point wise error estimate of a conservative difference scheme for the fractional schrodinger equation in this paper a new conservative difference scheme is proposed for solving the nonlinear space fractional schrodinger equation the riesz space fractional derivative is approximated by the second order accurate weighted and shifted grunwald difference operator building on a careful analysis of this operator the conservation properties including the mass and energy are analyzed and some norm equivalences are established then the numerical solution is shown to be bounded in various norms uniformly with respect to the discretization parameters and optimal order bounds on the global error of the scheme in the l 2 norm semi h 2 norm and l norm are derived the uniform bounds on the numerical solutions as well as the error bounds hold unconditionally in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed numerical tests are performed to support our theoretical results

interactive group decision making procedure under incomplete information this paper deals with multiple criteria decision making problem with incomplete information when multiple decision makers multiple criteria group decision making mcgdm are involved usually decision makers dms are willing or able to provide only incomplete information because of time pressure lack of knowledge or data and their limited expertise related with problem domain there have been just a few studies considering incomplete information in group settings this incompletely specified information constructs region of linear constraints and therefore pairwise dominance relationship between alternatives reduces to intractable nonlinear programmings hence to handle this difficulty we suggest a method utilizing individual decision results to form group consensus final group consensus ranking toward more agreement of participants can be built through solving a series of linear programmings using individual decision results under group members possibly different weight constraints

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

an implicit midpoint difference scheme for the fractional ginzburg landau equation this paper proposes and analyzes an efficient difference scheme for the nonlinear complex ginzburg landau equation involving fractional laplacian the scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted grunwald difference operator for the spatial fractional laplacian by virtue of a careful analysis of the difference operator some useful inequalities with respect to suitable fractional sobolev norms are established then the numerical solution is shown to be bounded and convergent in the l h 2 norm with the optimal order o 2 h 2 with time step and mesh size h the a priori bound as well as the convergence order holds unconditionally in the sense that no restriction on the time step in terms of the mesh size h needs to be assumed numerical tests are performed to validate the theoretical results and effectiveness of the scheme

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

assessing a set of additive utility functions for multicriteria decision making the uta method abstract the purpose of the method presented in this paper is to assess additive utility functions which aggregate multiple criteria in a composite criterion using the information given by a subjective ranking on a set of stimuli or actions weak order comparison judgments and the multicriteria evaluations of these actions it is an ordinal regression method using linear programming to estimate the parameters of the utility function stability and sensitivity analysis leads to the assessment of a set of utility functions by means of post optimality analysis techniques in linear programming finally a simple illustrative example is presented and some extensions of the method are proposed

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

irreducibly inconsistent systems of linear inequalities abstract irreducibly inconsistent systems of linear inequalities are considered from the point of view of applying simplex like methods for investigation we give necessary and sufficient conditions for a system to be irreducibly inconsistent which can be found by applying a simplex algorithm next we consider general inconsistent systems the simplex method can be used to identify irreducibly inconsistent subsystem such subsystems can give structural insight in the inconsistency and are used in trying to make the overall system consistent this can be done in an easy way e g by right hand side manipulations finally the problem of finding a cheapest way of doing this can be formulated and solved as an extended linear program

crank nicolson difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative in this paper the crank nicolson cn difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative is studied the existence of this difference solution is proved by the brouwer fixed point theorem the stability and convergence of the cn scheme are discussed in the l 2 norm when the fractional order is two all those results are in accord with the difference scheme developed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also presented

a linearly implicit conservative difference scheme for the space fractional coupled nonlinear schrodinger equations abstract in this paper a linearly implicit conservative difference scheme for the coupled nonlinear schrodinger equations with space fractional derivative is proposed this scheme conserves the mass and energy in the discrete level and only needs to solve a linear system at each step the existence and uniqueness of the difference solution are proved the stability and convergence of the scheme are discussed and it is shown to be convergent of order o 2 h 2 in the discrete l 2 norm with the time step and mesh size h when the fractional order is two all those results are in accord with the difference scheme proposed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also reported

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

an efficient split step quasi compact finite difference method for the nonlinear fractional ginzburg landau equations abstract in this paper we propose a split step quasi compact finite difference method to solve the nonlinear fractional ginzburg landau equations both in one and two dimensions the original equations are split into linear and nonlinear subproblems the riesz space fractional derivative is approximated by a fourth order fractional quasi compact method furthermore an alternating direction implicit scheme is constructed for the two dimensional linear subproblem the unconditional stability and convergence of the schemes are proved rigorously in the linear case numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method

on a fractional ginzburg landau equation and 1 2 harmonic maps into spheres this paper is devoted to the asymptotic analysis of a fractional version of the ginzburg landau equation in bounded domains where the laplacian is replaced by an integro differential operator related to the square root laplacian as defined in fourier space in the singular ginzburg landau limit we show that solutions with uniformly bounded energy converge weakly to sphere valued 1 2 harmonic maps i e the fractional analogues of the usual harmonic maps in addition the convergence holds in smooth functions spaces away from a n 1 rectifiable closed set of finite n 1 hausdorff measure the proof relies on the representation of the square root laplacian as a dirichlet to neumann operator in one more dimension and on the analysis of a boundary version of the ginzburg landau equation besides the analysis of the fractional ginzburg landau equation we also give a general partial regularity result for stationary 1 2 harmonic maps in arbitrary dimension

fourth order accurate scheme for the space fractional diffusion equations because of the nonlocal properties of fractional operators higher order schemes play a more important role in discretizing fractional derivatives than classical ones the striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first order schemes but greatly improve the accuracy nowadays there are already two types of second order discretization schemes for space fractional derivatives the first type is given and discussed in sousa and li arxiv 1109 2345v1 2011 chen and deng appl math model 38 2014 pp 3244 3259 chen deng and wu appl numer math 70 2013 pp 22 41 and the second type is a class of schemes presented in tian zhou and deng math comp to appear also available online from arxiv 1201 5949 2012 the core object of this paper is to derive a class of fourth order approximations called the weighted and shifted lubich difference operators for space fractional derivatives then we use the derived schemes t

a fast energy conserving finite element method for the nonlinear fractional schrodinger equation with wave operator abstract the main aim of this paper is to apply the galerkin finite element method to numerically solve the nonlinear fractional schrodinger equation with wave operator we first construct a fully discrete scheme combining the crank nicolson method with the galerkin finite element method two conserved quantities of the discrete system are shown meanwhile the prior bound of the discrete solutions are proved then we prove that the discrete scheme is unconditionally convergent in the senses of l 2 norm and h 2 norm moreover by the proposed iterative algorithm some numerical examples are given to verify the theoretical results and show the effectiveness of the numerical scheme finally a fast krylov subspace solver with suitable circulant preconditioner is designed to solve above toeplitz like linear system in each iterative step this method can effectively reduce the memory requirement of the proposed iterative finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m l o g m where m is the number of grid nodes several numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and lu factorization methods in terms of memory requirement and computational cost

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

linearized adi schemes for two dimensional space fractional nonlinear ginzburg landau equation abstract space and time approximations for two dimensional space fractional complex ginzburg landau equation are examined the schemes under consideration are discreted by the second order backward differential formula bdf2 in time and two classes of the fractional centered finite difference methods in space a linearized technique is employed by the extrapolation we prove the unique solvability and stability for both numerical methods the convergence of both numerical methods is analyzed at length utilizing the energy argument and the convergence orders under the optimal step size ratio are o 2 h 2 and o 2 h 4 in the sense of the discrete l 2 norm where is the time step size h max h x h y and h x h y are spatial grid sizes in the x direction and y direction respectively in addition we construct a multistep alternating direction implicit adi scheme and a multistep compact adi scheme based on bdf2 for the efficiently numerical implementation finally numerical examples are carried out to verify our theoretical results

time space fractional stochastic ginzburg landau equation driven by fractional brownian motion abstract the current paper is devoted to the time space fractional ginzburg landau equation driven by fractional brownian motion the spatial temporal regularity of the nonlocal stochastic convolution is firstly established and then the existence and uniqueness of the global mild solution are obtained by the banach fixed point theorem and mittag leffler functions finally the numerical simulations for the time fractional stochastic ginzburg landau equation are provided to verify the mathematical analysis results

a fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

decision making with incomplete information abstract decision situations with incomplete information are characterized by a decision maker without a precisely defined stable preference structure by probability distributions not known completely or by an inexact evaluation of consequences within the paper a general framework for decision making with incomplete information is presented which shows how to solve problems from descriptive as well as prescriptive decision theory based on this framework an overview of existing methods which are particularly suitable for handling incomplete information is given

resolving inconsistencies among constraints on the parameters of an mcda model we consider a framework where decision makers dms interactively define a multicriteria evaluation model by providing imprecise information i e a linear system of constraints to the model s parameters and by analyzing the consequences of the information provided dms may introduce new constraints explicitly or implicitly results that the model should yield if a new constraint is incompatible with the previous ones then the system becomes inconsistent and the dms must choose between removing the new constraint or removing some of the older ones we address the problem of identifying subsets of constraints which when removed lead to a consistent system identifying such subsets would indicate the reason for the inconsistent information given by dms there may exist several possibilities for the dms to resolve the inconsistency we present two algorithms to identify such possibilities one using 0 1 mixed integer linear programming and the other one using linear programming both approaches are based on the knowledge that the system was consistent prior to introducing the last constraint the output of these algorithms helps the dm to identify the conflicting pieces of information in a set of statements he she asserted the relevance of these algorithms for mcda is illustrated by an application to an aggregation disaggregation procedure for the electre tri method

attractors and inertial manifolds for finite difference approximations of the complex ginzburg landau equation a semidiscrete spatial finite difference approximation and two fully discrete finite difference approximations to the complex ginzburg landau equation are considered in this paper the existence of an inertial manifold is proved inside a discrete h1 absorbing ball for the semi discrete approximation by showing that certain spectral properties hold for the linear operator and certain lipschitz properties hold for the nonlinear term convergence of the true and discrete inertial manifolds is then shown by considering the contraction mappings used to construct the inertial manifolds a fully implicit scheme is shown to form a continuous semigroup in a discrete l2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete l2 and h1 spaces are constructed existence of a global attractor follows furthermore the existence of an inertial manifold is proved by showing that certain spectral and lipschitz properties hold for a semi implicit scheme existence of absorbing balls in the discrete space l2 of radii independent of the spatial and temporal mesh is proved existence of a global attractor follows

ascending chain condition for log canonical thresholds and termination of log flips we prove that the ascending chain condition acc for log canonical lc thresholds in dimension d existence of log flips in dimension d and the log minimal model program lmmp in dimension d 1 imply termination of any sequence of log flips starting with a d dimensional effective lc pair and also imply termination of flops in dimension d in particular the latter terminations in dimension 4 follow from the alexeev borisov conjecture in dimension 3

dissipative or conservative finite difference schemes for complex valued nonlinear partial differential equations abstract we propose a new procedure for designing finite difference schemes that inherit energy conservation or dissipation property from complex valued nonlinear partial differential equations pdes such as the nonlinear schrodinger equation the ginzburg landau equation and the newell whitehead equation the procedure is a complex version of the procedure that furihata has recently presented for real valued nonlinear pdes furthermore we show that the proposed procedure can be modified for designing linearly implicit finite difference schemes that inherit energy conservation or dissipation property

