variational formulations of differential equations and asymmetric fractional embedding abstract variational formulations for classical dissipative equations namely friction and diffusion equations are given by means of fractional derivatives in this way the solutions of those equations are exactly the extremal of some fractional lagrangian actions the formalism used is a generalization of the fractional embedding developed by cresson fractional embedding of differential operators and lagrangian systems j math phys 48 2007 033504 where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives moreover this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle

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

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

lower bounds for the minimum eigenvalue of hadamard product of an m matrix and its inverse for the hadamard product a a 1 of an m matrix a and its inverse a 1 we give new lower bounds for the minimum eigenvalue of a a 1 these bounds are strong enough to prove the conjecture of fiedler and markham an inequality for the hadamard product of an m matrix and inverse m matrix linear algebra appl 101 1988 1 8

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

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

time scale differential integral and variational embeddings of lagrangian systems we introduce differential integral and variational delta embeddings we prove that the integral delta embedding of the euler lagrange equations and the variational delta embedding coincide on an arbitrary time scale in particular a new coherent embedding for the discrete calculus of variations that is compatible with the least action principle is obtained

mixed convolved action for classical and fractional derivative dissipative dynamical systems the principle of mixed convolved action provides a new rigorous weak variational formalism for a broad range of initial value problems in mathematical physics and mechanics here the focus is initially on classical single degree of freedom oscillators incorporating either kelvin voigt or maxwell dissipative elements and then subsequently on systems that utilize fractional derivative constitutive models in each case an appropriate mixed convolved action is formulated and a corresponding weak form is discretized in time using temporal shape functions to produce an algorithm suitable for numerical solution several examples are considered to validate the mixed convolved action principles and to investigate the performance of the numerical algorithms for undamped systems the algorithm is found to be symplectic and unconditionally stable with respect to the time step in the case of dissipative systems the approach is shown to be robust and to be accurate with good convergence characteristics for both classical and fractional derivative based models as part of the derivations some interesting results in the calculus of caputo fractional derivatives also are presented

new lower bounds on eigenvalue of the hadamard product of an m matrix and its inverse abstract some new lower bounds for the minimum eigenvalue of the hadamard product of an m matrix and its inverse are given these bounds improve the results of h b li t z huang s q shen h li lower bounds for the minimum eigenvalue of hadamard product of an m matrix and its inverse linear algebra appl 420 2007 235 247

stochastic perturbation theory in this paper classical matrix perturbation theory is approached from a probabilistic point of view the perturbed quantity is approximated by a first order perturbation expansion in which the perturbation is assumed to be random this permits the computation of statistics estimating the variation in the perturbed quantity up to the higher order terms that are ignored in the expansion these statistics tend to be more realistic than perturbation bounds obtained in terms of norms the technique is applied to a number of problems in matrix perturbation theory including least squares and the eigenvalue problem

numerical solution of fractional oscillator equation abstract we focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval this type of equation includes a complex form of left and right sided fractional derivatives its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations here we elaborated two numerical schemes being dependent on a fractional order of the equation the results of numerical calculations are compared with analytical solutions then we illustrate convergence and stability of our schemes

nilpotency in automorphic loops of prime power order abstract a loop is automorphic if its inner mappings are automorphisms using so called associated operations we show that every commutative automorphic loop of odd prime power order is centrally nilpotent starting with suitable elements of an anisotropic plane in the vector space of 2 2 matrices over the field of prime order p we construct a family of automorphic loops of order p 3 with trivial center

a paradigm for jointed hamiltonian and dissipative systems a paradigm for describing dynamical systems that have both hamiltonian and dissipative parts is presented features of generalized hamiltonian systems and metric systems are combined to produce what are called metriplectic systems the phase space for metriplectic systems is equipped with a bracket operator that has an antisymmetric poisson bracket part and a symmetric dissipative part flows are obtained by means of this bracket together with a quantity called the generalized free energy which is composed of an energy and a generalized entropy the generalized entropy is some function of the casimir invariants of the poisson bracket two examples are considered 1 a relaxing free rigid body and 2 a plasma collision operator that can be tailored so that the equilibrium state is an arbitrary monotonic function of the energy

spectral and discontinuous spectral element methods for fractional delay equations we first develop a spectrally accurate petrov galerkin spectral method for fractional delay differential equations fddes this scheme is developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently presented in m zayernouri and g e karniadakis j comput phys 252 2013 pp 495 517 specifically we obtain solutions to fddes in terms of new nonpolynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of the first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of the second kind fslp ii we prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the pg spectral method in contrast to standard nondelay fractional differential equations the delay character of fddes might induce solutions which are either nonsmooth or piecewise smooth in order to effectively trea

