extensions of the analytic hierarchy process in fuzzy environment abstract the aim of this paper is to extend the analytic hierarchy process ahp to the fuzzy environment firstly starting from the viewpoint of actual application considering the method of group decision making in complex systems and the fuzziness of judgment in pairwise comparison of projects we construct the fuzzy judgement matrix by using a set valued statistics method on continuous judgment matrix and prove that every element of the fuzzy judgment scale can be represented by a positive bounded closed fuzzy number after a discussion of the properties of a positive bounded closed fuzzy number the fuzzy weight vector of projects can be introduced by applying the fuzzy extension principle to the altered gradient eigenvector method thus the rank ordering of projects is given finally we also give an example to apply our model

split step orthogonal spline collocation methods for nonlinear schrodinger equations in one two and three dimensions abstract split step orthogonal spline collocation osc methods are proposed for one two and three dimensional nonlinear schrodinger nls equations with time dependent potentials firstly the nls equation is split into two nonlinear equations and one or more one dimensional linear equations commonly the nonlinear subproblems could be integrated directly and accurately but it fails when the time dependent potential cannot be integrated exactly in this case we propose three approximations by using quadrature formulae but the split order is not reduced discrete time osc schemes are applied for the linear subproblems in numerical experiments many tests are carried out to prove the reliability and efficiency of the split step osc ssosc methods solitons in one two and three dimensions are well simulated and conservative properties and convergence rates are demonstrated we also apply the ways of solving the nonlinear subproblems to the split step finite difference ssfd methods and the time splitting spectral tssp methods and the approximate ways still work well finally we apply the ssosc methods to solve some problems of bose einstein condensates

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

a fuzzy simple additive weighting system under group decision making for facility location selection with objective subjective attributes this work presents a new fuzzy multiple attributes decision making fmadm approach i e fuzzy simple additive weighting system fsaws for solving facility location selection problems by using objective subjective attributes under group decision making gdm conditions the proposed system integrates fuzzy set theory fst the factor rating system frs and simple additive weighting saw to evaluate facility locations alternatives the fsaws is applied to deal with both qualitative and quantitative dimensions the fsaws process considers the importance of each decision maker and the total scores for alternative locations are then derived by homo heterogeneous group of decision makers finally a numerical example illustrates the procedure of the proposed fsaws

a two phase multi attribute decision making approach for new product introduction this study aims at improving the quality and effectiveness of decision making in new product introduction new product development has long been recognized as one of the corporate core functions to be competitive on an increasingly competitive global market however developing new products is a process involving risk and uncertainty in order to solve this stochastic problem companies need to evaluate their new product initiatives carefully and make accurate decisions for this reason a systematic decision process for selecting more rational new product ideas is proposed basically two stages of decision making are described the identification of nondominated new product candidates and the selection of the best new product idea these stages are composed of an integrated approach based on a fuzzy heuristic multi attribute utility method and a hierarchical fuzzy topsis method finally an application is given to demonstrate the potential of the methodology

finite difference spectral approximations for the fractional cable equation fujian nsf s0750017 national nsf of china 10531080 ministry of education of china 973 high performance scientific computation research program 2005cb321703

difference schemes for solving the generalized nonlinear schrodinger equation this paper studies finite difference schemes for solving the generalized nonlinear schrodinger gnls equationiut uxx q u 2 u f x t u a new linearlized crank nicolson type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the gnls equation several schemes including crank nicolson type schemes hopscotch type schemes split step fourier scheme and pseudospectral scheme are adopted for solving three model problems of gnls equation which arise from many physical problems withq s s2 q s ln 1 s andq s 4s 1 s respectively the numerical results demonstrate that the linearized crank nicolson scheme is efficient and robust

godel type space times in induced matter gravity theory five dimensional 5d generalized godel type manifolds are examined in the light of the equivalence problem techniques as formulated by cartan the necessary and sufficient conditions for local homogeneity of these 5d manifolds are derived the local equivalence of these homogeneous riemannian manifolds is studied it is found that they are characterized by three essential parameters k m2 and identical triads k m2 correspond to locally equivalent 5d manifolds an irreducible set of isometrically nonequivalent 5d locally homogeneous riemannian generalized godel type metrics are exhibited a classification of these manifolds based on the essential parameters is presented and the killing vector fields as well as the corresponding lie algebra of each class are determined it is shown that the generalized godel type 5d manifolds admit maximal group of isometry gr with r 7 r 9 or r 15 depending on the essential parameters k m2 and the breakdown of causality in all these classes of homogeneous

