdiffusing territorial boundaries inspired by the collective phenomenon of territorial emergence whereby animals move and interact through the scent marks they deposit we study the dynamics of a 1d brownian walker in a random environment consisting of confining boundaries that are themselves diffusing anomalously we show how to reduce in certain parameter regimes the non markovian many body problem of territoriality to the analytically tractable one body problem studied here the mean square displacement msd of the 1d brownian walker within subdiffusing boundaries is calculated exactly and generalizes well known results when the boundaries are immobile furthermore under certain conditions if the boundary dynamics are strongly subdiffusive we show the appearance of an interesting nonmonotonicity in the time dependence of the msd giving rise to transient negative diffusion

generalized continuous time random walks master equations and fractional fokker planck equations continuous time random walks which generalize random walks by adding a stochastic time between jumps provide a useful description of stochastic transport at mesoscopic scales the continuous time random walk model can accommodate certain features such as trapping which are not manifest in the standard macroscopic diffusion equation the trapping is incorporated through a waiting time density and a fractional diffusion equation results from a power law waiting time a generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force here we have derived the master equations for continuous time random walks with space and time dependent forcing for two cases when the force is evaluated at the start of the waiting time and at the end of the waiting time the differences persist in low order spatial continuum approximations however the two processes are shown to be governed by the same fokker planck equations in the diffusi

the fourier bessel method for solving the cauchy problem connected with the helmholtz equation this paper is concerned with the cauchy problem connected with the helmholtz equation we investigate the denseness of fourier bessel functions and propose a numerical method for approximating the solution to the problem the convergence and stability are analyzed with truncated singular value decomposition method numerical experiments are also presented to show the effectiveness of our method

generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains the ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena these models often involve fractional derivatives the important physical extension of this work to processes occurring in growing materials has proven highly nontrivial here we derive evolution equations for modeling subdiffusive transport in a growing medium the derivation is based on a continuous time random walk the concise formulation of these evolution equations requires the introduction of a new comoving fractional derivative the implementation of the evolution equation is illustrated with a simple model of subdiffusing proteins in a growing membrane

uncertainty observer based robust tracking control for high order nonlinear systems using forwarding technique based on uncertainty observers we develop a robust tracking control approach for high order nonlinear systems subject to uncertainties satisfying mismatched condition in this study by combining

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

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

fractional diffusion in a periodic potential overdamped and inertia corrected solutions for the spectrum of the velocity correlation function anomalous diffusion of a particle in a cosine periodic potential is treated using fractional diffusion equations in both phase and configuration space exact solutions of two distinct forms of the fractional klein kramers fokker planck equation for the distribution function in phase space are obtained via matrix continued fractions yielding the average velocity the velocity autocorrelation function its spectrum etc in the overdamped limit the results yielded by both equations agree with those from a fractional probability density diffusion equation in configuration space a simple analytic solution for the spectrum of the velocity correlation function is also given using the effective eigenvalue approximation the results represent generalizations of the conventional solutions for the normal diffusion of a brownian particle in a cosine potential to fractional dynamics giving rise to anomalous diffusion

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

existence and uniqueness for a problem involving hilfer fractional derivative we consider an initial value problem for a class of nonlinear fractional differential equations involving hilfer fractional derivative we prove existence and uniqueness of global solutions in the space of weighted continuous functions the stability of the solution for a weighted cauchy type problem is also analyzed

regional controllability analysis of fractional diffusion equations with riemann liouville time fractional derivatives abstract this paper is concerned with the concepts of regional controllability for the riemann liouville time fractional diffusion systems of order 0 1 the characterizations of strategic actuators to achieve regional controllability are investigated when the control inputs emerge in the differential equations as distributed inputs in the end an approach to guarantee the regional controllability of the problems under consideration in the considered subregion with minimum energy control is described and successfully tested through two applications

