definition of the riesz derivative and its application to space fractional quantum mechanics we investigate and compare different representations of the riesz derivative which plays an important role in anomalous diffusion and space fractional quantum mechanics in particular we show that a certain representation of the riesz derivative r x that is generally given as also valid for 1 behaves no differently than the other definition given in terms of its fourier transform in the light of this we discuss the 1 limit of the space fractional quantum mechanics and its consistency

dynamic analysis of continuous plane trusses with equidistant supports the natural and forced vibrations of a continuous warren truss resting on equidistant roller supports are studied in the analyses the longitudinal inertial forces are neglected in the equation of motion and the mass of the truss is replaced by two sets of lumped masses at the nodes in the forced vibration case the truss is subjected to transverse harmonic loads acting at the nodes the truss can be considered as an equivalent system with cyclic bi periodicity periodicities due to the repetitive units of the truss and the supports respectively therefore the governing equation for such a system can be uncoupled by applying u transformation twice to form a set of single degree of freedom equations such equations can be used to solve for the natural frequencies of the trusses or their responses due to external excitations typical warren trusses are taken as examples to demonstrate the advantages of the proposed method and the accuracy that can be achieved the response of a truss with six substructures and four supports with a harmonic load acting at the centre node is investigated

on the nonlocality of the fractional schrodinger equation a number of papers over the past eight years have claimed to solve the fractional schrodinger equation for systems ranging from the one dimensional infinite square well to the coulomb potential to one dimensional scattering with a rectangular barrier however some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal preventing the fractional schrodinger equation from being solved in the usual piecewise fashion we focus on the one dimensional infinite square well and show that the purported ground state which is based on a piecewise approach is definitely not a solution of the fractional schrodinger equation for the general fractional parameter on a more positive note we present a solution to the fractional schrodinger equation for the one dimensional harmonic oscillator with 1

transmission through locally periodic potentials in space fractional quantum mechanics abstract we study transmission through locally periodic potentials in the framework of space fractional quantum mechanics sfqm in particular we calculate the transmission probabilities t n for a particle in a chain of n dirac barriers and for the other case a chain of n square barriers we use a transfer matrix m matrix approach in the context of sfqm to obtain equations for t n for both systems bandlike structures emerge even for a small number of barriers of as low as n 4 and are more apparent for lower values of levy index considered

convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions convergence of a shock capturing streamline diffusion nite element method for a conservation law in two space dimensions

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

comment on on the consistency of the solutions of the space fractional schrodinger equation j math phys 53 042105 2012 recently we have reanalyzed the consistency of the solutions of the space fractional schrodinger equation found in a piecewise manner and showed that an exact and a proper treatment of the relevant integrals prove that they are consistent in this comment for clarity we present additional information about the critical integrals and describe how their analytic continuation is accomplished

the nonrelativistic limit in radiation hydrodynamics i weak entropy solutions for a model problem abstract this paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions the system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integral type source term to a family of radiation transport equations we first show existence of entropy solutions to cauchy problems of the model system in the framework of functions of bounded variation this is done by using difference schemes and discrete ordinates then we establish strong convergence of the entropy solutions indexed with c as c goes to infinity the limit function satisfies a scalar integro differential equation

exact analysis of localized modes in two dimensional bi periodic mass spring systems with a single disorder abstract the frequency equation and localized modes in two dimensional bi periodic mass spring systems with one disordered subsystem are exactly analyzed by means of double u transformation at first a plane distributed mass spring system with 2 n 1 2 n 2 subsystems and cyclic periodicity in x and y directions is considered then by adopting a limiting process with n 1 n 2 approaching to infinity the limiting solution is applicable for the plane distributed bi periodic mass spring systems with boundary at infinity the explicit frequency equation and localized modes are derived some specific systems are taken as examples to demonstrate how to apply the formulas and equations obtained in the present paper in order to find the localized modes and corresponding frequencies

dynamics of bi periodic structures abstract many structures considered for space applications are bi periodic in their construction bi periodicity means that two types of structural subassemblies alternating in one or more directions make up the structure to gain insight into the dynamics of bi periodic space structures a variety of one and two dimensional bi periodic structures are considered results indicate that bands in which natural frequencies lie for periodic structures are further subdivided as a consequence of the bi periodicity analytical solutions for the modes and frequencies of finite length one dimensional bi periodic structures are obtained for general boundary conditions a transmission method is developed to simplify the application of boundary conditions it is found that some modes occur at frequencies which are outside the frequency bands predicted for bi periodic structures two dimensional bi periodic crossed beam grillage and truss structures are considered in this study for cases where a separation of variables solution is possible the two dimensional structures exhibit similar properties to the one dimensional bi periodic structures analytical solutions for the one and two dimensional bi periodic structures considered above lead to a compact solution form similar to that of periodic structures analysis

simplified analytical models for transient uniaxial waves in a layered periodic stack abstract a physical understanding is gained of some results acquired in the analysis of transient propagation of uniaxial elastic waves in weakly coupled periodic stacks using simple analytical models three simplified models are examined a mass spring chain a single mass spring attached to a delayed moving base and high frequencies of an elastic mass reacted upon by a spring closed form expressions and asymptotic behavior are obtained for attenuation of maximum stress characteristic velocities internal stress distribution and transmission or suppression of highfrequency oscillations the results provide insights in the design of impact resistant structural systems using layered periodic stacks

