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

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

a convergent numerical scheme for integrodifferential kinetic models of angiogenesis abstract we study a robust finite difference scheme for integrodifferential kinetic systems of fokker planck type modeling tumor driven blood vessel growth the scheme is of order one and enjoys positivity features we analyze stability and convergence properties and show that soliton like asymptotic solutions are correctly captured we also find good agreement with the solution of the original stochastic model from which the deterministic kinetic equations are derived working with ensemble averages a numerical study clarifies the influence of velocity cut offs on the solutions for exponentially decaying data

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

dynamic least squares kernel density modeling of fokker planck equations with application to neural population the dynamic least squares kernel density lsqkd model c pantano and b shotorban phys rev e 76 066705 2007 is used to solve the fokker planck equations in this model the probability density function pdf is approximated by a linear combination of basis functions with unknown parameters whose governing equations are determined by a global least squares approximation of the pdf in the phase space in this work basis functions are set to be gaussian for which the mean variance and covariances are governed by a set of partial differential equations pdes or ordinary differential equations odes depending on what phase space variables are approximated by gaussian functions three sample problems of univariate double well potential bivariate bistable neurodynamical system g deco and d mart i phys rev e 75 031913 2007 and bivariate brownian particles in a nonuniform gas are studied the lsqkd is verified for these problems as its results are compared against the results of the method of characteristics in nondiffusive cases and the stochastic particle method in diffusive cases for the double well potential problem it is observed that for low to moderate diffusivity the dynamic lsqkd well predicts the stationary pdf for which there is an exact solution a similar observation is made for the bistable neurodynamical system in both these problems least squares approximation is made on all phase space variables resulting in a set of odes with time as the independent variable for the gaussian function parameters in the problem of brownian particles in a nonuniform gas this approximation is made only for the particle velocity variable leading to a set of pdes with time and particle position as independent variables solving these pdes a very good performance by lsqkd is observed for a wide range of diffusivities

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

simulation of the solution of a viscous porous medium equation by a particle method to solve a particular partial differential equation namely a viscous porous medium equation we discuss a particle method which is based on the concept of moderately interacting many particle systems our approach may be classified as a combination of a smoothed particle hydrodynamics method and a particle mesh method quantitatively it is assessed in terms of estimates on the approximation error

accuracy of the random vortex method for a problem with non smooth initial conditions abstract the accuracy of the random vortex method in approximating the solution of the navier stokes equation is investigated using the model problem of a circular vortex as suggested by milinazzo and saffman 13 the method consists of partitioning the vorticity into vortex blobs these blobs are moved via two actions first a blob is deterministically moved under the action of the velocity field associated with the other blobs then to simulate viscosity a random component is added to the position of the blob for this model problem the nonlinear terms of the navier stokes equation vanish thus the major error inherent in the deterministic component of the method vanishes consequently for this model problem concentration is on the interaction of the deterministic and random components of the method results show that the accuracy of the method depends heavily on the initial distribution and strength of the computational elements i e the vortex blobs with the right choice of initial conditions it is found that e1 t l t a t a t is o r 1 2 n 1 2 where l t and a t are respectively the exact and computed angular moment of vorticity distribution at time t for n vortex blobs at a reynolds number r

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

convergence of a semi lagrangian scheme for the one dimensional vlasov poisson system a semi lagrangian scheme is proposed for solving the periodic one dimensional vlasov poisson system in phase space on unstructured meshes the distribution function f t x v and the electric field e t x are shown to converge to the exact solution values in the l infty norm the rate of convergence is in o h4 3

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

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

the convergence theory of particle in cell methods for multidimensional vlasov poisson systems in this work a convergence theory of particle in cell methods for multidimensional vlasov poisson systems is provided such methods replace the continuous distribution of masses or charges by disc

higher order time splitting for the linear vlasov equation many semi lagrangian schemes for solving the vlasov equation use a second order time splitting method proposed by cheng and knorr in this paper a fourth order time splitting method is proposed this is combined with cubic spline interpolation in x and v and a rigorous ell 2 error bound is obtained for the linear vlasov equation in arbitrary space dimension also this method and that of cheng and knorr are implemented and compared for a test case in one space dimension

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

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

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

hydrodynamics in two dimensions and vortex theory we consider the navier stokes equation for a viscous and incompressible fluid inr2 we show that such an equation may be interpreted as a mean field equation vlasov like limit for a system of particles called vortices interacting via a logarithmic potential on which in addition a stochastic perturbation is acting more precisely we prove that the solutions of the navier stokes equation may be approximated in a suitable way by finite dimensional diffusion processes with the diffusion constant related to the viscosity as a particular case when the diffusion constant is zero the finite dimensional theory reduces to the usual deterministic vortex theory and the limiting equation reduces to the euler equation

convergence of a difference scheme for the vlasov poisson fokker planck system in one dimension a finite difference scheme is proposed for the periodic one dimensional vlasov poisson fokker planck system which is related to the scheme in c cheng and g knorr em j comp phys 22 1976 pp 330 351 the density f left t x v right and the electric field e left t x right are shown to converge to the exact solution values in l 2 and l infty respectively the rate of convergence is essentially first order this scheme has been implemented in j schaeffer center for nonlinear analysis research report 97 na 004 carnegie mellon university 1997 and compares favorably with a random particle method

convergence of a high order semi lagrangian scheme with propagation of gradients for the one dimensional vlasov poisson system in this paper we give a proof of convergence of a new numerical method introduced in n besse and e sonnendrucker j comput phys 191 2003 pp 341 376 for the vlasov equation the numerical method is based on the semi lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high order and stable reconstruction these kinds of new schemes have been successfully implemented on unstructured meshes of four dimensional phase space cf n besse etude mathematique et numerique de l equation de vlasov sur des maillages non structures de l espace des phases these de l universite louis pasteur strasbourg france 2003 n besse j segre and e sonnendrucker transport theory statist phys 34 2005 pp 311 332 in order to make a rigorous proof of convergence of this method and simplify the convergence analysis we have considered the periodic one dimensional vlasov poisson system in phase space on a grid the distribution f t x v and the electric field are shown to converge to the exact solution values in h 1 norm the rate of convergence is of mathcal o delta t 2 frac delta x 4 alpha delta t frac delta v 4 alpha delta t alpha in n 2 alpha leq 1

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

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

random vortex methods for the navier stokes equations two random vortex methods of runge kutta type are presented for solving the two dimensional navier stokes equations we intesgitate the accuracy of these methods by considering the model problem of a rotating flow with initial vorticity concentrated uniformly on a disk of finite radius functionals of the numerical solution are computed by monte carlo estimates with efficient variance reduction and the results are compared to those obtained from euler s method the numerical results show that both the methods produce errors smaller by one power of the time step size than euler s method one seemingly even better than the other these runge kutta methods are derivations of similar schemes proposed by us in an earlier time for solving stochastic differential equations with constant diffusion coefficients copyright 1988 academic press inc

the numerical analysis of random particle methods applied to vlasov poisson fokker planck kinetic equations we devise and study a random particle method for approximating vlasov poisson fokker planck systems such a proposed scheme takes into account the fact that the trajectories of a particle undergoing brownian motion due to collisions with the medium or background particles can be obtained as the solutions of stochastic differential equations i e the langevin equations these equations are the precise analogs of the deterministic hamiltonian system in the collisionless model the particle approximation in particular simulates the action of viscosity by the use of independent wiener processes brownian motions the error analysis generalizes that of ganguly and victory siam j numer anal 26 1989 pp 249 288 by incorporating into the consistency estimates the sampling errors generated by the random motion of the particles these particular errors are the dominant component of the consistency errors and are estimated by applying bennett s inequality for estimating tail probabilities for standard

grid free simulation of diffusion using random walk methods abstract the simulation of diffusion of a continuum field by the random walk displacement of a set of particles is investigated in detail computational particles are used to transport elements of the gradients of the diffusive concentration one dimensional and quasi one dimensional cases are treated for a generalized diffusion variable different types of boundary conditions of the diffusion equation are considered as well as the extensions to a system of coupled diffusion equations the reaction diffusion equations and the convection diffusion set of equations it is shown that by using concentration gradients in the random walk process statistical errors are reduced and each realization of the numerical solution is a representation of the exact solution moreover transport of higher order derivatives can be utilized to improve the smoothness the algorithm is grid free and the computational elements move to follow the gradients thus it is self adaptive and uniform resolution is attained for all times the method is particularly suitable for the simulation of diffusion in systems which involve more than one transport mechanism when the diffusivity is small and when lagrangian elements are used to model the other mechanisms

a splitting fourier pseudospectral method for vlasov poisson fokker planck system abstract in this paper we propose a splitting fourier pseudospectral method for vlasov poisson fokker planck vpfp system which describes the motion of charged particles in plasma the numerical integration for the system is performed by employing the splitting method in time fourier galerkin method in space direction and fourier collocation method in phase direction respectively the algorithm has spectral accuracy in both space and phase directions and can be implemented efficiently with the fast fourier transform and technique of diagonalization respectively extensive numerical results in one dimensional phase space or 1 x 1 v from the proposed numerical method are shown and have proven the good agreement with the theory and previous studies numerical algorithm for the vpfp system in two dimensional phase space or 2 x 2 v has been summarized in appendix

numerical solution of stochastic differential equations with constant diffusion coefficients we present runge kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients we analyze l sub 2 convergence of these methods and present convergence proofs for scalar equations a second order method is derived and for systems a method of order one and one half is derived we further consider a variance reduction technique based on hermite expansions for evaluating expectations of functions of sample solutions numerical examples in two dimensions are presented

a deterministic particle method for the vlasov fokker planck equation in one dimension the vlasov fokker planck equation is a model for a collisional electrostatic plasma the approximation of this equation in one spatial dimension is studied the equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function the approximation method applied is the deterministic particle method described in wollman and ozizmir numerical approximation of the vlasov poisson fokker planck system in one dimension j comput phys 202 2005 602 644 for the present linear problem an analysis of the stability and convergence of the numerical method is carried out in addition computations are done that verify the convergence of the numerical solution it is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in bouchut and dolbeault on long time asymptotics of the vlasov fokker planck equation and of the vlasov poisson fokker planck system with coulombic and newtonian potentials differential integral equations 8 3 1995 487 514

the convergence analysis of fully discretized particle methods for solving vlasov poisson systems error estimates for fully discretized particle or simulation methods for multidimensional vlasov poisson systems are provided the treatment requires stability results which generalize to the vlasov poisson context those of j t beale and a majda math comp 39 1982 pp 29 52 and those of c anderson and c greengard siam j numer anal 22 1985 pp 413 440 for studying the convergence of vortex methods with these stability results the total error at each timestep can be studied this approach produces estimates better than those which would be obtained by considering the errors in the semidiscrete approximation siam j numer anal 26 1989 pp 249 288 separate from the errors incurred in the time discretization of the mollified hamiltonian trajectory equations

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

existence and uniqueness of a global smooth solution for the vlasov poisson fokker planck system in three dimensions abstract studying precisely the regularity of the force field created by the solution of the linear vlasov fokker planck equation we prove existence and uniqueness of a smooth solution to the vlasov poisson fokker planck system in three dimensions the attractive and repulsive cases are treated the method involves an analysis of the effect of the fokker planck kernel without any force field on non linear expressions coming from the vlasov term

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

concentrations in the one dimensional vlasov poisson equations ii screening and the necessity for measure valued solutions in the two component case abstract weak and measure valued solutions for the two component vlasov poisson equations in a single space dimension are proposed and studied here as a simpler analogue problem for the limiting behavior of approximations for the two dimensional euler equations with general vorticity of two signs from numerical experiments and mathematical theory it is known that much more complex behavior can occur in limiting processes for vortex sheets with general vorticity of two signs as compared with non negative vorticity here such behavior is confirmed rigorously for the simpler analogue problem through explicit examples involving singular charge concentration for the two component vlasov poisson equations the concepts of measure valued and weak solution are introduced explicit examples with charge concentration establish that the limit of weak solutions in a dynamic process is necessarily a measure valued solution in some cases rather than the anticipated weak solution i e no concentration cancellation occurs the limiting behavior of computational regularizations involving high resolution particle methods is presented here both for the instances with measure valued solutions and also for new examples with non unique weak solutions the authors demonstrate that different computational regularizations can exhibit completely different limiting behavior in situations with measure valued and or non unique weak solutions

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

numerical approximation of the vlasov poisson fokker planck system in two dimensions a numerical method is developed for approximating the solution to the vlasov poisson fokker planck system in two spatial dimensions the method generalizes the approximation for the system in one dimension given in s wollman e ozizmir numerical approximation of the vlasov poisson fokker planck system in one dimension j comput phys 202 2005 602 644 the numerical procedure is based on a change of variables that puts the convection diffusion equation into a form so that finite difference methods for parabolic type partial differential equations can be applied the computational cycle combines a type of deterministic particle method with a periodic interpolation of the solution along particle trajectories onto a fixed grid computational work is done to demonstrate the accuracy and effectiveness of the approximation method parts of the numerical procedure are adapted to run on a parallel computer