generalized chapman kolmogorov equation a unifying approach to the description of anomalous transport in external fields the generalized chapman kolmogorov equation v m kenkre e w montroll and m f shlesinger j stat phys 9 45 1973 is discussed it is demonstrated that this equation unifies recently proposed kinetic equations of fractional order that describe anomalous transport in external fields as well as continuous time random walks the conditions under which the individual models can be established are discussed

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 diffusion waiting time distributions and cattaneo type equations we discuss a generalized diffusion equation resulting from an additive two state process in combination with an asymptotically fractal asymptotic power law waiting time distribution the obtained equation is an extension to previously discussed fractional diffusion equations our description leads to a mean squared displacement which describes enhanced subballistic transport for long times the short time behavior however is of a ballistic nature this separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting time distribution this is also mirrored by the observation that for small times our generalized diffusion equation reduces to the standard cattaneo equation the asymptotic probability density is of compressed gaussian type and thus differs from the l evy tail generally found for these kinds of processes

holderian variational problems subject to integral constraints abstract we prove a necessary stationary condition for non differentiable isoperimetric variational problems with scale derivatives defined on the class of holder continuous functions

fractional variational problems depending on indefinite integrals abstract we obtain necessary optimality conditions for variational problems with a lagrangian depending on a caputo fractional derivative a fractional and an indefinite integral main results give fractional euler lagrange type equations and natural boundary conditions which provide a generalization of the previous results found in the literature isoperimetric problems problems with holonomic constraints and depending on higher order caputo derivatives as well as fractional lagrange problems are considered

noether s theorem of fractional birkhoffian systems abstract in this paper we study noether type symmetry theorem to fractional birkhoffian systems with riemann liouville derivatives this theorem provides an explicit algorithmic way to compute a constant for any birkhoffian systems admitting a symmetry finally we extend our noether s theorem to fractional birkhoffian systems base on caputo or riesz derivatives

levy flights in quenched random force fields l evy flights characterized by the microscopic step index f are for fl2 the case of rare events considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f ensuremath 2 in the short range case and the critical fall off exponent 2f ensuremath 2 in the long range case by means of a dynamic renormalization group analysis based on the momentum shell integration method we determine flows fixed point and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient unlike the case of ordinary brownian motion in a quenched force field characterized by a single critical dimension or fall off exponent d 2 two critical dimensions appear in the l evy case a critical dimension or fall off exponent d f below which the diffusion coefficient exhibits anomalous scaling behavior i e algebraic spatial behavior and long time tails and a critical dimension or fall off exponent d 2f ensuremath 2 below which the force correlations characterized by a nontrivial fixed point become relevant as a general result we find in all cases that the dynamic exponent z characterizing the mean square displacement locks onto the l evy index f independent of dimension and independent of the presence of weak quenched disorder

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

non differentiable variational principles abstract we develop a calculus of variations for functionals which are defined on a set of non differentiable curves we first extend the classical differential calculus in a quantum calculus which allows us to define a complex operator called the scale derivative which is the non differentiable analogue of the classical derivative we then define the notion of extremals for our functionals and obtain a characterization in term of a generalized euler lagrange equation we finally prove that solutions of the schrodinger equation can be obtained as extremals of a non differentiable variational principle leading to an extended hamilton s principle of least action for quantum mechanics we compare this approach with the scale relativity theory of nottale which assumes a fractal structure of space time

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

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

linear response in complex systems ctrw and the fractional fokker planck equations abstract we consider the linear response of systems modelled by continuous time random walks ctrw and by fractional fokker planck equations under the influence of time dependent external fields we calculate the corresponding response functions explicitly the ctrw curve exhibits aging i e it is not translationally invariant in the time domain this is different from what happens under fractional fokker planck conditions

the stochastic wave equation in two spatial dimensions keywords stochastic wave equation gaussian noise process solution reference prob article 1998 001view record in web of science record created on 2008 12 01 modified on 2017 05 12

conservation laws for linear equations on braided linear spaces the properties of linear equations on braided linear spaces are investigated and conservation laws for them are derived the conserved currents are given in the explicit form the procedure is then applied to scalar wave equations on a quantum plane and on q minkowski space

conservation laws for linear equations on quantum minkowski spaces the general linear equations with constant coefficients on quantum minkowski spaces are considered and the explicit formulae for their conserved currents are given the proposed procedure can be simplified for invariant equations the derived method is then applied to klein gordon dirac and wave equations on different classes of minkowski spaces in the examples also symmetry operators for these equations are obtained they include quantum deformations of classical symmetry operators as well as an additional operator connected with deformation of the leibnitz rule in non commutative differential calculus

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

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

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

fractional variational problems from extended exponentially fractional integral in this letter we study the fractional variational problems from extended exponentially fractional integral both the lagrangian and hamiltonian formulations are explored and discussed in some details some interesting consequences are revealed

the euler lagrange and legendre equations for functionals involving distributed order fractional derivatives abstract in this paper we extend some fractional calculus of variations results by considering functionals depending on distributed order fractional derivatives using variational techniques we deduce optimal necessary conditions of euler lagrange and legendre type we also study the case where integral and holonomic constraints are imposed finally a numerical procedure is given to solve some proposed problems

a counterexample to a frederico torres fractional noether type theorem abstract in this note we present a counterexample to theorem 24 in g s f frederico and d f m torres 2010 3

higher order infinite horizon variational problems in discrete quantum calculus we obtain necessary optimality conditions for higher order infinite horizon problems of the calculus of variations via discrete quantum operators

discrete direct methods in the fractional calculus of variations finite differences as a subclass of direct methods in the calculus of variations consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem this article generalizes the same idea for fractional variational problems we consider a minimization problem with a lagrangian that depends on the left riemann liouville fractional derivative using the grunwald letnikov definition we approximate the objective functional in an equispaced grid as a multi variable function of the values of the unknown function on mesh points the problem is then transformed to an ordinary static optimization problem the solution to the latter problem gives an approximation to the original fractional problem on mesh points

fractional variational calculus for nondifferentiable functions we prove necessary optimality conditions in the class of continuous functions for variational problems defined with jumarie s modified riemann liouville derivative the fractional basic problem of the calculus of variations with free boundary conditions is considered as well as problems with isoperimetric and holonomic constraints

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

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

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

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

noether symmetries and conserved quantities for fractional forced birkhoffian systems abstract in this paper noether symmetries and conserved quantities for fractional forced birkhoffian systems are studied firstly a new fractional pfaff birkhoff variational principle with riemann liouville derivatives and generalized force is established and from which the fractional forced birkhoff equations are derived secondly the definitions and the criteria of the invariance of a functional under an e parameter group of infinitesimal transformations are given finally the fractional noether theorems for forced birkhoffian systems are obtained and an illustrative example is presented

stationarity conservation laws for certain linear fractional differential equations the leibniz rule for the fractional riemann liouville derivative is studied in the algebra of functions defined by laplace convolution this algebra and the derived leibniz rule is used in construction of an explicit form of stationary conserved currents for linear fractional differential equations the examples of fractional diffusion in 1 1 and fractional diffusion in d 1 dimensions are discussed in detail the results are generalized to the mixed fractional differential and mixed sequential fractional differential systems for which the stationarity conservation laws are obtained the derived currents are used in construction of stationary nonlocal charges

leitmann s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales abstract the fundamental problem of the calculus of variations on time scales concerns the minimization of a delta integral over all trajectories satisfying given boundary conditions this includes the discrete time the quantum and the continuous classical calculus of variations as particular cases in this note we follow leitmann s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale

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

random non linear wave equations smoothness of the solutions we show existence and uniqueness for the solution of a onedimensional wave equation with non linear random forcing then we give sufficient conditions for the solution at a given time and a given point to have a density and for this density to be smooth the proof uses the extension of the malliavin calculus to the two parameters wiener functionals

generalized variational problems and euler lagrange equations this paper introduces three new operators and presents some of their properties it defines a new class of variational problems called generalized variational problems or gvps in terms of these operators and derives euler lagrange equations for this class of problems it is demonstrated that the left and the right fractional riemann liouville integrals and the left and the right fractional riemann liouville caputo riesz riemann liouville and riesz caputo derivatives are special cases of these operators and they are obtained by considering a special kernel further the euler lagrange equations developed for functional defined in terms of the left and the right fractional riemann liouville caputo riesz riemann liouville and riesz caputo derivatives are special cases of the euler lagrange equations developed here examples are considered to demonstrate the applications of the new operators and the new euler lagrange equations the concepts of adjoint differential operators and adjoint differential equations defined in terms of the new operators are introduced a new class of generalized lagrangian hamiltonian and action principles are presented in special cases these formulations lead to fractional adjoint differential operators and adjoint differential equations and fractional lagrangian hamiltonian and action principle thus the new operators introduce a generalized approach to many problems in classical mechanics in general and variational calculus in particular possible extensions of the subject and the concepts discussed here are also outlined

scaling limits for cumulative input processes we study different scaling behavior of very general telecommunications cumulative input processes the activities of a telecommunication system are described by a marked point process tn zn n z where tn is the arrival time of a packet brought to the system or the starting time of the activity of an individual source and the mark zn is the amount of work brought to the system at time tn this model includes the popular on off process and the infinite source poisson model in addition to the latter models one can flexibly model dependence of the interarrival times tn tn 1 clustering behavior due to the arrival of an impulse generating a flow of activities but also dependence between the arrival process tn and the marks zn similarly to the on off and infinite source poisson model we can derive a multitude of scaling limits for the input process of one source or for the superposition of an increasing number of such sources the memory in the input process depends on a variety of factors such as the tails of the interarrival times or the tails of the distribution of activities initiated at an arrival tn or the number of activities starting at tn it turns out that as in standard results on the scaling behavior of cumulative input processes in telecommunications fractional brownian motion or infinite variance levy stable motion can occur in the scaling limit however the fractional brownian motion is a much more robust limit than the stable motion and many other limits may occur as well

generalized euler lagrange equations for fractional variational problems with free boundary conditions in this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems fvps with completely free boundary conditions have an extremum the fractional derivatives are defined in the caputo sense first we present the necessary conditions for the problem with only one dependent variable and then we generalize them to problems with multiple dependent variables we also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables it is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives the new results reduce to the known necessary conditions some examples are presented to demonstrate the applicability of the new formulations

the conservation laws and integrals of motion for a certain class of equations in discrete models we derive the conservation laws for a wide class of discrete equations including the q deformations of classical field theoretic equations the explicit expressions for the conserved currents are used in the construction of the integrals of motion the procedure is applied to the nonlinear difference equation and to the q wave equations in d 3 and d 4

the sharp markov property of levy sheets keywords brownian sheet two parameter levy processes sharp field poisson sheet splitting field reference prob article 1992 002doi 10 1214 aop 1176989793 record created on 2008 12 01 modified on 2017 05 12

fractional variational calculus with classical and combined caputo derivatives abstract we give a proper fractional extension of the classical calculus of variations by considering variational functionals with a lagrangian depending on a combined caputo fractional derivative and the classical derivative euler lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

higher order hahn s quantum variational calculus abstract we prove a necessary optimality condition of euler lagrange type for quantum variational problems involving hahn s derivatives of higher order

lattice fractional diffusion equation in terms of a riesz caputo difference abstract a fractional difference is defined by the use of the right and the left caputo fractional differences the definition is a two sided operator of riesz type and introduces back and forward memory effects in space difference then a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains a lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders the result shows that the riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

heavy tails and long range dependence in on off processes and associated fluid models on off models are common inputs for a variety of communication network models as well as storage and inventory models a stationary renewal process alternating periods of activity active transmission fluid buildup etc with periods of inactivity silence no transmission inputs off etc induces a stationary indicator process which indicates if the system is active or not heavy tails for the on periods induces a covariance function for the indicator process which decreases slowly at a rate characteristic of long range dependence this has dramatic consequences for fluid models where fluid flows in at constant rate and there is a constant rate of release

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

generalized fractional calculus with applications to the calculus of variations we study operators that are generalizations of the classical riemann liouville fractional integral and of the riemann liouville and caputo fractional derivatives a useful formula relating the generalized fractional derivatives is proved as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual such results are explored in the context of dynamic optimization by considering problems of the calculus of variations with general fractional operators necessary optimality conditions of euler lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of riemann liouville or caputo as an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem illustrative optimization problems are considered

direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method this paper presents an adaptive legendre gauss pseudospectral method for solving a type of constrained fractional variational problems fvps the fractional derivative is defined in the caputo sense in the presented method by dividing the domain of the problem into a uniform mesh the given fvp reduces to a nonlinear mathematical programming problem and there is no need to solve the complicated fractional euler lagrange equations the method developed in this paper adjusts both the mesh spacing and the number of collocation points on each subinterval in order to improve the accuracy the method is easy to implement and yields very accurate results some error estimates and convergence properties of the method are discussed numerical examples are included to confirm the efficiency and convergence of the proposed method

the sharp markov property of the brownian sheet and related processes keywords brownian sheet sharp markov property processes with independent planar increments reference prob article 1992 001doi 10 1007 bf02392978 record created on 2008 12 01 modified on 2017 05 12