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

the evidential reasoning approach for mada under both probabilistic and fuzzy uncertainties abstract many multiple attribute decision analysis mada problems are characterised by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er approach has been developed in the 1990s and in the recent years to support the solution of mada problems with ignorance a kind of probabilistic uncertainty in this paper the er approach is further developed to deal with mada problems with both probabilistic and fuzzy uncertainties in this newly developed er approach precise data ignorance and fuzziness are all modelled under the unified framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a utility based grade match method is proposed to transform both numerical data and qualitative fuzzy assessment information of various formats into the fuzzy belief structure a new fuzzy er algorithm is developed to aggregate multiple attributes using the information contained in the fuzzy belief matrix resulting in an aggregated fuzzy distributed assessment for each alternative different from the existing er algorithm that is of a recursive nature the new fuzzy er algorithm provides an analytical means for combining all attributes without iteration thus providing scope and flexibility for sensitivity analysis and optimisation a numerical example is provided to illustrate the detailed implementation process of the new er approach and its validity and wide applicability

a numerical method for two dimensional schrodinger equation using collocation and radial basis functions in this paper we propose a numerical scheme to solve the two dimensional 2d time dependent schrodinger equation using collocation points and approximating the solution using multiquadrics mq and the thin plate splines tps radial basis function rbf the scheme works in a similar fashion as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

fourth order compact and energy conservative difference schemes for the nonlinear schrodinger equation in two dimensions abstract in this paper a fourth order compact and energy conservative difference scheme is proposed for solving the two dimensional nonlinear schrodinger equation with periodic boundary condition and initial condition and the optimal convergent rate without any restriction on the grid ratio at the order of o h 4 2 in the discrete l 2 norm with time step and mesh size h is obtained besides the standard techniques of the energy method a new technique and some important lemmas are proposed to prove the high order convergence in order to avoid the outer iteration in implementation a linearized compact and energy conservative difference scheme is derived numerical examples are given to support the theoretical analysis

truncation error and oscillation property of the combined compact difference scheme in this work we study a sixth order combined compact difference scheme for a one dimensional convection diffusion equation we derive the truncation error representation and analyze its oscillation property numerical experiments are performed to illustrate and support our analysis

dynamic properties of cubic nonlinear schrodinger equation with varying nonlinear parameter the dynamic properties of the cubic nonlinear schrodinger equation are investigated numerically using the symplectic scheme euler centred scheme we discuss the dynamic behaviour of the cubic nonlinear schrodinger equation with varying nonlinear parameter the results show that the system exhibits regular recurrence for weakly nonlinearity we also illustrate that the system will exhibit varying dynamic behaviour with increasing nonlinear parameter i e the system will show the homoclinic orbit hmo crossing quasi recurrence pseudorecurrence irregular motion or stochastic motion for a strongly nonlinear constants

riemannian space times of godel type in five dimensions the five dimensional 5d riemannian godel type manifolds are examined in light of the equivalence problem techniques as formulated by cartan the necessary and sufficient conditions for local homogeneity of these 5d manifolds are derived the local equivalence of these homogeneous riemannian manifolds is studied it is found that they are characterized by two essential parameters m2 and identical pairs m2 correspond to locally equivalent 5d manifolds an irreducible set of isometrically nonequivalent 5d locally homogeneous riemannian godel type metrics are exhibited a classification of these manifolds based on the essential parameters is presented and the killing vector fields as well as the corresponding lie algebra of each class are determined it is shown that apart from the m2 4 2 0 and m2 0 0 classes the homogeneous riemannian godel type manifolds admit a seven parameter maximal group of isometry g7 the special class m2 4 2 0 and the degenerated godel type class m2 0 0 a

numerical solution to the unsteady two dimensional schrodinger equation using meshless local boundary integral equation method a meshless local boundary integral equation lbie method is proposed to solve the unsteady two dimensional schrodinger equation the method is based on the lbie with moving least squares mls approximation for the mls approximation nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables a time stepping method is employed to deal with the time derivative an efficient method for dealing with singular domain integrations that appear in the discretized equations is presented finally numerical results are considered for some examples to demonstrate the accuracy efficiency and high rate of convergence of this method copyright 2008 john wiley sons ltd

the use of a fuzzy decision making method in a large scale computer system choice problem abstract we recently proposed a fuzzy decision making method for multiple objectives problems in an uncertain and ill defined environment this paper presents a real and large scale application of the method the problem is to decide what sort of computer system for research and education should be renewed in the department of computer engineering one member of a renewal committee appears as a decision maker after describing the outline of the fuzzy decision making method the decision analysis process objective hierarchy identification of decision maker s preference structure rating and ranking of alternatives and sensitivity analysis is presented

modeling subjective evaluation for fuzzy group multicriteria decision making this paper presents a new fuzzy multicriteria decision making mcdm approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers a pairwise comparison process is used to help individual decision makers make comparative judgements and a linguistic rating method is used for making absolute judgements a hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons to reflect the inherent imprecision of subjective judgements individual assessments are aggregated as a group assessment using triangular fuzzy numbers to obtain a cardinal preference value for each decision alternative a new fuzzy mcdm algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement an empirical study of aircraft selection is presented to illustrate the effectiveness of the approach

three point combined compact difference schemes for time fractional advection diffusion equations with smooth solutions a ccd method with the l1 formula is proposed for solving time fractional advection diffusion equations the method is globally at least fifth order accurate in space and 2 th order accurate in time the unconditional stability is proved by means of fourier analysis numerical experiments are conducted to demonstrate the efficiency the present method preserves the higher order accuracy for advection dominated problems with the smooth solutions in this paper we study the numerical solution for the time fractional advection diffusion equation tfade a high order accurate three point combined compact difference scheme with the l1 formula is proposed to solve a class of tfades the method is globally 2 th order accurate in time and at least fifth order accurate in space for the constant coefficient tfades subject to periodic boundary conditions where is the order of time fractional derivative in the governing equation the unconditional stability and the high order convergence are proved using the fourier analysis method the proposed method can be extended to treat the variable coefficient tfades subject to other local boundary conditions several numerical examples are computed to validate the numerical accuracy effectiveness and robustness of the present method for tfades with smooth solutions and different boundary conditions moreover the proposed algorithm preserves the higher order accuracy for advection dominated problems with the smooth solutions

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

a new fuzzy multiple attributive group decision making methodology and its application to propulsion manoeuvring system selection problem in this paper a new fuzzy multiple attribute decision making fmadm method which is suitable for multiple attributive group decision making gdm problems in fuzzy environment is proposed to deal with the problem of ranking and selection of alternatives since the subjectivity imprecision and vagueness in the estimates of a performance rating enter into multiple attribute decision making madm problems fuzzy set theory provides a mathematical framework for modelling vagueness and imprecision in the proposed approach an attribute based aggregation technique for heterogeneous group of experts is employed and used for dealing with fuzzy opinion aggregation for the subjective attributes of the decision problem the propulsion manoeuvring system selection as a real case study is used to demonstrate the versatility and potential of the proposed method for solving fuzzy multiple attributive group decision making problems the proposed method is a generalised model which can be applied to great variety of practical problems encountered in the naval architecture from propulsion manoeuvring system selection to warship requirements definition

computer aided study of a class of riemannian space times the riemannian godel type manifolds are examined in the light of the equivalence problem techniques by using algebraic computing the conditions for space time homogeneity of a riemannian manifold with a godel type metric are derived generalizing previous work on this subject matter a classification of the godel type riemannian manifolds based on the two relevant parameters m and is formulated it is shown that apart from the m2 4 2 case they are all petrov type d with a five parameter group of motions the special m2 4 2 manifold is shown to be conformally flat and to admit a seven parameter group of isometries an algebraic classification of the ricci spinor of all godel type riemannian manifolds is discussed possible sources for these space times are examined a generalization of the reboucas tiomno theorem on godel type manifolds is given

multicriteria decision analysis with fuzzy pairwise comparisons abstract we present a modification of the fuzzy multi criteria method proposed by van laarhoven and pedrycz 1983 first the weights of the decision criteria are calculated by the minimization of a logarithmic regression function next the weights of the decision alternatives are calculated for each criterion separately lastly the fuzzy final scores of the alternatives are determined by an appropriate aggregationm of the calculated weights we will show that our modified method in contrast with the original one yields weights which are optimal with respect to the logarithmic regression function we also present a version of our method with geometric ratio scales rather than saaty s equi distant scale to quantify the gradations of human comparative judgement this approach also enables us to develop an efficient procedure for investigating the scale sensitivity of the calculated weights and the final scores of the alternatives

ordering fuzzy sets over the real line an approach based on decision making under uncertainty abstract the ordering of fuzzy sets over the real line is approached from the point of view of ordering intervals rather than ordering numbers first the maximax and maximin criteria which are commonly used for ordering intervals are expressed in terms of characteristic functions these criteria and the hurwicz criterion for decision making under complete ignorance are then reformulated in a manner that allows for their generalization to the fuzzy case transitivity is established for these ordering rules a criterion based on the principle of diminishing marginal utility is also presented

fuzzy credibility relation method for multiple criteria decision making problems combining the concepts of fuzzy set theory entropy ideal and grey relation analysis a fuzzy grey relation method for multiple criteria decision making problems is proposed first triangular fuzzy numbers and linguistic values characterized by triangular fuzzy numbers are used to evaluate the importance weights of all criteria and the superiority of all alternatives versus various criteria above the alternative level then the concept of entropy is utilized to solve the adjusted integration weight of all objective criteria above the alternative level furthermore using the concepts of ideal the grey ration grades of various alternatives versus ideal solution are ranked to determine the best alternative finally a numerical example of selecting most appropriate company to build a new highway is used to demonstrate the applicability of proposed method the study results show that this method is an effective means for tackling mcdm problems in fuzzy and grey environments

ranking fuzzy numbers with index of optimism abstract fuzzy numbers are convenient for representing imprecise numerical quantities in a vague environment and their comparison or ranking is very important for application purposes despite many methods suggested in the literature there is no single measure that is universally applicable to a wide variety of situations this paper suggests a new method for comparing fuzzy numbers based on the combination of maximizing possibility and minimizing possibility using an index of optimism in 0 1 reflecting the decision maker s risk taking attitude the method is simple but has many comparative advantages

the use of compact boundary value method for the solution of two dimensional schrodinger equation in this paper a high order and accurate method is proposed for solving the unsteady two dimensional schrodinger equation we apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulting linear system of ordinary differential equations the proposed method has fourth order accuracy in both space and time variables moreover this method is unconditionally stable due to the favorable stability property of boundary value methods the results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature these results show that the combination of a compact finite difference approximation of fourth order and a fourth order boundary value method gives an efficient algorithm for solving the two dimensional schrodinger equation

a compact finite difference scheme for the nonlinear schrodinger equation with wave operator abstract in this paper a compact finite difference scheme is presented for an periodic initial value problem of the nonlinear schrodinger nls equation with wave operator this is a scheme of three levels with a discrete conservation law the unconditional stability and convergence in maximum norm with order o h 4 2 are proved by the energy method a numerical experiment is presented to support our theoretical results

a semi discrete higher order compact scheme for the unsteady two dimensional schrodinger equation in this study an implicit semi discrete higher order compact hoc scheme with an averaged time discretization has been presented for the numerical solution of unsteady two dimensional 2d schrodinger equation the scheme is second order accurate in time and fourth order accurate in space the results of numerical experiments are presented and are compared with analytical solutions and well established numerical results of some other finite difference schemes in all cases the present scheme produces highly accurate results with much better computational efficiency

a fuzzy multi criteria decision making model by associating technique for order preference by similarity to ideal solution with relative preference relation generally classical multi criteria decision making mcdm methods were extended to encompass uncertainty and vagueness of messages under fuzzy environment for solving decision making problems especially for technique for order preference by similarity to ideal solution topsis in the fuzzy extension of topsis fuzzy numbers comparison and aggregation based on fuzzy preference relation are important issues to compute distance values between alternatives and ideal or anti ideal solution or rank feasible alternatives because lots of messages are reserved by fuzzy preference relation however fuzzy preference relation on pair wise comparison is commonly too complex to calculate to avoid the drawback we use a relative preference relation improved from fuzzy preference relation in the fuzzy extension of topsis for computing distance values between alternatives and ideal or anti ideal solution or obtaining relative closeness coefficients of alternatives thus the relative preference relation on fuzzy numbers will be associated with topsis under fuzzy environment to develop a fuzzy multi criteria decision making fmcdm model through the association above fmcdm problems can be easily solved by the model further we compare the proposed model with other methods to demonstrate the model s feasibility and rationality

designing a model of fuzzy topsis in multiple criteria decision making decision making is the process of finding the best option among the feasible alternatives in classical multiple attribute decision making madm methods the ratings and the weights of the criteria are known precisely due to vagueness of the decision data the crisp data are inadequate for real life situations since human judgments including preferences are often vague and cannot be expressed by exact numerical values the application of fuzzy concepts in decision making is deemed to be relevant we design a model of topsis for the fuzzy environment with the introduction of appropriate negations for obtaining ideal solutions here we apply a new measurement of fuzzy distance value with a lower bound of alternatives then similarity degree is used for ranking of alternatives examples are shown to demonstrate capabilities of the proposed model

a note on godel type space times the godel type space times are examined under different conditions on the algebraic structure of the energy momentum tensor a generalization of the bampi zordan theorem on godel type models is given chakraborty s results are recast as a special case of the generalized bampi zordan theorem the reboucas tiomno solution is shown to be the unique godel type metric with an algebraic structure of the tachyon fluid type

an unconditionally stable linearized ccdadi method for generalized nonlinear schrdinger equations with variable coefficients in two and three dimensions in this paper we propose a three level linearly implicit combined compact difference method ccd together with alternating direction implicit method adi for solving the generalized nonlinear schrdinger equation nlse with variable coefficients in two and three dimensions the method is sixth order accurate in space variable and second order accurate in time variable fourier analysis shows that the method is unconditionally stable comparing to the nonlinear ccdpradi scheme for solving the 2d cubic nlse with constant coefficients li etal 2015 current method is a linear scheme which generally requires much less computational cost moreover current method can handle 3d problems with variable coefficients naturally finally numerical results for both 2d and 3d cases are presented to illustrate the advantages of the proposed method

a new adi technique for two dimensional parabolic equation with an integral condition abstract nonclassical parabolic initial boundary value problems arise in the study of several important physical phenomena this paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions a new fourth order finite difference technique based upon the noye and hayman n h alternating direction implicit adi scheme is used as the basis to solve the two dimensional time dependent diffusion equation with an integral condition replacing one boundary condition this scheme uses less central processor time cpu than a second order fully implicit scheme based on the classical backward time centered space btcs method for two dimensional diffusion it also has a larger range of stability than a second order fully explicit scheme based on the classical forward time centered space ftcs method the basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyeet this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods the results of numerical experiments for the new method are presented the central processor times needed are also reported error estimates derived in the maximum norm are tabulated

evaluating weapon system by analytical hierarchy process based on fuzzy scales abstract in this paper we propose a new algorithm for evaluating weapon systems by analytical hierarchy process ahp based on fuzzy scales which is a multiple criteria decision making approach in a fuzzy environment we use the triangular fuzzy number to build the judgement matrices through the pair wise comparison technique from kaufmann and gupta fuzzy mathematical models in engineering and management science north holland amsterdam 1988 p 1 a 3 1 a 2 1 a 1 only approximates the inverse of triangular fuzzy number a 1 where a a 1 a 2 a 3 which is due to a 1 is maybe no longer a fuzzy number therefore we revise juang and lee s fuzzy number 9 ifes 1991 415 421 more accurately in order to estimate the fuzzy eigenvectors of this matrix we utilize interval arithmetic cuts together with optimism of index in this way performance scores of the alternatives and the weights of the difference attributes can finally be obtained and compared for easy computation we use the mathcad package to calculate all results at last we apply and verify this new algorithm to a weapon system evaluation and selection problem

derivation of the cubic non linear schrodinger equation from quantum dynamics of many body systems we prove rigorously that the one particle density matrix of three dimensional interacting bose systems with a short scale repulsive pair interaction converges to the solution of the cubic non linear schrodinger equation in a suitable scaling limit the result is extended to k particle density matrices for all positive integer k