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

an inverse source problem for a two parameter anomalous diffusion with local time datum abstract we determine the space dependent source term for a two parameter fractional diffusion problem subject to nonlocal non self adjoint boundary conditions and two local time distinct datum a bi orthogonal pair of bases is used to construct a series representation of the solution and the source term the two local time conditions spare us from measuring the fractional integral initial conditions commonly associated with fractional derivatives on the other hand they lead to delicate 2 2 linear systems for the fourier coefficients of the source term and of the fractional integral of the solution at t 0 the asymptotic behavior and estimates of the generalized mittag leffler function are used to establish the solvability of these linear systems and to obtain sufficient conditions for the existence of our construction analytical and numerical examples are provided

interpolation properties of generalized plane waves this paper is aimed at developing new shape functions adapted to the scalar wave equation with smooth possibly vanishing coefficients and investigates the numerical analysis of their interpolation properties the interpolation is local but high order convergence is shown with respect to the size of the domain considered the new basis functions are then implemented in a numerical method to solve a scalar wave equation problem with a mixed boundary condition the main theoretical result states that any given order of approximation can be achieved by an appropriate choice of parameters for the design of the shape functions the convergence is studied with respect to the size of the domain which is referred to in the literature as h h convergence

a generalized master equation approach to modelling anomalous transport in animal movement we present some models of random walks with internal degrees of freedom that have the potential to find application in the context of animal movement and stochastic search the formalism we use is based on the generalized master equation which is particularly convenient here because of its inherent coarse graining procedure whereby a random walker position is averaged over the internal degrees of freedom we show some instances in which non local jump probabilities emerge from the coupling of the motion to the internal degrees of freedom and how the tuning of one parameter can give rise to sub super and normal diffusion at long times remarks on the relation between the generalized master equation continuous time random walks and fractional diffusion equations are also presented

necessary conditions for asymptotic tracking in nonlinear systems in the literature it has been shown that if a single input single output analytic nonlinear plant 1 has a well defined relative degree and 2 is minimum phase it is possible to achieve asymptotic tracking for an open set of output trajectories containing the origin in c sup n 0 spl infin the space of n times continuously differentiable functions taking values in r when either of these sufficient conditions is not met various authors have investigated approximate analytic solutions discontinuous solutions and solutions for restricted sets of trajectories in this paper it is shown that conditions 1 and 2 are necessary for the existence of an analytic compensator which yields asymptotic tracking for an open set of output trajectories analogous results are established for multi input multi output systems

global stabilization of partially linear composite systems a linear stabilizable nonlinear asymptotically stable cascade system is globally stabilizable by smooth dynamic state feedback if a the linear subsystem is right invertible and weakly minimum phase and b the only linear variables entering the nonlinear subsystem are the output and the zero dynamics corresponding to this output both of these conditions are coordinate free and there is freedom of choice for the linear output variable this result generalizes several earlier sufficient conditions for stabilizability moreover the weak minimum phase condition for the linear subsystem cannot be relaxed unless a growth restriction is imposed on the nonlinear subsystem

trefftz approximations in complex media accuracy and applications abstract approximations by trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics by definition these functions satisfy locally in weak form the underlying differential equations of the problem which often results in high order or even exponential accuracy with respect to the size of the basis set we highlight two separate examples in applied electromagnetics and photonics i homogenization of periodic structures and ii numerical simulation of electromagnetic waves in slab geometries extensive numerical evidence and theoretical considerations show that trefftz approximations can be applied much more broadly than is traditionally done they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones two mechanisms underlying the high accuracy of trefftz approximations in such complex cases are pointed out the first one is related to trigonometric interpolation and the second one somewhat surprisingly to well posedness of random matrices

transport properties of random walks on scale free regular lattice hybrid networks we study numerically the mean access times for random walks on hybrid disor dered structures formed by embedding scale free networks into regular lattices considering different transition rates for steps across lattice bonds f and across network shortcuts f for fast shortcuts f f 1 and low shortcut densities traversal time data collapse onto a universal curve while a crossover behavior that can be related to the percolation threshold of the scale free network component is identified at higher shortcut densities in analogy to similar observations reported recently in newman watts small world networks further more we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution and we discuss access time distri butions as functions of the initial and final node degrees these findings are relevant in particular when considering the optimization of existing information networks by the addi tion of a small number of fast shortcut connections

hilfer prabhakar derivatives and some applications abstract we present a generalization of hilfer derivatives in which riemann liouville integrals are replaced by more general prabhakar integrals we analyze and discuss its properties furthermore we show some applications of these generalized hilfer prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations and in difference differential equations governing the dynamics of generalized renewal stochastic processes