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

higher order grunwald approximations of fractional derivatives and fractional powers of operators we give stability and consistency results for higher order grunwaldtype formulae used in the approximation of solutions to fractional in space partial differential equations we use a new carlson type inequality for periodic fourier multipliers to gain regularity and stability results we then generalise the theory to the point where the first derivative operator is replaced by the generator of a bounded group on an arbitrary banach space

fast iterative solvers and simulation for the space fractional ginzburg landau equations abstract in this paper we propose a new splitting iteration method for solving the toeplitz like complex linear system arising from the space fractional ginzburg landau equations of which the coefficient matrix is equal to the sum of a complex scaled toeplitz plus diagonal matrix and a symmetric toeplitz plus diagonal matrix the new splitting method enjoys computational advantage since circulant preconditioner and fast fourier transform fft can be used for solving the involved linear subsystem under some conditions the convergence properties of the corresponding iteration method are derived numerical examples are given to illustrate the effectiveness of the proposed method

on the continuum limit for discrete nls with long range lattice interactions we consider a general class of discrete nonlinear schrodinger equations dnls on the lattice h mathbb z with mesh size h 0 in the continuum limit when h 0 we prove that the limiting dynamics are given by a nonlinear schrodinger equation nls on mathbb r with the fractional laplacian as dispersive symbol in particular we obtain that fractional powers frac 1 2 alpha 1 arise from long range lattice interactions when passing to the continuum limit whereas the nls with the usual laplacian describes the dispersion in the continuum limit for short range or quick decaying interactions e g nearest neighbor interactions

high order algorithms for riesz derivative and their applications ii in this paper we firstly develop two high order approximate formulas for the riesz fractional derivative secondly we propose a temporal second order numerical method for a fractional reaction dispersion equation where we discretize the riesz fractional derivative by using two numerical schemes we prove that the numerical methods for a spatial riesz fractional reaction dispersion equation are both unconditionally stable and convergent and the orders of convergence are o 2 h 6 and o 2 h 8 in which and h are spatial and temporal step sizes respectively finally we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis

relational systems of preference with one or more pseudo criteria some new concepts and results this paper proposes new concepts and new results which could lead to a more realistic preference modeling than in classical decision theory sections 1 3 present four fundamental situations of preferences their combinations and the concept of relational system of preferences in 4 a particular case of relational system of preference is studied it is associated with the concept of pseudo criterion derived from the classical concept of criterion by adjunction of two thresholds some results are given generalizing the properties of such well known structures as complete preorders and semiorders sections 5 and 6 emphasize the possibilities given by the preceding concepts to take the imprecisions irresolutions and incomparabilities appearing in every concrete problem where several criteria must be considered into account

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

on tsertsvadze s difference scheme for the kuramoto tsuzuki equation in this paper we have proved the second order convergence in l norm of the tsertsvadze s difference scheme for the kuramoto tsuzuki equation the existence uniqueness and iterative algorithm are also discussed in detail furthermore a l second order convergent linearized difference scheme is given for inhomogeneous equation all results are obtained without any restrictions on the meshsizes at last a numerical example is presented

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

high order numerical approximation formulas for riemann liouville riesz tempered fractional derivatives construction and application i abstract in this paper we develop a new numerical algorithm for solving the riesz tempered space fractional diffusion equation the stability and convergence of the numerical scheme are discussed via the technique of matrix analysis finally numerical experiments are performed to confirm the effectiveness of our numerical algorithm

a fourth order compact adi scheme for two dimensional nonlinear space fractional schrodinger equation in this paper a novel compact operator is derived for the approximation of the riesz derivative with order alpha in 1 2 the compact operator is proved with fourth order accuracy combining the compact operator in space discretization a linearized difference scheme is proposed for a two dimensional nonlinear space fractional schrodinger equation it is proved that the difference scheme is uniquely solvable stable and convergent with order o tau 2 h 4 where tau is the time step size h max h 1 h 2 and h 1 h 2 are space grid sizes in the x direction and the y direction respectively based on the linearized difference scheme a compact alternating direction implicit scheme is presented and analyzed numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly