high order methods for low reynolds number flows around moving obstacles based on universal meshes summary we propose a family of methods for simulating two dimensional incompressible low reynolds number flow around a moving obstacle whose motion is prescribed the methods make use of a universal mesh a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the obstacle s boundary the resulting mesh provides a conforming triangulation of the fluid domain over which discretizations of any desired order of accuracy in space and time can be constructed using standard finite element spaces together with off the shelf time integrators we demonstrate the approach by using taylor hood elements to approximate the fluid velocity and pressure to integrate in time we consider implicit runge kutta schemes as well as a fractional step scheme we illustrate the methods and study their convergence numerically via examples that involve flow around obstacles that undergo prescribed deformations copyright 2015 john wiley sons ltd

hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

map of discrete system into continuous continuous limits of discrete systems with long range interactions are considered the map of discrete models into continuous medium models is defined a wide class of long range interactions that give the fractional equations in the continuous limit is discussed the one dimensional systems of coupled oscillators for this type of long range interactions are considered the discrete equations of motion are mapped into the continuum equation with the riesz fractional derivative

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

leitmann s direct method for fractional optimization problems abstract based on a method introduced by leitmann g leitmann a note on absolute extrema of certain integrals int j non linear mech 2 1967 55 59 we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control

fractional variations for dynamical systems hamilton and lagrange approaches fractional generalization of an exterior derivative for calculus of variations is defined the hamilton and lagrange approaches are considered fractional hamilton and euler lagrange equations are derived fractional equations are obtained by fractional variation of lagrangian and hamiltonian that have only integer derivatives

continuous limit of discrete systems with long range interaction discrete systems with long range interactions are considered continuous medium models as continuous limit of discrete chain system are defined long range interactions of chain elements that give the fractional equations for the medium model are discussed the chain equations of motion with long range interaction are mapped into the continuum equation with the riesz fractional derivative we formulate the consistent definition of continuous limit for the systems with long range interactions in this paper we consider a wide class of long range interactions that give fractional medium equations in the continuous limit the power law interaction is a special case of this class

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

necessary and sufficient conditions for pareto optimal solutions of cooperative differential games in this paper we present necessary as well as sufficient conditions for the existence of a pareto optimum for cooperative differential games the obtained results are used to analyze the regular indefinite linear quadratic differential game for the scalar case an algorithm is devised to find all pareto efficient solutions

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

high accuracy power series solutions with arbitrarily large radius of convergence for the fractional nonlinear schrodinger type equations fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories nonlinearity and fractional derivatives these effective descriptions thus appear commonly in physical and mathematical modeling we present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations including the fractional nonlinear schrodinger equation and the fractional nonlinear diffusion equation the method relies on spatially iterative use of power series expansions our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches in the specific case of the fractional nonlinear schrodinger equation we find fractional generalizations of cnoidal waves of jacobi elliptic functions as well as a fractional bright soliton for the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails albeit at a much lower density

a general form of the generalized taylor s formula with some applications in this article a new general form of fractional power series is introduced some results of classical power series are circulated and proved to this new form a new general form of the generalized taylor s formula is also obtained applications including fractional power series solutions for fdes are analyzed results reveal that the new expansion is effective for formulating the solutions in this article a new general form of fractional power series is introduced in the sense of the caputo fractional derivative using this approach some results of the classical power series are circulated and proved to this fractional power series whilst a new general form of the generalized taylor s formula is also obtained some applications including fractional power series solutions for higher order linear fractional differential equations subject to given nonhomogeneous initial conditions are provided and analyzed to guarantee and to confirm the performance of the proposed results the results reveal that the new fractional expansion is very effective straightforward and powerful for formulating the exact solutions in the form of a rapidly convergent series with easily computable components

fractional noether s theorem in the riesz caputo sense abstract we prove a noether s theorem for fractional variational problems with riesz caputo derivatives both lagrangian and hamiltonian formulations are obtained illustrative examples in the fractional context of the calculus of variations and optimal control are given

on analytic properties of periodic solutions for equation mathcal h u_x u u p 0 the paper focuses on 2l periodic solutions ul x of the nonlocal equation p 2 is integer is the hilbert transform we give numerical evidence for the existence of a continuous family of these solutions parametrized by l for p 3 4 5 the features of the solutions ul x are discussed in particular we offer strong numerical arguments that ul x can be continued analytically from the real axis to some strip in the complex plane ul x ul z the singularities which arise under analytical continuation of these solutions into the complex plane are considered it is proved theorem 1 that ul z cannot be any power of a meromorphic function the formula which describes the asymptotic behavior of ul z in a neighborhood of the singularity is given it agrees with the numerical results computations also allow us to locate the closest to the real axis singularity of ul z and estimate the width of the analyticity strip

ale supg finite element method for convection diffusion problems in time dependent domains conservative form a streamline upwind petrov galerkin supg finite element method for a convection dominated transient convection diffusion reaction equation in time dependent domains is proposed the time dependent domain is handled by the arbitrary lagrangian eulerian ale approach whereas the supg method is used for the spatial discretization further the first order modified backward euler and the second order modified crank nicolson methods are used for the temporal discretization it is shown that the stability of the semi discrete continuous in time conservative ale supg equation is independent of the mesh velocity whereas the stability of the fully discrete scheme with the implicit euler time discretization is unconditionally stable and is only conditionally stable time step depends on mesh velocity with the crank nicolson method numerical results are presented to support the stability estimates and to show the influence of the supg stabilization parameter in a time dependent domain further the proposed numerical scheme is applied to a boundary layer problem in a time dependent domain c 2017 elsevier inc all rights reserved

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

fractal geometry information growth and nonextensive thermodynamics abstract this is a study of the information evolution of complex systems by a geometrical consideration we look at chaotic systems evolving in fractal phase space the entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement due to the incompleteness of the state number counting at any scale on fractal support the incomplete normalization i p i q 1 is applied throughout the paper where q is the fractal dimension divided by the dimension of the smooth euclidean space in which the fractal structure of the phase space is embedded it is shown that the information growth is nonadditive and is proportional to the trace form i p i i p i q which can be connected to several nonadditive entropies this information growth can be extremized to give power law distributions for these nonequilibrium systems it can also be used for the study of the thermodynamics derived from tsallis entropy for nonadditive systems which contain subsystems each having its own q it is argued that within this thermodynamics the stefan boltzmann law of blackbody radiation can be preserved

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

relative pareto minimizers for multiobjective problems existence and optimality conditions in this paper we introduce and study enhanced notions of relative pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of pareto and weak pareto efficiency minimality using advanced tools of variational analysis and generalized differentiation we establish the existence of relative pareto minimizers for general multiobjective problems under a refined version of the subdifferential palais smale condition for set valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers as well as for conventional efficient and weak efficient counterparts that are new in both finite dimensional and infinite dimensional settings our proofs are based on variational and extremal principles of variational analysis in particular on new versions of the ekeland variational principle and the subdifferential variational principle for set valued and single valued mappings in infinite dimensional spaces

analysis and parameter identification of bilinear systems via shifted legendre polynomials the problems of analysis and parameter identification of lime delay systems have been recently studied using rectangular functions such as walsh and block pulse functions in this paper the continuous shifted legendre polynomials are first used to solve the above mentioned problems the approach followed is that of converting the delay differential equation to an algebraic form through using the operational matrices of integration and delay the key point is the derivation of a delay operational matrix d which relates the coefficient vector x of the shifted legendre series of a given function x t with the coefficient vector x of its delay form x t in the form x dx where the elements of a are obtained from the initial function of x t two computational examples are given to illustrate the utility of the method the availability of recursive algorithms for constructing the delay operational matrix d makes the method particularly attractive to practise

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

a fractional calculus of variations for multiple integrals with application to vibrating string we introduce a fractional theory of the calculus of variations for multiple integrals our approach uses the recent notions of riemann liouville fractional derivatives and integrals in the sense of jumarie the main results provide fractional versions of the theorems of green and gauss fractional euler lagrange equations and fractional natural boundary conditions as an application we discuss the fractional equation of motion of a vibrating string

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

time discrete higher order ale formulations stability arbitrary lagrangian eulerian ale formulations deal with pdes on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity this arbitrary extension has no effect on the stability of the pde but may influence that of a discrete scheme we examine this critical issue for higher order time stepping without space discretization we propose time discrete discontinuous galerkin dg numerical schemes of any order for a time dependent advection diffusion model problem in moving domains and study their stability properties the analysis hinges on the validity of the reynolds identity for dg exploiting the variational structure and assuming exact integration we prove that our conservative and nonconservative dg schemes are equivalent and unconditionally stable the same results remain true for piecewise polynomial ale maps of any degree and suitable quadrature that guarantees the validity of the reynolds identity this approach general