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

comment on on the consistency of solutions of the space fractional schrodinger equation j math phys 53 042105 2012 in bayin s paper j math phys 53 042105 2012 10 1063 1 4705268 he claims to prove the consistency of the purported piece wise solutions to the fractional schrodinger equation for an infinite square well however his calculation uses standard contour integral techniques despite the absence of an analytic integrand the correct calculation is presented and supports our earlier work proving that the purported piece wise solutions do not solve the fractional schrodinger equation for an infinite square well m jeng s l y xu e hawkins and j m schwarz j math phys 51 062102 2010 10 1063 1 3430552

fractional schrodinger equation for a particle moving in a potential well in this paper the fractional schrodinger equation that contains the quantum riesz fractional derivative instead of the laplace operator is revisited for the case of a particle moving in the infinite potential well in the recent papers m jeng s l y xu e hawkins and j m schwarz on the nonlocality of the fractional schrodinger equation j math phys 51 062102 2010 10 1063 1 3430552 and s s bayin on the consistency of the solutions of the space fractional schrodinger equation j math phys 53 042105 2012 10 1063 1 4705268 published in this journal controversial opinions regarding solutions to the fractional schrodinger equation for a particle moving in the infinite potential well that were derived by laskin fractals and quantum mechanics chaos 10 780 790 2000 10 1063 1 1050284 have been given in this paper a thorough mathematical treatment of these matters is provided the problem under consideration is reformulated in terms of three integral equations with the power

a fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrodinger equations abstract in this paper a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional schrodinger equations we prove that the scheme preserves both the mass and energy which are defined by virtue of some recursion relationships using the sobolev inequalities and then employing the mathematical induction the discrete scheme is proved to be unconditionally convergent in the sense of l 2 norm and h 2 norm which means that there are no any constraints on the grid ratios then the prior bound of the discrete solution in l 2 norm and l norm are also obtained moreover we propose an iterative algorithm by which the coefficient matrix is independent of the time level and thus it leads to toeplitz like linear systems that can be efficiently solved by krylov subspace solvers with circulant preconditioners this method can reduce the memory requirement of the proposed linearized finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m log m in each iterative step where m is the number of grid nodes finally numerical results are carried out to verify the correction of the theoretical analysis simulate the collision of two solitary waves and show the utility of the fast numerical solution techniques

fractional laplacians in bounded domains killed reflected censored and taboo levy flights the fractional laplacian 2 0 2 has many equivalent albeit formally different realizations as a nonlocal generator of a family of stable stochastic processes in r n on the other hand if the process is to be restricted to a bounded domain there are many inequivalent proposals for what a boundary data respecting fractional laplacian should actually be this ambiguity not only holds true for each specific choice of the process behavior at the boundary e g absorbtion reflection conditioning or boundary taboos but extends as well to its particular technical implementation dirichlet neumann etc problems the inferred jump type processes are inequivalent as well differing in their spectral and statistical characteristics which may strongly influence the ability of the formalism if uncritically adopted to provide an unambiguous description of real geometrically confined physical systems with disorder specifically that refers to their relaxation properties and the near equilibrium asymptotic behavior in the present paper we focus on levy flight induced jump type processes which are constrained to stay forever inside a finite domain this refers to a concept of taboo processes imported from brownian to levy stable contexts to so called censored processes and to reflected levy flights whose status still remains to be unequivocally settled as a by product of our fractional spectral analysis with reference to neumann boundary conditions we discuss disordered semiconducting heterojunctions as the bounded domain problem

on the consistency of the solutions of the space fractional schrodinger equation recently it was pointed out that the solutions found in the literature for the space fractional schrodinger equation in a piecewise manner are wrong except the case with the delta potential we re analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise we also discuss effective potential approach and present a free particle solution for the space and time fractional schrodinger equation in general coordinates in terms of fox s h functions

consistency problem of the solutions of the space fractional schrodinger equation recently consistency of the infinite square well solution of the space fractional schrodinger equation has been the subject of some controversy hawkins and schwarz j math phys 54 014101 2013 10 1063 1 4772533 objected to the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional schrodinger equation s s bayin j math phys 53 042105 2012 10 1063 1 4705268 s s bayin j math phys 53 084101 2012 10 1063 1 4739758 here we show for general n that as far as the integral representation of the solution in the momentum space is concerned there is no inconsistency to pinpoint the source of a possible inconsistency we also scrutinize the different representations of the riesz derivative that plays a central role in this controversy and show that they all have the same fourier transform when evaluated with consistent assumptions

upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2d systems of nonlinear hyperbolic conservation laws in two space dimensions are considered which are characterized by the fact that the coupling of the equations is only due to source terms to solve these weakly coupled systems numerically a class of explicit and implicit upwind finite volume methods on unstructured grids is presented provided an unique entropy solution of the system of conservation laws exists we prove that the approximations obtained by these schemes converge for vanishing discretization parameter to this entropy solution these results are applied to examples from combustion theory and hydrology where the existence of entropy solutions can be shown the proofs rely on an extension of a result due to diperna concerning measure valued solutions to the case of weakly coupled hyperbolic systems

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc