rational approximation solution of the fractional sharma tasso olever equation in the paper we implement relatively new analytical techniques the variational iteration method the adomian decomposition method and the homotopy perturbation method for obtaining a rational approximation solution of the fractional sharma tasso olever equation the three methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations in these schemes the solution takes the form of a convergent series with easily computable components the numerical results demonstrate the significant features efficiency and reliability of the three approaches

finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models this paper presents finite element formulations for transient dynamic analysis in structural systems with damping treatments in which viscoelastic materials are characterized by means of fractional derivative models in contrast to other formulations such as that of padovan employing the principle of virtual work the proposed formulations begin from the local equation of linear momentum and make use of the weighted residual method providing a matrix equation of motion involving fractional operators the numerical approximations of these are developed through the grunwald letnikov definition which allows to formulate explicit and implicit numerical schemes the principal advantage of the proposed formulations is that the history of the displacements and external forces must be stored but not that of the stress which reduces computational time and storage needs numerical applications are presented for a cantilever beam where a viscoelastic treatment has been applied using damping material modelled by a five parameter fractional derivative model the results of the proposed formulation are compared among them and with those of padovan for different damping values and for different load cases the influence of the truncation of grunwald coefficients and of the integration time step is also investigated copyright 2006 john wiley sons ltd

bayesian characterization of young s modulus of viscoelastic materials in laminated structures abstract this paper addresses an inverse approach to estimate the frequency dependent young s modulus of a viscoelastic polymer layer in a laminated structure the young s modulus is parameterized by a fractional derivative model and examined from a bayesian perspective with the consideration of measurement and modeling uncertainties the probabilistic bayesian identification is carried out based on an efficient surrogate model through the use of markov chain monte carlo sampling methods the proposed approach is experimentally validated on laminated glass

a note on optimal homotopy asymptotic method for the solutions of fractional order heat and wave like partial differential equations optimal homotopy asymptotic method oham is prolifically implemented to find the optimal solutions of fractional order heat and wave like equations we inspect the competence of the method by examining fractional order time dependent partial differential equations it is observed that oham is a prevailing and convergent method for the solutions of linear and nonlinear fractional order time dependent partial differential problems the numerical results rendering that the applied method is explicit effective and easy to use for handling more general fractional order heat and wave like models

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

closed form eigensolutions of nonviscously nonproportionally damped systems based on continuous damping sensitivity abstract in this paper nonviscous nonproportional vibrating structures are considered nonviscously damped systems are characterized by dissipative mechanisms which depend on the history of the response velocities via hereditary kernel functions solutions of the free motion equation lead to a nonlinear eigenvalue problem involving mass stiffness and damping matrices viscoelasticity leads to a frequency dependence of this latter in this work a novel closed form expression to estimate complex eigenvalues is derived the key point is to consider the damping model as perturbed by a continuous fictitious parameter assuming then the eigensolutions as function of this parameter the computation of the eigenvalues sensitivity leads to an ordinary differential equation from whose solution arises the proposed analytical formula the resulting expression explicitly depends on the viscoelasticity frequency derivatives of the damping function the nonproportionality influence of the modal damping matrix off diagonal terms eigenvectors are obtained using existing methods requiring only the corresponding eigenvalue the method is validated using a numerical example which compares proposed with exact ones and with those determined from the linear first order approximation in terms of the damping matrix frequency response functions are also plotted showing that the proposed approach is valid even for moderately or highly damped systems

analysis of four parameter fractional derivative model of real solid materials the introduction of fractional derivatives into the constitutive equation of the differential operator type of linear solid materials has led to the development of the so called fractional derivative models one of these models characterized by four parameters has been found usable for describing the variation of dynamics elastic and damping properties in a wide frequency range provided that there is only one loss peak in this paper this four parameter model is theoretically analyzed the effect of the parameters on the frequency curves is demonstrated and it is shown that there is a strict relation between the dispersion of the dynamic modulus the loss peak and the slope of the frequency curves the half value bandwidth of the loss modulus frequency curve is investigated and conditions are developed to establish the applicability of the model for a class of materials moreover it is shown that the model can be used to predict the frequency dependences of dynamic properties for a wide range even if measurements are made in only a narrow frequency range around the loss peak

group detection for synchronous gaussian code division multiple access channels the concept of group detection is introduced to address the design of suboptimum multiuser detectors for code division multiple access cdma channels a group detection scheme consists of a bank of p group detectors one each for detecting the information symbols of users in each group of a p group partition of the k simultaneously transmitting users in a parallel group detection scheme these group detectors operate independently whereas in a sequential scheme each group detector uses the decisions of the previous group detectors to successively cancel the interference from those users group detectors based on the generalized likelihood ratio test glrt are obtained for the synchronous gaussian cdma channel the complexity of these detectors is exponential in the group size whereas that of the optimum detector is exponential in k since the partition of users is a design parameter group sizes can be chosen to satisfy a wide range of complexity constraints a key performance result is that the glrt group detectors are optimally group near far resistant furthermore upper and lower bounds on the asymptotic efficiency of the sequential group detectors are derived these bounds reveal that the sequential group detectors can under certain conditions perform as well as glrt group detectors of much larger group sizes group detection provides a unifying approach to multiuser detection when the users are partitioned into k single user groups the glrt a modified form of glrt and the sequential group detectors reduce to previously proposed suboptimal detectors namely the decorrelator the two stage detector and the decorrelating decision feedback detector respectively for the other nontrivial partitions the group detectors are new and have a performance that is commensurate with their complexity

layout optimization of unconstrained viscoelastic layer on beams using fractional derivative model complex modulus f frequency i 1 k global stiffness matrix m global mass matrix t temperature t time u strain energy y eigenvector t shift factor ij kronecker delta e strain loss factor stress eigenvalue 2 f 2

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

consistency and asymptotic normality of the quasi maximum likelihood estimator in igarch 1 1 and covariance stationary garch 1 1 models this paper provides a proof of the consistency and asymptotic normality of the quasi maximum likelihood estimator in garch 1 1 and igarch 1 1 models in contrast to the case of a unit root in the conditional mean the presence of a unit root in the conditional variance does not affect the limiting distribution of the estimators in both models estimators are normally distributed in addition a consistent estimator of the covariance matrix is available enabling the use of standard test statistics for inference

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

analysis of the error performance of trellis coded modulations in rayleigh fading channels this work presents an exact expression for the pairwise error event probability of trellis coded modulation tcm transmitted over rayleigh fading channels it includes phase shift keying psk and multilevel quadrature amplitude modulation qam codes as well as coherent and partially coherent e g differential pilot tone etc detection due to the form of the exact pairwise error event probabilities this calculation technique cannot be used with the transfer function technique to obtain an upper union bound on the overall bit error probability for this reason the authors estimate the bit error probability by considering only a small number of short error events through simulations they found that the estimation is usually very accurate at high signal to noise ratios but not as accurate at lower signal to noise ratios they study several coded modulation schemes this way among the results are the fact that tcm provides significant improvement in the error floor when detected differentially and an asymmetry in the pairwise error event probability for 16 qam

approximate solutions of fractional zakharov kuznetsov equations by vim this paper presents the approximate analytical solution of a fractional zakharov kuznetsov equation with the help of the powerful variational iteration method the fractional derivatives are described in the caputo sense several examples are given and the results are compared to exact solutions the results show that the variational iteration method is very effective convenient and simple to use

the multi component coupled burgers hierarchy of soliton equations and its multi component integrable couplings system with two arbitrary functions abstract a new loop algebra g m is constructed which is devoted to establishing an isospectral problem by making use of tu scheme the multi component coupled burgers hierarchy is obtained furthermore an expanding loop algebra f m of the loop algebra g m is presented it follows that the multi component integrable coupling system of the multi component coupled burgers hierarchy is worked out in general the method can be applied to other evolution equations hierarchies

thermorheologically simple materials a bayesian framework for model calibration and validation abstract the dynamic properties of viscoelastic materials show highly frequency temperature dependency and numerical methods for structural systems containing this type of material require accurate mathematical models to describe their dynamical behaviour the material behaviour here is modelled using a constitutive equation based on fractional derivative operators and considering the temperature dependence of the material under the thermorheologically simple postulate the quest for information about the constitutive model parameters is phrased as a statistical inverse problem under the bayesian framework a markov chain monte carlo mcmc method is used to explore the posterior density of model parameters using measured data from dynamic tests at different temperatures the agreement between measured data and the predictive capabilities of sixteen models were quantitatively assessed using two validation metrics based on the validation metrics analysis it is possible to conclude that the range of temperature of the calibration data set is a key point into the implementation of the frequency temperature superposition principle ftsp this was verified defining some scenarios for assessing the agreement of model predictions and the set of available experimental data the results are quite compelling due to the fact that the proposed approach is easy handed furthermore this approach could be applied on any generic constitutive model using the ftsp

computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed point iteration abstract linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions the free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency dependent damping matrix in the present paper a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non proportional viscoelastic structures is developed the central idea is the construction of two complex valued functions of a complex variable whose fixed points are precisely the eigenvalues this important property allows the use of these functions in a fixed point iterative scheme with help of some results in fixed point theory necessary conditions for global and local convergence are provided it is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping in addition under certain conditions the recursive method can also be used for the calculation of non viscous real eigenvalues in order to validate the mathematical results two numerical examples are analyzed one for single degree of freedom systems and another for multiple ones

the fractional boltzmann transport equation the fractional boltzmann transport equation is derived making use of the fractional hamilton s equations based on the fractional actionlike variational approach by simply defining a distribution function and inspecting its time derivative many important results in statistical physics can be derived

application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives two nonlinear anelastic models with fractional derivatives describing the properties of a series of materials as polymers and polycrystalline materials are presented in this paper these models are studied analytically using a variational iteration method the paper clarifies the different ways in which the fractional differentiation operator can be defined a volterra series method of model parameters identification from the experimental data is also presented

powerful tests for structural changes in volatility abstract detecting structural changes in volatility is important for understanding volatility dynamics and stylized facts observed for financial returns such as volatility persistence we propose modified cusum and lm tests that are built on a robust estimator of the long run variance of squared series we establish conditions under which the new tests have standard null distributions and diverge faster than standard tests under the alternative the theory allows smooth and abrupt structural changes that can be small the smoothing parameter is automatically selected such that the proposed test has good finite sample size and meanwhile achieves decent power gain

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

weak solvability of a fractional viscoelastic frictionless contact problem abstract the goal of this paper is to study a quasistatic frictionless contact problem for a viscoelastic body in which the constitutive equation is modeled with the fractional kelvin voigt law and the contact condition is described by the clarke subdifferential of a nonconvex and nonsmooth functional the variational formulation of this problem is provided in the form of a fractional hemivariational inequality in order to solve this inequality we apply the rothe method and prove that the associated abstract volterra inclusion has at least one solution

dispersion curves of viscoelastic plane waves and rayleigh surface wave in high frequency range with fractional derivatives abstract the moduli of conventional elastic structural materials are extended to one of the viscoelastic materials through a modification whereby the dynamic moduli converge to the static moduli of elasticity as the fractional order approaches zero by plotting phase velocity curves and group velocity curves of plane waves and rayleigh surface wave for a viscoelastic material polyvinyl chloride foam the influence of the fractional order of viscoelasticity is examined the phase and group velocity curves in the high frequency range were derived for longitudinal transverse and rayleigh waves inherent to the viscoelastic material in addition the equation for the phase velocity was mathematically derived on the complex plane too and graphically illustrated a phenomenon was found that at the moment when the fractional order of the time derivative reaches an integer value 1 the curve on the complex plane becomes completely different exhibiting snap through behavior we examined the mechanism of the snap through mathematically numerical calculation examples were solved and good agreement was confirmed between the numerical calculation and the analytical expression mentioned above from the results of the numerical example regularities were derived for the absolute value of the complex phase and group velocities on the complex plane

the variational iteration method for solving riesz fractional partial differential equations in this paper the variational iteration method is applied to obtain the solution for space fractional partial differential equations where the space fractional derivative is in the riesz sense on the basis of the properties and definition of the fractional derivative the iterative technique is carried out in a straightforward manner without the need for transforms or numerical approximations examples demonstrate that the series solution obtained shows agreement with the exact solutions of the problems solved

five parameter fractional derivative model for polymeric damping materials fractional derivative models offer a powerful tool to describe the dynamic behaviour of real viscoelastic materials a version of the fractional derivative models characterized by five parameters is presented and investigated in this paper in order to describe asymmetrical loss factor peak and the high frequency behaviour of polymeric damping materials the speculative derivation of the model constitutive equation containing time derivatives of stress and strain of different orders is given the model behaviour is investigated in the frequency domain the physical meaning of the model parameters is defined and constraints on the parameter values are made it is shown that the asymmetry of loss peak and the high frequency behaviour of the model are governed by the difference between the order of time derivatives of stress and strain moreover it is shown that this difference is related to the high frequency limit value of the loss factor the model is fitted to experimental data on some polymeric damping materials to verify its behaviour

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

an adaptation of the gear scheme for fractional derivatives abstract the gear scheme is a three level step algorithm backward in time and second order accurate for the approximation of classical time derivatives in this contribution the formal power of this scheme is proposed to approximate fractional derivative operators in the context of finite difference methods some numerical examples are presented and analysed in order to show the effectiveness of the present gear scheme at the power g scheme when compared to the classical grunwald letnikov approximation in particular for a fractional damped oscillator problem the combined g newmark scheme is shown to be second order accurate

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

fractional generalization of gradient and hamiltonian systems we consider a fractional generalization of hamiltonian and gradient systems we use differential forms and exterior derivatives of fractional orders we derive fractional generalization of helmholtz conditions for phase space examples of fractional gradient and hamiltonian systems are considered the stationary states for these systems are derived

verification of local kramers kronig relations for complex modulus by means of fractional derivative model abstract the local kramers kronig k k relations which link the damping properties of solid materials at one frequency to the rate of frequency variation of dynamic modulus are not exact the validity and accuracy of the local k k relations is theoretically investigated in this paper by means of material models especially the fractional zener model it is shown that the local k k relations qualitatively always properly predict the relation between the damping and the frequency dependence of dynamic modulus for any type of deformation and any linear mechanism of energy loss determining the frequency variations nevertheless the accuracy depends on the rate of frequency variation of dynamic properties mainly of the loss modulus and loss factor and the weaker the frequency dependence the better the accuracy the accuracy is better than 10 if the slope of frequency increase or decrease of loss functions plotted in a log log system is smaller than 0 35 the application of the local k k relations to some experimental data is presented

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

neglecting parameter changes in garch models if a garch model is estimated on a time series that contains parameter changes in the conditional volatility process and these parameter changes are not accounted for a distinct error in the estimation occurs the sum of the estimated autoregressive parameters of the conditional variance converges to one in finite samples the sum of the estimated autoregressive parameters is heavily biased towards one this paper shows that this convergence holds for all common estimators of garch simulations of the garch model show that the effect occurs for realistic parameter changes and sample sizes for financial volatility data

hierarchical analogues to fractional relaxation equations recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena such as stress strain relationships in polymeric materials in the rheological constitutive equations this implies the replacement of the first order derivatives by fractional derivatives here we show that such procedures have hierarchical mechanical analogues we focus on the generalized dashpot and the generalized maxwell model and display the corresponding arrangements our models allow a transparent interpretation of the parameters which enter the fractional equations and reveal that the internal dynamics are hierarchically constrained

truncation for low complexity mimo signal detection joint maximum likelihood jml detector may be used in memoryless multiple input multiple output mimo systems to obtain optimal detection performance however jml detector performs an exhaustive search and has prohibitively large decoding complexity to reduce the complexity of mimo signal detection minimum mean square error mmse linear detector ld decision feedback detector dfd group detector and sphere detector sd may be used in this correspondence we propose a truncation based detector for low complexity mimo signal detection and give theoretical insight into the design and performance of such a detector we study bitruncation in detail and present two bitruncation approaches these approaches have low complexity and computer simulation results show that they outperform mmse ld and mmse dfd

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

comments on nonlinear vibration of viscoelastic beams described using fractional order derivatives abstract this discussion seeks to address some of the inconsistencies and concerns regarding fractional derivatives when applied to vibration analysis of viscoelastically damped beams 1 detailed numerical assessments of the major fractional derivatives of concern have been carried out with their relationships numerically examined numerical results have shown that two of the key equations of 1 are mathematically inappropriate and their applications could lead to serious modelling errors a simplified yet representative nonlinear system with fractional viscoelastic damping and cubic stiffness nonlinearity is used to illustrate the existence of periodic steady state solutions as well as amplitude frequency characteristics of such fractional nonlinear systems

application of he s variational iteration method for solving the cauchy reaction diffusion problem in this paper the solution of cauchy reaction diffusion problem is presented by means of variational iteration method reaction diffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields application of variational iteration technique to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution moreover this technique does not require any discretization linearization or small perturbations and therefore it reduces significantly the numerical computations

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

a numerical study based on an implicit fully discrete local discontinuous galerkin method for the time fractional coupled schrodinger system in this paper we develop and analyze an implicit fully discrete local discontinuous galerkin ldg finite element method for solving the time fractional coupled schrodinger system the method is based on a finite difference scheme in time and local discontinuous galerkin methods in space through analysis we show that our scheme is unconditionally stable and the l 2 error estimate has the convergence rate o h k 1 dt 2 dt a 2h k 1 2 for the linear case extensive numerical results are provided to demonstrate the efficiency and accuracy of the scheme

dynamics of a rod made of generalized kelvin voigt visco elastic material abstract in this paper we develop a new mathematical model for the lateral vibration of an axially compressed visco elastic rod as the basis for this model we use a fractional derivative type of stress strain relation we show that the dynamics of the lateral vibration is governed by two coupled linear differential equations with fractional derivatives for a special case of the generalized kelvin voigt body this system is reduced to a single fractional derivative differential equation eq 19 for a class of problems to which 19 belongs the questions of the existence of a solution and its regularity are analyzed both continuous and impulsive loading are treated

stability of an elastic rod on a fractional derivative type of foundation the lateral vibration of an axially loaded elastic rod positioned on a fractional derivative type of foundation is studied it is shown that the dynamics of the problem is governed by a system of two coupled linear differential equations with fractional derivatives for this system of equations the questions of existence regularity and the stability of solution are analysed the results are compared with the stability bound for an elastic rod on winkler elastic type of foundation

a causal and fractional all frequency wave equation for lossy media this work presents a lossy partial differential acoustic wave equation including fractional derivative terms it is derived from first principles of physics mass and momentum conservation and an equation of state given by the fractional zener stress strain constitutive relation for a derivative order in the fractional zener relation the resulting absorption k obeys frequency power laws as k 1 in a low frequency regime k 1 2 in an intermediate frequency regime and k 1 in a high frequency regime the value 1 corresponds to the case of a single relaxation process the wave equation is causal for all frequencies in addition the sound speed does not diverge as the frequency approaches infinity this is an improvement over a previously published wave equation building on the fractional kelvin voigt constitutive relation

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

a multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries in this paper we propose a flexible model to describe nonlinearities and long range dependence in time series dynamics the new model is a multiple regime smooth transition extension of the heterogeneous autoregressive har model which is specifically designed to model the behavior of the volatility inherent in financial time series the model is able to simultaneously approximate long memory behavior as well as describe sign and size asymmetries a sequence of tests is developed to determine the number of regimes and an estimation and testing procedure is presented monte carlo simulations evaluate the finite sample properties of the proposed tests and estimation procedures we apply the model to several dow jones industrial average index stocks using transaction level data from the trades and quotes database that covers ten years of data we find strong support for long memory and both sign and size asymmetries furthermore the new model when combined with the linear har model is viable and flexible for purposes of forecasting volatility

a method for modeling polymer viscoelastic data and the temperature shift function dynamic viscoelastic polymer data is traditionally time temperature shifted to obtain a temperature shift function tsf and then fitted to various analytical models the process of obtaining the tsf can introduce considerable procedural or operator bias nevertheless the havriliak and negami hn model using the tsf methodology can generally describe polymers that are rheologically simple in this paper the wicket plot is utilized as an important tool in analyzing data as it is completely independent of time temperature shifting tts using the wicket plot the data is fit to the hn equation to determine the four material hn constants independent of tts having obtained the complete spectra of dynamic properties the specific relaxation time frequency at each temperature is obtained by matching the hn curve to the experimental data at that temperature thus determining the tsf the procedure is illustrated by analyzing computer generated data with random error in modulus and loss and finally real data

alternative models for stock price dynamics abstract this paper evaluates the role of various volatility specifications such as multiple stochastic volatility sv factors and jump components in appropriate modeling of equity return distributions we use estimation technology that facilitates nonnested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns we consider two broad families of models 1 the multifactor loglinear family and 2 the affine jump family both classes of models have attracted much attention in the derivatives and econometrics literatures there are various tradeoffs in considering such diverse specifications if pure diffusion sv models are chosen over jump diffusions it has important implications for hedging strategies if logarithmic models are chosen over affine ones it may seriously complicate option pricing comparing many different specifications of pure diffusion multifactor models and jump diffusion models we find that 1 log linear models have to be extended to two factors with feedback in the mean reverting factor 2 affine models have to have a jump in returns stochastic volatility or probably both models 1 and 2 are observationally equivalent on the data set in hand in either 1 or 2 the key is that the volatility can move violently as we obtain models with comparable empirical fit one must make a choice based on arguments other than statistical goodness of fit criteria the considerations include facility to price options to hedge and parsimony the affine specification with jumps in volatility might therefore be preferred because of the closed form derivatives prices

dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractionalderivatives frame structures with viscoelastic dampers mounted on them are considered in this paper viscoelastic ve dampers are modelled using two three parameter fractional rheological models the structures are treated as elastic linear systems the equation of motion of the whole system structure with dampers is written in terms of state space variables the resulting matrix equation of motion is the fractional differential equation the proposed state space formulation is new and does not require matrices with huge dimensions the paper is devoted to determine the dynamic properties of the considered structures the nonlinear eigenvalue problem is formulated from which the dynamic parameters of the system can be determined the continuation method is used to solve the nonlinear eigenvalue problem moreover results of typical calculations are presented

fractional visco elastic timoshenko beam deflection via single equation summary this paper deals with the response determination of a visco elastic timoshenko beam under static loading condition and taking into account fractional calculus in particular the fractional derivative terms arise from representing constitutive behavior of the visco elastic material further taking advantages of the mellin transform method recently developed for the solution of fractional differential equation the problem of fractional timoshenko beam model is assessed in time domain without invoking the laplace transforms as usual further solution provided by the mellin transform procedure will be compared with classical central difference scheme one based on the grunwald letnikov approximation of the fractional derivative moreover timoshenko beam response is generally evaluated by solving a couple of differential equations in this paper expressing the equation of the elastic curve just through a single relation a more general procedure which allows the determination of the beam response for any load condition and type of constraints is developed copyright 2015 john wiley sons ltd

analysis of free vibrations of a viscoelastic oscillator via the models involving several fractional parameters and relaxation retardation times free damped vibrations of a linear viscoelastic oscillator based on the fractional derivative model involving more than one different fractional parameters and several relaxation retardation times are investigated the analytical solution is obtained in the form of two terms one of which governs the drift of the system s equilibrium position and is defined by the dynamic relaxation retardation processes occurring in the system and the other term describes damped vibrations around the equilibrium position and is determined by the system s inertia and energy dissipation the drift is governed by an improper integral taken along two sides of the cut of the complex plane while damped vibrations are determined according to the two complex conjugate roots of the characteristic equation which are located on the left half of the complex plane the behaviour of the characteristic equation roots as functions of the system s rheological parameters which enable the control of the dynamic response of the oscillator is shown in the complex plane

dispersion curves for a viscoelastic timoshenko beam with fractional derivatives abstract the kramers kronig dispersion relation often used as a viscoelastic constitutive law for polymeric materials is based on purely mathematical properties of linearity convergence of improper integrals and causality thus it may also be valid as a viscoelastic constitutive law for general structural materials accordingly the motion equation of a timoshenko beam composed of conventional elastic structural materials is extended to one composed of viscoelastic materials from the derived governing equation a dispersive equation is derived for a viscoelastic timoshenko beam by plotting phase velocity curves and group velocity curves for a beam of solid circular cross section composed of a viscoelastic material polyvinyl chloride foam the influence of the fractional order of viscoelasticity is examined as a result it is found that in the high frequency range only the first mode of a timoshenko beam converged to the propagation velocity of the rayleigh wave which takes account of the fractional order of viscoelasticity in addition the phase velocity and the group velocity were found to increase as the fractional order approaches 0 and to decrease as the fractional order approaches 1 furthermore the rate of velocity change becomes greater as the fractional order approaches 0 and becomes smaller as the fractional order approaches 1

fractional differential forms a generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders that is a fractional exterior derivative is defined this is found to generate new vector spaces of finite and infinite dimension fractional differential form spaces the definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case coordinate transformation rules are also computed the transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative the metric for the fractional form spaces is given based on the coordinate transformation rules all results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one

fractional actionlike variational problems fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems in this note we address multidimensional fractional actionlike problems of the calculus of variations

a variational method for identification of viscoelastic parameters from experimental data an identification method based on a variational algorithm is proposed for the determination of the viscoelastic properties of plates from a set of experimental data a pulse transmission configuration with parametric sources is used for the experimental measurements whereby two electrically mixed high frequency pulses are combined to generate a low frequency narrow beam signal to minimize the edge diffraction effect from a finite sized panel the variational method is based on an optimization technique adapted from an approach normally used in dynamics for the identification of motion parameters the fluid panel fluid system is initially represented by a transmission matrix model the model and the known experimental data are then used to build an optimization functional to be minimized by means of a collocation method based on the hamilton principle analogy the plate viscoelastic parameters are then calculated by application of well established numerical procedures to the resulting set of integral differential equations

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

convergence of the variational iteration method for solving multi order fractional differential equations in this paper the variational iteration method vim is applied to obtain approximate solutions of multi order fractional differential equations m fdes we can easily obtain the satisfying solution just by using a few simple transformations and applying the vim a theorem for convergence and error estimates of the vim for solving m fdes is given moreover numerical results show that our theoretical analysis are accurate and the vim is a powerful method for solving m fdes

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

fractional systems and fractional bogoliubov hierarchy equations we consider the fractional generalizations of the phase volume volume element and poisson brackets these generalizations lead us to the fractional analog of the phase space we consider systems on this fractional phase space and fractional analogs of the hamilton equations the fractional generalization of the average value is suggested the fractional analogs of the bogoliubov hierarchy equations are derived from the fractional liouville equation we define the fractional reduced distribution functions the fractional analogs of the vlasov equation and the debye radius are considered

simulation of fractionally damped mechanical systems by means of a newmark diffusive scheme a newmark diffusive scheme is presented for the time domain solution of dynamic systems containing fractional derivatives this scheme combines a classical newmark time integration method used to solve second order mechanical systems obtained for example after finite element discretization with a diffusive representation based on the transformation of the fractional operator into a diagonal system of linear differential equations which can be seen as internal memory variables the focus is given on the algorithm implementation into a finite element framework the strategies for choosing diffusive parameters and applications to beam structures with a fractional zener model

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

higher order eigensensitivities based numerical method for the harmonic analysis of viscoelastically damped structures this paper presents an efficient numerical method for the harmonic analysis of viscoelastically damped structural systems characterized by a frequency dependent structural damping matrix making use of the complex mode superposition method departing from the undamped eigensolution the proposed numerical method updates the complex and frequency dependent eigenpair avoiding the solution of a complex eigenproblem for each computational frequency the complex eigenvalues and eigenvectors are updated within the desired tolerance by an adaptive step size control scheme using the first and higher order eigenderivatives the influence on the computation time of the considered number of eigenderivatives and the tolerance is discussed and the efficiency of the proposed numerical method for the harmonic analysis of viscoelastically damped large ordered structural systems is proved finally a practical application is presented where the harmonic response of a constrained layer damping cantilever beam subjected to a base motion is analyzed the complex and frequency dependent eigenvalues and eigenvectors are computed the modal contributions to the total response are determined and the total response is approximated by the complex mode superposition method finally the approximated response is validated with the exact one computed by the direct frequency method and with that experimentally measured copyright 2011 john wiley sons ltd

nonlinear hamiltonian equations with fractional damping existence and uniqueness are proved for a class of hamiltonian nonlinear differential equations of second order with a caputo fractional derivative of order 0 2 for noninteger the rate of convergence of solutions to a critical point is algebraic for 0 1 a non negative dissipation function is constructed for 2 the equation is hamiltonian the solution is an analytic function of for 0 1 and for 1 2 numerical studies of nonlinear pendulum and the duffing equation with a fractional derivative term replacing the usual damping show striking differences between ordinary differential and fractional differential equations

compact structures in a class of nonlinearly dispersive equations with time fractional derivatives in this work variants of the kdv equation with time fractional derivatives which exhibits compactons solitons with compact support are investigated new solitary wave solutions with compact support are developed the homotopy perturbation method is employed to derive compact solutions for these variants the chosen initial solution or trial function plays a major role in changing the physical structure of the solution the proposed scheme can be used to construct compact and noncompact solutions for a wide class of nonlinear fractional evolution equations

a generalized fractional kn equation hierarchy and its fractional hamiltonian structure a generalized hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders we construct the generalized fractional trace identity through the riemann liouville fractional derivative an example of the fractional kn soliton equation hierarchy and hamiltonian structure is presented which is a new integrable hierarchy and possesses hamiltonian structure

finite element analysis of vibrating linear systems with fractional derivative viscoelastic models abstract fractional derivative rheological models are known to be very effective in describing the viscoelastic behaviour of materials especially of polymers and when applied to dynamic problems the resulting equations of motion after a fractional state space expansion can still be studied in terms of modal analysis but the growth in matrix dimensions carried by this expansion is in general so fast to make the calculations exceedingly cumbersome this paper presents a method for reducing the computational effort due to finite element fe analysis of vibrating linear systems with a fractional derivative viscoelastic model namely the fractional kelvin voigt the proposed method may be applied also to problems involving other fractional derivative linear models and it takes under control the computational effort by reducing the main eigenproblem of large dimension to the solution of two standard related eigenproblems of lower size numerical examples are provided in order to validate both the accuracy and the efficiency of the proposed methodology