brief paper stability and resonance conditions of elementary fractional transfer functions elementary fractional transfer functions are studied in this paper some basic properties of elementary transfer functions of the first kind are recalled then two main results are presented regarding elementary fractional transfer functions of the second kind written in a canonical form and characterized by a commensurate order a pseudo damping factor and a natural frequency first stability conditions are established in terms of the pseudo damping factor and the commensurate order as a corollary to matignon s stability theorem they extend the previous result into conditions that are simpler to check then resonance conditions are established numerically in terms of the commensurate order and the pseudo damping factor and give interesting information on the frequency behavior of fractional systems it is shown that elementary transfer functions of the second kind might have up to two resonant frequencies moreover three abaci are given allowing to determine the pseudo damping factor and the commensurate order for respectively a desired normalized gain at each resonance a desired phase at each resonance and a desired normalized first or second resonant frequency

relationships between power law long range interactions and fractional mechanics abstract we investigate the relationships between models of power law long range interactions and mechanics based on fractional derivatives we present the fractional lagrangian density which gives the euler lagrange equation that serves as the equation of motion for fractional power law long range interactions we derive this equation by the fractional variational method in addition we derive a noether like current from the fractional lagrangian density

multiobjective fractional variational calculus in terms of a combined caputo derivative abstract the study of fractional variational problems in terms of a combined fractional caputo derivative is introduced necessary optimality conditions of euler lagrange type for the basic isoperimetric and lagrange variational problems are proved as well as transversality and sufficient optimality conditions this allows to obtain necessary and sufficient pareto optimality conditions for multiobjective fractional variational problems

higher dimensional mixed fractional rotation groups as a basis for dynamic symmetries generating the spectrum of the deformed nilsson oscillator abstract based on the riemann and caputo definitions of the fractional derivative we use the fractional extensions of the standard rotation group s o 3 to construct a higher dimensional representation of a fractional rotation group with mixed derivative types an extended symmetric rotor model is derived which predicts the sequence of magic proton and neutron numbers accurately the ground state properties of nuclei are correctly reproduced within the framework of this model

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

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

on a jump type stochastic fractional partial differential equation with fractional noises abstract we study a class of stochastic fractional partial differential equations of order 1 driven by a pure jump levy space time white noise and a fractional noise we prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions

on a stochastic fractional partial differential equation driven by a levy space time white noise we study the existence and uniqueness of the global mild solution for a stochastic fractional partial differential equation driven by a levy space time white noise moreover the flow property for the solution is also studied

fractional hamiltonian analysis of higher order derivatives systems the fractional hamiltonian analysis of 1 1 dimensional field theory is investigated and the fractional ostrogradski s formulation is obtained the fractional path integral of both simple harmonic oscillator with an acceleration squares part and a damped oscillator are analyzed the classical results are obtained when fractional derivatives are replaced with the integer order derivatives

lattice fractional calculus integration and differentiation of non integer orders for n dimensional physical lattices with long range particle interactions are suggested the proposed lattice fractional derivatives and integrals are represented by kernels of lattice long range interactions such that their fourier series transformations have a power law form with respect to components of wave vector continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non integer orders with respect to coordinates lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics

numerical solution of the higher order linear fredholm integro differential difference equation with variable coefficients the main aim of this paper is to apply the legendre polynomials for the solution of the linear fredholm integro differential difference equation of high order this equation is usually difficult to solve analytically our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted legendre polynomials with unknown coefficients the operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted legendre polynomials illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results

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

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

time discrete higher order ale formulations a priori error analysis we derive optimal a priori error estimates for discontinuous galerkin dg time discrete schemes of any order applied to an advection diffusion model defined on moving domains and written in the arbitrary lagrangian eulerian ale framework our estimates hold without any restrictions on the time steps for dg with exact integration or reynolds quadrature they involve a mild restriction on the time steps for the practical runge kutta radau methods of any order the key ingredients are the stability results shown earlier in bonito et al time discrete higher order ale formulations stability 2013 along with a novel ale projection numerical experiments illustrate and complement our theoretical results

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

nonholonomic constraints with fractional derivatives we consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples the corresponding equations of motion are derived using variational principle we prove that fractional constraints can be used to describe the evolution of dynamical systems in which some coordinates and velocities are related to velocities through a power law memory function

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

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

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

psi series solution of fractional ginzburg landau equation one dimensional ginzburg landau equations with derivatives of noninteger order are considered using psi series with fractional powers the solution of the fractional ginzburg landau fgl equation is derived the leading order behaviours of solutions about an arbitrary singularity as well as their resonance structures have been obtained it was proved that fractional equations of order with polynomial nonlinearity of order s have the noninteger power like behaviour of order 1 s near the singularity

high order finite element methods for moving boundary problems with prescribed boundary evolution abstract we introduce a framework for the design of finite element methods for two dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy both in space and in time at the core of our approach is the use of a universal mesh a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary the resulting method maintains an exact representation of the prescribed moving boundary at the discrete level or an approximation of the appropriate order yet is immune to large distortions of the mesh under large deformations of the domain the framework is general making it possible to achieve any desired order of accuracy in space and time by selecting a preferred and suitable finite element space on the universal mesh for the problem at hand and a preferred and suitable time integrator for ordinary differential equations we illustrate our approach by constructing a particular class of methods and apply them to a prescribed boundary variant of the stefan problem we present numerical evidence for the order of accuracy of our schemes in one and two dimensions

a general finite element formulation for fractional variational problems abstract this paper presents a general finite element formulation for a class of fractional variational problems fvps the fractional derivative is defined in the riemann liouville sense for fvps the euler lagrange and the transversality conditions are developed in the fractional finite element formulation ffef presented here the domain of the equations is divided into several elements and the functional is approximated in terms of nodal variables minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme three examples are considered to show the performance of the algorithm results show that as the number of discretization is increased the numerical solutions approach the analytical solutions and as the order of the derivative approaches an integer value the solution for the integer order system is recovered for unspecified boundary conditions the numerical solutions satisfy the transversality conditions this indicates that for the class of problems considered the numerical solutions can be obtained directly from the functional and there is no need to solve the fractional euler lagrange equations thus the formulation extends the traditional finite element approach to fvps

a direct approach using the shifted legendre series expansion for near optimum control of linear time varying systems with multiple state and control delays a computationally efficient method which uses finite orthogonal expansions to approximate state and control variables is developed to obtain suboptimal control for time varying systems with multiple state and control delays and with quadratic cost shifted legendre polynomials form the basis of expansions as a consequence of their useful properties the galerkin method is used to reduce the problem to one of minimizing a quadratic form with linear algebraic constraints the solution of a coupled two point boundary value problem with both delayed and advanced terms which is always required in applying the pontryagin s maximum principle to the optimization of delay systems is thus avoided

analysis of a stabilized finite element approximation of the transient convection diffusion equation using an ale framework in this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an arbitrary lagrangian eulerian ale framework as basic numerical strategy we discretize the equation in time using first and second order backward differencing bdf schemes whereas space is discretized using a stabilized finite element method the orthogonal subgrid scale formulation to deal with convection dominated flows the semidiscrete problem continuous in space is first analyzed in this situation it is easy to identify the error introduced by the ale approach after that the fully discrete method is considered we obtain optimal error estimates in both space and time in a mesh dependent norm the analysis reveals that the ale approach introduces an upper bound for the time step size for the results to hold the results obtained for the fully discretized second order scheme in time are associated to a weaker norm than the one used for the first order method nevertheless optimal convergence results have been proved for fixed domains we recover stability and convergence results with the strong norm for the second order scheme stressing the aspects that make the analysis of this method much more involved

incomplete normalization of probability on multifractals abstract this work is an extension of the incomplete probability theory from the simple case of monofractals previously studied to the more general case of multifractals that can occur in the phase space without equiprobable partition

breathers on lattices with long range interaction we analyze the properties of breathers time periodic spatially localized solutions on chains in the presence of algebraically decaying interactions 1 r s we find that the spatial decay of a breather shows a crossover from exponential short distances to algebraic large distances decay we calculate the crossover distance as a function of s and the energy of the breather next we show that the results on energy thresholds obtained for short range interactions remain valid for s 3 and that for s 3 fractal dispersion at the band edge nonzero thresholds occur for cases where the short range interaction system would yield zero threshold values

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

on the solution of hyperbolic two dimensional fractional systems via discrete variational schemes of high order of accuracy abstract in this work we consider a general class of damped wave equations in two spatial dimensions the model considers the presence of weyl space fractional derivatives as well as a generic nonlinear potential the system has an associated positive energy functional when damping is not present in which case the model is capable of preserving the energy throughout time meanwhile the energy of the system is dissipated in the damped scenario in this work the weyl space fractional derivatives are approximated through second order accurate fractional centered differences a high order compact difference scheme with fourth order accuracy in space and second order in time is proposed some associated discrete quantities are introduced to estimate the energy functional we prove that the numerical method is capable of conserving the discrete variational structure under the same conditions for which the continuous model is conservative the positivity of the discrete energy of the system is also discussed the properties of consistency solvability stability and convergence of the proposed method are rigorously proved we provide some numerical simulations that illustrate the agreement between the physical properties of the continuous and the discrete models