stochastic calculus for uncoupled continuous time random walks the continuous time random walk ctrw is a pure jump stochastic process with several applications not only in physics but also in insurance finance and economics a definition is given for a class of stochastic integrals driven by a ctrw which includes the it and stratonovich cases an uncoupled ctrw with zero mean jumps is a martingale it is proved that as a consequence of the martingale transform theorem if the ctrw is a martingale the it integral is a martingale too it is shown how the definition of the stochastic integrals can be used to easily compute them by monte carlo simulation the relations between a ctrw its quadratic variation its stratonovich integral and its it integral are highlighted by numerical calculations when the jumps in space of the ctrw have a symmetric levy alpha stable distribution and its waiting times have a one parameter mittag leffler distribution remarkably these distributions have fat tails and an unbounded quadratic variation in the diffusive limit of vanishing scale parameters the probability density of this kind of ctrw satisfies the space time fractional diffusion equation fde or more in general the fractional fokker planck equation which generalizes the standard diffusion equation solved by the probability density of the wiener process and thus provides a phenomenologic model of anomalous diffusion we also provide an analytic expression for the quadratic variation of the stochastic process described by the fde and check it by monte carlo

fractional diffusion equation with a generalized riemann liouville time fractional derivative in this paper the solution of a fractional diffusion equation with a hilfer generalized riemann liouville time fractional derivative is obtained in terms of mittag leffler type functions and fox s h function the considered equation represents a quite general extension of the classical diffusion heat conduction equation the methods of separation of variables laplace transform and analysis of the sturm liouville problem are used to solve the fractional diffusion equation defined in a bounded domain by using the fourier laplace transform method it is shown that the fundamental solution of the fractional diffusion equation with a generalized riemann liouville time fractional derivative defined in the infinite domain can be expressed via fox s h function it is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the caputo and riemann liouville sense are special cases of those diffusion equations with the hilfer generalized riemann liouville time fractional derivative the asymptotic behaviour of the solutions are found for large values of the spatial variable the fractional moments of the fundamental solution of the fractional diffusion equation are obtained the obtained results are relevant in the context of glass relaxation and aquifer problems

numerical algorithm for calculating the generalized mittag leffler function a numerical algorithm for calculating the generalized mittag leffler mathrm e alpha beta z function for arbitrary complex argument z and real parameters alpha 0 and beta in mathbb r is presented the algorithm uses the taylor series the exponentially improved asymptotic series and integral representations to obtain optimal stability and accuracy of the algorithm special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm

short note on the emergence of fractional kinetics abstract in the present short note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non markovian master equations particle trajectories are assumed to be solely markovian and described by the continuous time random walk model but as a consequence of the complexity of the medium each trajectory is supposed to scale in time according to a particular random timescale the link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed in particular when a stationary distribution is considered the timescale distribution is uniquely determined as a function related to the fundamental solution of the space time fractional diffusion equation in contrast when the non stationary case is considered the timescale distribution is no longer unique two distributions are here computed one related to the m wright mainardi function which is green s function of the time fractional diffusion equation and another related to the mittag leffler function which is the solution of the fractional relaxation equation

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

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 class of difference schemes with flexible local approximation solutions of many physical problems have salient local features that are qualitatively known a priori for example singularities at point sources edge and corners boundary layers derivative jumps at material interfaces strong dipole field components near polarized spherical particles cusps of electronic wavefunctions at the nuclei electrostatic double layers around colloidal particles etc the known methods capable of providing flexible local approximation of such features include the generalized finite element partition of unity method special variational difference schemes in broken sobolev spaces and a few other specialized techniques in the proposed new class of flexible local approximation methods flame a desirable set of local approximating functions such as cylindrical or spherical harmonics plane waves harmonic polynomials etc defines a finite difference scheme on a chosen grid stencil one motivation is to minimize the notorious staircase effect at curved and slanted interface boundaries however the new approach has much broader applications as illustrative examples the paper presents arbitrarily high order 3 point schemes for the 1d schrodinger equation and a 1d singular equation schemes for electrostatic interactions of colloidal particles electromagnetic wave propagation and scattering plasmon resonances moreover many classical finite difference schemes including the collatz mehrstellen schemes are direct particular cases of flame

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london