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

integrals involving complete elliptic integrals we give a closed form evaluation of a number of erdelyi kober fractional integrals involving elliptic integrals of the rst and second kind in terms of the 3f2 generalized hypergeometric function reduction formulae for 3f2 enable us to simplify the solutions for a number of particular cases c 1999 elsevier science b v all rights reserved

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

scale calculus and the schrodinger equation this paper is twofold in a first part we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus the scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f t e such that f t e f t when e goes to zero this led to several new notions as representations fractal functions and e differentiability the basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative we then discuss some algebraic properties of these operators we define a natural bialgebra called quantum bialgebra associated with them finally we discuss a convenient geometric object associated with our study in a second part we define a first quantization procedure of classical mechanics following the scale relativity theory developed by nottale we obtain a nonlinear schrodinger equation v

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

solution of the fractional langevin equation and the mittag leffler functions we introduce the fractional generalized langevin equation in the absence of a deterministic field with two deterministic conditions for a particle with unitary mass i e an initial condition and an initial velocity are considered for a particular correlation function that characterizes the physical process and using the methodology of the laplace transform we obtain the solution in terms of the three parameter mittag leffler function as particular cases some recent results are also presented

the b spectral sequence lagrangian formalism and conservation laws ii the nonlinear theory notations et faits provenant du calcul differentiel dans des algebres commutatives operateurs adjoints complexes de spencer et formule de green lagrangiens quadratiques et l operateur d euler lois de conservation en theorie lineaire automorphismes et le theoreme de n ether lineaire

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

random nonlinear wave equations propagation of singularities on etudie les proprietes de lissite des solutions d equations d ondes unidimensionnelles avec forcing aleatoire non lineaire on definit les singularites comme anomalies dans le modulo local de continuite des solutions

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

on the existence of optimal solutions to fractional optimal control problems abstract in the paper we study control systems containing a fractional riemann liouville derivative the main result is a theorem on the existence of optimal solutions for such problems with a nonlinear integral performance index

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

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

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces the conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived for discrete models the conserved charges are constructed explicitly the applications of the general method include equations on quantum plane supersymmetric equations for chiral and antichiral supermultiplets auxiliary equations of integrable models which means various cases of nonlinear toda lattice equations and anomalous diffusion equation

on generalized renewal measures and certain first passage times by relating it to the family x t infln 1 sn t t 0 one of the major results is that up t oo for all t e r iff 0 t u0 t 1 a 1 ua i1as t a oo iff e xj 2 x1 oo provided 4 is ultimately increasing a 2 0 a related result is proved for u t h u t and up t e n 2 o4 n p mn t where mn max0 i n s our results form extensions of earlier ones by heyde kalma gut and others who either considered more specific functions 4 or used stronger moment assumptions the proofs are based on a regeneration technique from renewal theory and two martingale inequalities by burkholder davis and gundy

stochastic embedding of dynamical systems most physical systems are modeled by an ordinary or a partial differential equation like the n body problem in celestial mechanics in some cases for example when studying the long term behavior of the solar system or for complex systems there exist elements which can influence the dynamics of the system which are not well modeled or even known one way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic in this paper we develop a theory for the stochastic embedding of ordinary differential equations we apply this method to lagrangian systems in this particular case we extend many results of classical mechanics namely the least action principle the euler lagrange equations and noether s theorem we also obtain a hamiltonian formulation for our stochastic lagrangian systems many applications are discussed at the end of the paper

explicit solutions of fractional integral and differential equations involving erdelyi kober operators by means of fractional calculus techniques we find explicit solutions of volterra integral equations of second kind and fractional differential equations involving erdelyi kober fractional integrals or derivatives we use the transmutation method to reduce the solutions of these equations to known solutions of simpler riemann liouville equations of the same type some examples are given

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function