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

multiplication groups of commutative automorphic p loops of odd order are p groups abstract an automorphic loop or a loop is a loop whose inner mappings are automorphisms it was recently proved see p jedlicka m k kinyon p vojt chovsk 2012 8 a finite commutative a loop of order a power of an odd prime is centrally nilpotent in this paper we prove without using the nilpotence that the multiplication group of a finite commutative a loop of order a power of an odd prime is a p group which implies the central nilpotence of such loop too

on analogues of exponential functions for antisymmetric fractional derivatives an equation with the antisymmetric fractional derivative of order email protected 1 2 containing the t b potential is solved using the mellin transform method the solutions are analogues of exponential functions of a new type they are represented as meijer g function series in a finite time interval in the classical limit email protected 1 the eigenfunction equation for a derivative of the first order and its solution an exponential function are recovered then an analogy between the derivation of euler lagrange equations in fractional mechanics and in classical mechanics is discussed the results are applied to a simple fractional euler lagrange equation containing an antisymmetric fractional derivative and its general solution is obtained

approximation of solutions to fractional integral equation in this paper we shall study a fractional integral equation in an arbitrary banach space x we used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem we also prove the existence of global solution the existence and convergence of the faedo galerkin solution to the given problem is also proved in a separable hilbert space with some additional assumptions on the operator a finally we give an example to illustrate the applications of the abstract results

fractional differential equations and the schrodinger equation in a previous paper we defined following a previous work of kolvankar and gangal a notion of a derivative 0 a 1 in this article we study a differential equations associated to our fractional calculus we then discuss a fundamental problem concerning the schrodinger equation in the framework of nottale s scale relativity theory

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

a spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion wave equations in this paper an efficient and accurate spectral numerical method is presented for solving second fourth order fractional diffusion wave equations and fractional wave equations with damping the proposed method is based on jacobi tau spectral procedure together with the jacobi operational matrix for fractional integrals described in the riemann liouville sense the main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought for spectral approximations the validity and effectiveness of the method are demonstrated by solving five numerical examples numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier an efficient numerical scheme for time fractional diffusion wave equations is proposed a time space jacobi tau approximation is developed for such equations several tau spectral methods can be achieved as special cases the validity and applicability of the proposed method are demonstrated accurate numerical results are obtained by selecting limited collocation nodes

the fractional quadratic form identity and hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy starting from a general isospectral problem of fractional order we propose a fractional quadratic form identity from which the hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy is derived the method can be generalized to other fractional integrable couplings

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

mixed convolved action a stationary principle is developed for dynamical systems by formulating the concept of mixed convolved action which is written in terms of displacement and force variables using temporal convolutions and fractional derivatives the classical canonical single degree of freedom dynamical system is considered as an initial application with this new formulation a single real scalar functional provides the governing differential equations along with all the pertinent initial conditions as the euler lagrange equations emanating from the stationarity of the mixed convolved action both conservative and non conservative processes can be considered within a common framework thus resolving a long standing limitation of variational approaches for dynamical systems several results in fractional calculus also are developed

inverse m matrices ii this is an update of the 1981 survey by the first author in the meantime a considerable amount has been learned about the very special structure of the important class of inverse m matrices developments since the earlier survey are emphasized but we have tried to be somewhat complete and some results have not previously been published some proofs are given where appropriate and references are given for others after some elementary preliminaries results are grouped by certain natural categories

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

lie s correspondence for commutative automorphic formal loops abstract we develop lie s correspondence for commutative automorphic formal loops which are natural candidates for non associative abelian groups to show how linearization techniques based on hopf algebras can be applied to study non linear structures over fields of characteristic zero we prove that the category of commutative automorphic formal loops is equivalent to certain category of lie triple systems an explicit baker campbell hausdorff formula for these loops is also obtained with the help of formal power series with coefficients in the algebra of 3 by 3 matrices our formula is strongly related to the function e 2 s e 2 t s t 2 e 2 s t 1

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

a constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation abstract this paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations an algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed several illustrative examples with linear and nonlinear fractional differential equations are given to demonstrate the effectiveness of the method

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

the structure of commutative automorphic loops an automorphic loop or a loop is a loop whose inner mappings are automorphisms every element of a commutative a loop generates a group and xy 1 x 1 y 1 holds let q be a finite commutative a loop and p a prime the loop q has order a power of p if and only if every element of q has order a power of p the loop q decomposes as a direct product of a loop of odd order and a loop of order a power of 2 if q is of odd order it is solvable if a is a subloop of q then a divides q if p divides q then q contains an element of order p if there is a finite simple nonassociative commutative a loop it is of exponent 2

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

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

non differentiable embedding of lagrangian systems and partial differential equations abstract we develop the non differentiable embedding theory of differential operators and lagrangian systems using a new operator on non differentiable functions we then construct the corresponding calculus of variations and we derive the associated non differentiable euler lagrange equation and apply this formalism to the study of pdes first we extend the characteristics method to the non differentiable case we prove that non differentiable characteristics for the navier stokes equation correspond to extremals of an explicit non differentiable lagrangian system second we prove that the solutions of the schrodinger equation are non differentiable extremals of the newton s lagrangian

an efficient jacobi pseudospectral approximation for nonlinear complex generalized zakharov system in this paper we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized zakharov system gzs subject to initial boundary conditions the jacobi pseudospectral approximation is investigated for spatial approximation of the gzs it possesses the spectral accuracy in space the jacobi gauss lobatto quadrature rule is established to treat the boundary conditions and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable this scheme has the advantage of allowing us to obtain the spectral solution in terms of the jacobi parameters and which therefore means that the algorithm holds a number of collocation methods as special cases finally two illustrative examples are implemented to assess the efficiency and high accuracy of the jacobi pseudo spectral scheme

about non differentiable functions we develop the notion of local fractional derivative introduced by kolvankar and gangal it allows a fine study of the local structure of irregular fractal functions using this tool we extend classical theorems of analysis extrema rolle to non differentiable functions in particular we prove a generalized taylor expansion theorem we introduce a new derivative of real order and discuss its properties

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

proof of a conjecture of fiedler and markham let a be an n n nonsingular m matrix for the hadamard product a a 1 m fiedler and t l markham conjectured in linear algebra appl 10l 1988 1 that q a a 1 2 n where q a a 1 is the smallest eigenvalue in modulus of a a 1 we considered this conjecture in linear algebra appl 288 1999 259 having observed an incorrect proof in linear algebra appl 144 1991 171 and obtained that q a a 1 2 n n 1 n the present paper gives a proof for this conjecture

on some new bounds for eigenvalues of the hadamard product and the fan product of matrices abstract let a and b be nonnegative matrices a new upper bound on the spectral radius a b is obtained meanwhile a new lower bound on the smallest eigenvalue q a b for the fan product and a new lower bound on the minimum eigenvalue q b a 1 for the hadamard product of b and a 1 of two nonsingular m matrices a and b are given these bounds improve some results of li et al 9 and generalize some corresponding results the new estimating formulae are only depending on the entries of matrices a and b therefore they are easy to calculate

convergence analysis of space time jacobi spectral collocation method for solving time fractional schrodinger equations abstract in this paper the space time jacobi spectral collocation method jsc method is used to solve the time fractional nonlinear schr o dinger equations subject to the appropriate initial and boundary conditions at first the considered problem is transformed into the associated system of nonlinear volterra integro partial differential equations pdes with weakly singular kernels by the definition and related properties of fractional derivative and integral operators therefore by collocating the associated system of integro pdes in both of the space and time variables together with approximating the existing integral in the equation using the jacobi gauss type quadrature formula then the problem is reduced to a set of nonlinear algebraic equations we can consider solving the system by some robust iterative solvers in order to support the convergence of the proposed method we provided some numerical examples and calculated their l norm and weighted l2 norm at the end of the article

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

derivation of the postulates of quantum mechanics from the first principles of scale relativity quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have up to now not been derived from first principles in the present work we suggest such a derivation in the framework of the theory of scale relativity after having analyzed the actual status of the various postulates rules and principles that underlie the present axiomatic foundation of quantum mechanics in terms of main postulates secondary rules and derived principles we attempt to provide the reader with an exhaustive view of the matter by both gathering here results which are already available in the literature and deriving new ones which complete the postulate list

stochastic derivatives for fractional diffusions in this paper we introduce some fundamental notions related to the so called stochastic derivatives with respect to a given fieldq in our framework we recall well known results about markov wiener diffusions we then focus mainly on the case where x is a fractional diffusion and where q is the past the future or the present of x we treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of x when x solves a stochastic differential equation driven by a fractional brownian motion with hurst index h 1 2 we give explicit formulas 1 introduction there exist various ways to generalize the notion of differentiation on deterministic functions we may think of fractional derivatives or differentiation in the sense of the theory of distributions in both cases we lose a dynamical or geometric interpretation of tangent vectors velocities e g in the present work we seek to construct derivatives on stochastic processes which conserve a dynamical meaning our goal is motivated by the stochastic embedding of dynamical systems introduced in 2 this procedure aims at comprehending the following question how can we write an equation which contains the dynamical meaning of an initial ordinary differential equation and which extends this dynamical meaning to stochastic processes we refer to 3 for more details unfortunately for most of the stochastic processes used in physical models the limit zt h zt h