unified analysis of finite element methods for problems with moving boundaries we present a unified analysis of finite element methods for problems with prescribed moving boundaries in particular we study an abstract parabolic problem posed on a moving domain with prescribed evolution discretized in space with a finite element space that is associated with a moving mesh that conforms to the domain at all times the moving mesh is assumed to evolve smoothly in time except perhaps at a finite number of remeshing times where the solution is transferred between finite element spaces via a projection a key result of our analysis is an abstract estimate for the l 2 norm of the error between the exact and semidiscrete solutions at a fixed positive time expressed in terms of the total variation in time of a quantity that measures the difference between the exact solution at time t and its elliptic projection onto the finite element space at time t specializing the abstract estimate to particular choices of the mesh motion strategy finite element space and projector leads to e

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

numerical solution of a class of two dimensional nonlinear volterra integral equations using legendre polynomials in this paper a method for finding an approximate solution of a class of two dimensional nonlinear volterra integral equations is discussed the properties of two dimensional shifted legendre functions are presented the operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations some results concerning the error analysis are obtained we also consider the application of the method to the solution of certain partial differential equations illustrative examples are included to demonstrate the validity and applicability of the technique

stability and geometric conservation laws for ale formulations the aim of this paper is to investigate a model ale scheme with respect to various possible choices of time discretizations for each time scheme we investigate the relationships between stability and the so called geometric conservation laws gcl we shall see that gcl condition proves neither necessary nor sufficient for stability in doing so we review some known theoretical results and we prove some new stability results for space time ale discretizations some new error estimates are also presented several numerical experiments confirm the theory

fully discrete spectral methods for solving time fractional nonlinear sine gordon equation with smooth and non smooth solutions abstract we consider the initial boundary value problem of the time fractional nonlinear sine gordon equation and the fractional derivative is described in caputo sense with the order 1

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

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

fokker planck equation with fractional coordinate derivatives abstract using the generalized kolmogorov feller equation with long range interaction we obtain kinetic equations with fractional derivatives with respect to coordinates the method of successive approximations with averaging with respect to a fast variable is used the main assumption is that the correlation function of probability densities of particles to make a step has a power law dependence as a result we obtain a fokker planck equation with fractional coordinate derivative of order 1 2

transfer function matrix identification in mimo systems via shifted legendre polynomials a general algorithm is presented for the identification of the parameters in the transfer function matrix of a multi input multi output mimo system the approach adopted is that of expanding the system input and output variables in shifted legendre series the feasibility of the method lies in the generation of linear algebraic equations in the unknown parameters and initial conditions by means of an elegant operational matrix which relates shifted legendre polynomials to their integrals an example is included to illustrate the applicability of the proposed method

fractional dynamics of systems with long range space interaction and temporal memory abstract field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power law memory function and long range interaction in systems the method is applied to obtain a fractional generalization of the ginzburg landau and nonlinear schrodinger equations as another example dynamical equations for particle chains with power law interaction and memory are considered in the continuous limit the obtained fractional equations can be applied to complex media with without random parameters or 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

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

fractional liouville equation on lattice phase space abstract in this paper we propose a lattice analog of phase space fractional liouville equation the liouville equation for phase space lattice with long range jumps of power law types is suggested we prove that the continuum limit transforms this lattice equation into liouville equation with conjugate riesz fractional derivatives of non integer orders with respect to coordinates of continuum phase space an application of the fractional liouville equation with these riesz fractional derivatives to describe properties of plasma like nonlocal media is considered

controlling bifurcation in a delayed fractional predator prey system with incommensurate orders this paper investigates an issue of bifurcation control for a novel incommensurate fractional order predator prey system with time delay firstly the associated characteristic equation is analyzed by taking time delay as the bifurcation parameter and the conditions of creation for hopf bifurcation are established it is demonstrated that time delay can heavily effect the dynamics of the proposed system and each order has a major influence on the creation of bifurcation simultaneously then a linear delayed feedback controller is introduced to successfully control the hopf bifurcation for such system it is shown that the control effort is markedly influenced by feedback gain it is further found that the onset of the bifurcation can be delayed as feedback gain decreases finally two illustrative examples are exploited to verify the validity of the obtained newly results

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

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

generalized natural boundary conditions for fractional variational problems in terms of the caputo derivative this paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a lagrangian depending on the free end points the fractional derivatives are defined in the sense of caputo

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