discrete spline methods for solving two point fractional bagley torvik equation a new discrete spline method is developed to solve the two point fractional bagley torvik equation the method is based on discrete spline function and a nonstandard grunwald letnikov difference nsgd and the weighted and shifted grunwald letnikov difference wsgd operators to approximate the fractional derivative bounds for grunwald letnikov weights are considered convergence analysis is discussed and a class of second order and third order methods are obtained illustrative examples are presented to validate the practical usefulness of the methods

a consistent characteristic length for smeared cracking models a numerical scheme for crack modelling by means of continuous displacement fields is presented in two dimensional problems a crack is modelled as a limiting case of two singular lines with continuous displacements but discontinuous displacement gradients across them which tend to coincide with each other an analysis of the energy dissipated inside the band bounded by both lines allows one to obtain an expression for the characteristic length as the ratio between the energy dissipated per unit surface area fracture energy and the energy dissipated per unit volume specific energy at a point the application of these mathematical expressions to the finite element discretized medium allow one to obtain a general spatial and directional expression for the characteristic length which guarantees the objectivity of the results with respect to the size of the finite element mesh the numerical results presented show the reliability of the proposed expressions

a return mapping algorithm for plastic damage models 3 d and plane stress formulation three dimensional and plane stress formulations of the return mapping algorithm for a class of plastic damage models are derived using the spectral decomposition form of the stress an efficient plane stress computation scheme based on the spectral return mapping algorithm is developed the consistent algorithmic tangent stiffness for the present algorithm is formulated the validation and performance of the present return mapping algorithm is demonstrated by numerical examples copyright 2001 john wiley sons ltd

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

a plastic damage constitutive model for composite materials in this paper a generalized elasto plastic damage model for the analysis of multiphase frictional composite materials is presented details of the derivation of the secant and tangent constitutive equations are given mixing theory is used to insert the basic constitutive expressions for each substance on the multiphase composite solid details of the numerical implementation of the model into a general non linear finite element solution scheme are presented some examples of linear and non linear behaviour of composites are given

ergodic properties of the multidimensional rayleigh gas with a semipermeable barrier we consider a multidimensional system consisting of a particle of massm and radiusr molecule surrounded by an infinite ideal gas of point particles of massm atoms the molecule is confined to the unit ball and interacts with its boundary barrier via elastic collision while the atoms are not affected by the boundary we obtain convergence to equilibrium for the molecule from almost every initial distribution on its position and velocity furthermore we prove that the infinite composite system of the molecule and the atoms is bernoulli

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

identifications of parameters in ill posed linear parabolic equations abstract in this paper we study the identification of conductivities q in linear parabolic equation y t q y f in 0 t with boundary condition y x t 0 on 0 t and without initial data where the observations are taken in an open subset of we obtain the existence and introduce an approximation process for the identification problem

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

on the solutions of certain fractional kinetic equations fractional kinetic equations have gained importance during the last decade due to their occurrence in certain problems in science and engineering this article gives a brief survey of various fractional kinetic equations solved by a number of authors we also present a new method for solving the fractional kinetic equations that is different from that using laplace transform the results presented are in compact forms suitable for numerical computation some figures are given to show the behaviour of the derived solutions

acoustics in a two deck viscothermal boundary layer over an impedance surface the acoustics of a mean flow boundary layer over an impedance surface or acoustic lining is considered by considering a thick mean flow boundary layer possibly due to turbulence the boundary la

a 3d fractional elastoplastic constitutive model for concrete material abstract the core mission to establish elastoplastic constitutive model is how to determine the magnitude and direction of plastic strain increment for material under the given load fractional derivative provides a powerful tool for determining the nonorthogonal direction of a certain differentiable function a three dimensional 3d fractional plastic flow rule suitable for concrete material is presented for the first time utilizing fractional derivative the direction of plastic strain increment is determined by the proposed fractional plastic flow rule a closed form yield function is developed in the transformed stress ts space based on the nonlinear unified strength criterion and further used to describe the plastic strain induced by hydrostatic loading a united hardening softening function proposed by the authors in previous works is adopted to describe the pre peak hardening and post peak softening behaviour of concrete furthermore a 3d fractional elastoplastic constitutive model for concrete material is presented only nine material parameters with the clear physical meaning are need for the proposed model and the determination method of each material parameter is given a series of test results of concrete with different strength under biaxial and triaxial stress conditions are used to assess the performance of proposed model

identification of nonlinearity in k approximate periodic parabolic equations abstract this work is concerned with the identification of nonlinearities in k approximate periodic parabolic equations with observations on a subdomain of the physical domain some approximation processes for the identification problems are given

a latency fractional order model for hiv dynamics we study a fractional order model for hiv infection where latent t helper cells are included we compute the reproduction number of the model and study the stability of the disease free equilibrium we observe that the reproduction number varies with the order of the fractional derivative in terms of epidemics this suggests that varying induces a change in the patients epidemic status moreover we simulate the variation of relevant parameters such as the fraction of uninfected cd4 t cells that become latently infected and the ctls proliferation rate due to infected cd4 t cells the model produces biologically reasonable results

identification of discontinuous parameters in flow equations a problem of identifying possibly discontinuous diffusion coefficients in parabolic equations is considered general theorems on existence and convergence of galerkin approximations are proved in l 1 setting classes of functions of bounded variation are discussed and the variation estimates are obtained a double discretization method with the variations constraints is used in two and three dimensional problems and the numerical experiments are presented

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

some boundary value problems of temperature fields in oil strata we consider two boundary value problems related to the temperature fields in oil strata laplace and a generalized hankel transform are used to obtain the desired solutions efro s theorem which is a modified form of the convolution theorem for the laplace transform is generalized using confluent hypergeometric function particular cases of our results established here are in agreement with some known results given in the literature

a plastic damage model for concrete in this paper a constitutive model based on an internal variable formulation of plasticity theory for the non linear analysis of concrete is presented the model uses a new yield criterion which matches experimental data quite well and it accounts for both elastic and plastic stiffness degradations effects onset and amount of cracking can be studied by a simple postprocessing of the finite element plasticity solution the accuracy of the model is checked with some examples of application

coupled plastic damaged model abstract a constitutive model that couples plasticity and damage is presented the model is thermodynamically consistent and comes from a generalization of classical plasticity theory and isotropic damage theory of kachanov coupling between plasticity and damage is achieved through a simultaneous solution of the plastic and the damage problem after a description of the model a numerical algorithm for the integration of the resulting constitutive equations is presented it is an euler backward type of algorithm that is particularly suitable to solve plain stress non linear problems with a 2d finite element program the consistent stiffness matrix is also derived the paper is completed with some application examples that show that the model presented accurately reproduces the behaviour of elastic plastic damaged materials

towards a unified dynamical theory of the brownian particle in an ideal gas we consider the trajectoryqm t of a brownian particle of massm in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 the dynamics is uniform motion plus elastic collisions with the brownian particle our theory in dimension one describes a variety of limiting processes containing the wiener process and the ornstein uhlenbeck process fora 1 2qm a at depending on the asymptotic behaviour ofm a part of the theory is hypothetical while another part relies upon known results we also prove that ifa1 2 e m a a thena 1 2qm a at converges to a wiener process whose variance is known from papers of sinai soloveichik and of the present authors

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

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

a boundary element formulation for a class of non local damage models this paper proposes and discusses a boundary element formulation for a particular class of non local damage models the formulation as well as the boundary element computational code developed during this research have proven to be very simple and efficient providing reliable information on the strains and stresses in damage softening models the numerical approach uses a finite grid to estimate the damage values some illustrative numerical examples which show the simplicity and versatility of the proposed approach are included and discussed in detail

fractional sliding mode based on rbf neural network observer application to hiv infection mathematical model abstract a novel fractional observer based on sliding mode method combined with radial basis function neural network rbfnn is presented to distinguish the uncertainties of fractional order hiv human immunodeficiency virus mathematical dynamic model considering that the dynamical model of hiv infection varies from person to person including the symptoms and the complex conditions the mathematical model would be affected by uncertainties and disturbances therefore in the proposed method system uncertainties are estimated by the universal approximation ability of the neural network and external disturbance is vanquished by the robust attribute of sliding mode observer the objectives are firstly to propose a method for estimating the states for nonlinear fractional order system based on a novel fractional update law for rbfnn secondly by applying the fractional extension of lyapunov stability called mittag leffler stability guarantee the finite time stability of the proposed observer finally applying matlab simulation validate the performance of the proposed observer on hiv system

a new variational theory and a computational algorithm for coupled elastoplastic damage models a model of elastoplasticity with hardening coupled with ductile damage which allows anisotropic change of the shape of the damage domain is presented the assumption for the development of the constitutive model consists in assigning the elastic and damage domains and the free energy the general variational formulation for the elastoplastic model coupled with damage is derived as well as a mixed principle which is useful from the computational standpoint accordingly a solution algorithm is then derived and examples are presented

an implicit projection algorithm for simultaneous flow of plasticity and damage in standard generalized materials reference lmaf article 1996 002view record in web of science record created on 2005 09 14 modified on 2017 05 10

a strain based plastic viscous damage model for massive concrete structures abstract within the framework of continuum damage mechanics a new constitutive damage model for massive concrete is presented mainly intended for the seismic analysis of gravity and arch dams consistent with thermodynamic requirements a straindriven formalism is adopted improving the algorithmic efficiency as much as required for the analysis of large scale problems to become feasible two scalar damage variables are introduced as internal variables as well as a plastic strain tensor an extension to account for the concrete strain rate dependency suitable for seismic analysis is presented at the end the efficiency of numerical predictions from the constitutive model is illustrated through numerical applications and algorithmic implementation is also detailed

exact and discretized stability of the bagley torvik equation abstract the paper discusses stability and asymptotic properties of a two term linear fractional differential equation involving the bagley torvik equation as the particular case these properties are analysed for the exact as well as numerical solutions obtained from the grunwald letnikov discretization of the studied differential equation as the main results precise descriptions of the exact and discretized stability regions are presented including the decay rate of the solutions these results enable us among others to observe similarities and distinctions between the asymptotic behaviour of the classical and fractional damping models

bounds for the limiting variance of the heavy particle inr1 we consider a one dimensional system consisting of a tagged particle of massm surrounded by a gas of unit mass hard point particles in thermal equilibrium denoting byq t the displacement of the tagged particle we give lower and upper bounds independent ofm for it results from the proof that the correct nontrivial norming ofq t if any is sqrt t

sustaining stable dynamics of a fractional order chaotic financial system by parameter switching in this paper a simple parameter switching ps methodology is proposed for sustaining the stable dynamics of a fractional order chaotic financial system this is achieved by switching a controllable parameter of the system within a chosen set of values and for relatively short periods of time the effectiveness of the method is confirmed from a computer aided approach and its applications to chaos control and anti control are demonstrated in order to obtain a numerical solution of the fractional order financial system a variant of the grunwald letnikov scheme is used extensive simulation results show that the resulting chaotic attractor well represents a numerical approximation of the underlying chaotic attractor which is obtained by applying the average of the switched values moreover it is illustrated that this approach is also applicable to the integer order financial system

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

central limit theorems for the one dimensional rayleigh gas with semipermeable barriers a version of the one dimensional rayleigh gas is considered a point particle of massm molecule confined to the unit interval 0 1 is surrounded by an infinite ideal gas of point particles of mass 1 atoms the molecule interacts with the atoms and with the walls via elastic collision central limit theorems are proved for a wide class of additive functionals of this system e g the number of collisions with the walls and the total length of the molecular path

viscous damage model for timoshenko beam structures abstract a local damage constitutive model based on kachanov s theory is used within a finite element frame and applied to the case of 2d and 3d timoshenko beam elements the model takes into account viscous effects thus allowing damping to be considered in a rigorous way a damage index based on potential energy criteria useful in evaluating the behaviour of structures or of parts of structures is proposed the procedure is applied to estimate the damage produced by seismic actions in reinforced concrete building structures whose response is computed by using a non linear newmark type incremental time integration scheme three numerical examples are included one of them compares results obtained by using the proposed model with results of a laboratory test

a numerical solution for a variable order reaction diffusion model by using fractional derivatives with non local and non singular kernel abstract a reaction diffusion system can be represented by the gray scott model the reaction diffusion dynamic is described by a pair of time and space dependent partial differential equations pdes in this paper a generalization of the gray scott model by using variable order fractional differential equations is proposed the variable orders were set as smooth functions bounded in 0 1 and specifically the liouville caputo and the atangana baleanu caputo fractional derivatives were used to express the time differentiation in order to find a numerical solution of the proposed model the finite difference method together with the adams method were applied the simulations results showed the chaotic behavior of the proposed model when different variable orders are applied

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

an energetic formulation of a gradient damage model for concrete and its numerical implementation abstract the energetic formulation of a rate independent system assumes that the evolution of the system is driven by two scalar valued functions the storage energy functional and the dissipation pseudo potential the evolution of the material system is then characterized by two energetic principles the global stability condition and the energy balance between stored and dissipated energies with the work of external loading rate independent generalized standard materials are endowed with such structure whereas the existence of such energetic structure is not so apparent in materials with non associated flow rule in this paper we consider a rate independent gradient damage model for concrete where the evolution of the damage does not follow the normality rule we will show that such model can be nevertheless derived by a dissipation potential at the expense of having a state variable dependent potential and therefore the energetic formulation can be obtained also in this case after introducing the incremental minimization problem consistent with such formulation we obtain a discrete version of the stability condition and establish lower and upper a priori energy bounds met by the energetic solution these are fundamental results for the analysis of the formulation the actual numerical solution of the incremental minimization problem is realized by considering first a variable splitting in order to treat the gradient of the damage field as an independent variable and then by applying the augmented lagrangian method to tackle with the resulting constrained optimization problem we solve the first order stationarity conditions of the augmented lagrangian functional by a path following newton s method based on the energy dissipation rate control we show that we are able to describe highly non linear responses of the material such as softening branches and snap back responses several numerical tests are performed to verify the objectivity of the formulation and of the proposed numerical method details of the numerical implementation are also given