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

fractional order evolutionary integral equations the topic of fractional calculus derivatives and integrals of arbitrary orders is enjoying growing interest not only among mathematicians but also among physicists and engineers also the evolutionary integral equations have many applications in applied mathematics in this work we are concerned with a class of evolutionary integral equations of arbitrary fractional orders the existence uniqueness and some other properties of the solutions will be proved also the continuation of the problems will be established

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

an inequality for the hadamard product of an m matrix and an inverse m matrix abstract an estimate from below the smallest eigenvalue q a c of the hadamard product a c of an m matrix a and an inverse m matrix c is proved in particular if a is an m matrix of order n we obtain the inequality q a a 1 1 n

commutator theory for loops abstract using the freese mckenzie commutator theory for congruence modular varieties as the starting point we develop commutator theory for the variety of loops the fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group we specialize the fundamental theorem into several varieties of loops and also discuss the commutator of two normal subloops consequently we argue that some standard definitions of loop theory such as elementwise commutators and associators should be revised and linked more closely to inner mappings using the new definitions we prove several natural properties of loops that could not be so elegantly stated with the standard definitions of loop theory for instance we show that the subloop generated by the new associators defined here is automatically normal we conclude with a preliminary discussion of abelianess and solvability in loops

homogeneous fractional embeddings fractional equations appear in the description of the dynamics of various physical systems for lagrangian systems the embedding theory developed by cresson fractional embedding of differential operators and lagrangian systems j math phys 48 033504 2007 provides a univocal way to obtain such equations stemming from a least action principle however no matter how equations are obtained the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects in this paper we show that it is necessary to introduce an extrinsic constant of time then we use it to construct two equivalent fractional embeddings which retains homogeneity the notion of fractional constant is also discussed through this formalism finally an illustration is given with natural lagrangian systems and the case of the harmonic oscillator is entirely treated

the structure of automorphic loops automorphic loops are loops in which all inner mappings are automorphisms this variety of loops includes for instance groups and commutative moufang loops we study uniquely 2 divisible automorphic loops particularly automorphic loops of odd order from the point of view of the associated bruck loops motivated by glauberman s work on uniquely 2 divisible moufang loops and the associated lie rings motivated by a construction of wright we prove that every automorphic loop q of odd order is solvable contains an element of order p for every prime p dividing q and s divides q for every subloop s of q there are no finite simple nonassociative commutative automorphic loops and there are no finite simple nonassociative automorphic loops of order less than 2500 we show that if q is a finite simple nonassociative automorphic loop then the socle of the multiplication group of q is not regular the existence of a finite simple nonassociative automorphic loop remains open let p be an odd prime automorphic loops of order p or p are groups but there exist nonassociative automorphic loops of order p some with trivial nucleus center and of exponent p we construct nonassociative dihedral automorphic loops of order 2n for every n 2 and show that there are precisely p 2 nonassociative automorphic loops of order 2p all of them dihedral

semigroups and some nonlinear fractional differential equations equations of the formd au t dt a au t f t b 1 t u t b r t u t are considered where 0 0 is a family of linear closed operators defined on dense sets in e into e and f is a given abstract nonlinear function defined on 0 t xe r with values in e t 0 it is assumed that a generates an analytic semigroup under suitable conditions on the family of operators b i t i 1 r t 0 and on f we study the existence and uniqueness of the solution of the cauchy problem for the considered equation some properties concerning the stability of solutions are obtained we also give an application for nonlinear partial differential equations of fractional orders

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

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

multiscale functions scale dynamics and applications to partial differential equations modeling phenomena from experimental data always begins with a choice of hypothesis on the observed dynamics such as determinism randomness and differentiability depending on these choices different behaviors can be observed the natural question associated to the modeling problem is the following with a finite set of data concerning a phenomenon can we recover its underlying nature from this problem we introduce in this paper the definition of multi scale functions scale calculus and scale dynamics based on the time scale calculus see bohner m and peterson a dynamic equations on time scales an introduction with applications springer science business media 2001 which is used to introduce the notion of scale equations these definitions will be illustrated on the multi scale okamoto s functions scale equations are analysed using scale regimes and the notion of asymptotic model for a scale equation under a particular scale regime the introduced formalism explains why a single scale equation can produce distinct continuous models even if the equation is scale invariant typical examples of such equations are given by the scale euler lagrange equation we illustrate our results using the scale newton s equation which gives rise to a non linear diffusion equation or a non linear schrodinger equation as asymptotic continuous models depending on the particular fractional scale regime which is considered

a fully spectral collocation approximation for multi dimensional fractional schrodinger equations a shifted legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one and two dimensional time fractional schrodinger equations tfses subject to initial boundary and non local conditions the first step depends mainly on shifted legendre gauss lobatto collocation sl gl c method for spatial discretization an expansion in a series of shifted legendre polynomials for the approximate solution and its spatial derivatives occurring in the tfse is investigated in addition the legendre gauss lobatto quadrature rule is established to treat the nonlocal conservation conditions thereby the expansion coefficients are then determined by reducing the tfse with its nonlocal conditions to a system of fractional differential equations sfdes for these coefficients the second step is to propose a shifted legendre gauss radau collocation sl gr c scheme for temporal discretization to reduce such system into a system of algebraic equations which is far easier to be solved the proposed collocation scheme both in temporal and spatial discretizations is successfully extended to solve the two dimensional tfse numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms by selecting relatively limited legendre gauss lobatto and gauss radau collocation nodes we are able to get very accurate approximations demonstrating the utility and high accuracy of the new approach over other numerical methods

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

numerical solution of volterra fredholm integral equations using legendre collocation method a numerical method for solving the volterra fredholm integral equations is presented the method is based upon shifted legendre polynomials approximation the properties of shifted legendre polynomials are first presented these properties together with the shifted gauss legendre nodes are then utilized to reduce the volterra fredholm integral equations to the solution of a matrix equation an estimation of the error is given illustrative examples are included to demonstrate the validity and applicability of the technique

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 local fractional derivative abstract we present the necessary conditions for the existence of the kolwankar gangal local fractional derivatives kg lfd and introduce more general but weaker notions of lfds by using limits of certain integral averages of the difference quotient by applying classical results due to stein and zygmund 1965 16 we show that the kg lfd is almost everywhere zero in any given intervals we generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing

a lower bound for the minimum eigenvalue of the hadamard product of matrices abstract suppose both a and b are n n nonsingular m matrices an estimate from below for the smallest eigenvalue a b 1 in modulus of the hadamard product a b 1 of a and b 1 is derived as a special case we obtain the inequality a a 1 2 n n 2

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

fractional spectral collocation method we develop an exponentially accurate fractional spectral collocation method for solving steady state and time dependent fractional pdes fpdes we first introduce a new family of interpolants called fractional lagrange interpolants which satisfy the kronecker delta property at collocation points we perform such a construction following a spectral theory recently developed in m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 for fractional sturm liouville eigenproblems subsequently we obtain the corresponding fractional differentiation matrices and we solve a number of linear fodes in addition to linear and nonlinear fpdes to investigate the numerical performance of the fractional collocation method we first examine space fractional advection diffusion problem and generalized space fractional multiterm fodes next we solve fpdes including the time and space fractional advection diffusion equation time and space fractional multiterm fpdes and finally the space

on a conjecture of fiedler and markham abstract for the hadamard product a a 1 of an m matrix a and its inverse a 1 fiedler and markham conjectured that q a a 1 2 n see m fiedler and t l markham linear algebra appl 101 1988 1 8 where q a a 1 is the smallest eigenvalue in modulus of a a 1 the present paper studies this conjecture an incorrect proof is given in li ching and chen ji cheng linear algebra appl 144 1991 171 178 and establishes q a a 1 2 n n 1 n for some special matrices the conjecture is proved

a new computational approach for solving nonlinear local fractional pdes abstract in this article we propose a new factorization technique for nonlinear odes involving local fractional derivatives for the first time by making use of the traveling wave transformation the exact solutions for nonlinear local fractional fitzhugh nagumo and newell whitehead equations are given the obtained results illustrate that the proposed method is efficient and accurate for finding the exact solutions for a class of local fractional pdes occurring in mathematical physics

on an inequality for the hadamard product of an m matrix and its inverse abstract when a r n n is an m matrix we prove the following inequality q a a 1 1 for n 2 2 n for n 2 where a b is the hadamard product of two matrices a and b q b p b 1 1 and p c denotes the perron eigenvalue of a nonnegative matrix c this also gives a positive answer to the conjecture posed by fiedler and markham

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

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