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership functions

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

parallel in time multigrid for space time finite element approximations of two dimensional space fractional diffusion equations abstract the paper investigates a non intrusive parallel time integration with multigrid for space fractional diffusion equations in two spatial dimensions which is discretized by the space time finite element method to propagate solutions we develop a multigrid reduction in time mgrit algorithm with time dependent time grid propagators and provide its two level convergence theory under the assumptions of the stability and simultaneous diagonalizability on time grid propagators numerical results show that the proposed method possesses the saturation error order theoretical results of the two level variant deliver good predictions for our model problems and significant speedups of the mgrit can be achieved when compared to the two level variant with f relaxation an equivalent version of the parareal algorithm and the sequential time stepping approach

generalizing topsis for fuzzy multiple criteria group decision making in this paper we generalize topsis to fuzzy multiple criteria group decision making fmcgdm in a fuzzy environment topsis is one of the well known methods for multiple criteria decision making mcdm most of the steps of topsis can be easily generalized to a fuzzy environment except max and min operations in finding the ideal solution and negative ideal solution thus we propose two operators up and lo which satisfy the partial ordering relation on fuzzy numbers to the generalization of topsis in generalized topsis these two operations up and lo are employed to find ideal and negative ideal solutions under a fuzzy environment then the fmcgdm problem can be solved effectively and efficiently

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

a fast solver of the shallow water equations on a sphere using a combined compact difference scheme the spectral like three point combined compact difference sp ccd scheme is developed for the shallow water models on a sphere which is originally proposed by chu and fan j comput phys 140 1998 370 a standard test set proposed by williamson et al j comput phys 102 1992 211 is used to evaluate the accuracy of the new method it is shown that the accuracy of the method is comparable to that of the spherical harmonics model as a whole the relative efficiency of the method with respect to the spherical harmonics model appears from the resolution of 512 256 and it becomes significant for the higher resolutions the test results suggest that the method could be extended to numerical atmospheric models

notes on a class of homogeneous space times the breakdown of causality in homogeneous godel type space time manifolds is examined an extension of reboucas tiomno rt and accioly goncalves studies is made the existence of noncausal curves is also investigated under two different conditions on the energy momentum tensor an integral representation of the infinitesimal generators of isometries is obtained extending previous works on the rt geometry

a study of the ranking function approach through mean values abstract the average value was introduced to help in the ordering of fuzzy numbers and it was defined by means of an integrating process of a parametric function representing the position of every cut in the real line by considering the relation between a particular case of the average value and the notion of expectation for fuzzy numbers we interpret the general ranking function approach as a ranking process using a mean value and an interval relation some well known indices are included in this schema using the general average value a more general mean value for a fuzzy number is defined finally some properties of the general mean value are studied as well as its behaviour on several particular cases

a three point sixth order nonuniform combined compact difference scheme a three point nonuniform combined compact difference nccd scheme with sixth order accuracy is proposed for numerical models the nccd scheme is a generalization of the previously proposed combined compact difference ccd scheme with a global hermitan polynomial spline and has major improved features such as error and computational cpu time reduction for nonperiodic boundaries additional sixth or fifth order nonuniform boundary conditions are proposed the nccd scheme with either sixth or fifth order additional boundary conditions can increase the accuracy and decrease the cpu time about 1 2 orders of magnitude compared to the ccd scheme

interval valued fuzzy multi criteria decision making based on simple additive weighting and relative preference relation abstract to encompass imprecise messages traditional multi criteria decision making mcdm was generalized into fuzzy multi criteria decision making fmcdm in solving decision making problems in numerous mcdm methods simple additive weighting saw is the simplest one but fuzzy multiplication for generalizing saw under fuzzy environment is still complicated especially interval valued fuzzy numbers in the past wang proposed a model based on saw and relative preference relation to resolve fuzzy multiplication tie practically wang s model is useful to fuzzy generalization of saw however the model is only used in triangular fuzzy numbers recently more and varied messages are obtained and processed for decision making problems triangular fuzzy numbers are insufficient to express the variance whereas interval valued fuzzy numbers are suitable therefore saw had to be generalized under interval valued fuzzy environment in this paper we combine saw with a relative preference relation into interval valued fmcdm to avoid fuzzy multiplication and grasp more messages our proposed relative preference relation improved from lee s fuzzy preference relation is similar to wang s relative preference relation the main difference is that the proposed relation is utilized in interval valued fuzzy numbers whereas wang s is used in triangular fuzzy numbers to sum up we provide the fmcdm to solve decision making problems with interval valued fuzzy numbers easily and quickly

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

compact homogeneous universes a thorough classification of the topologies of compact homogeneous universes is given except for the hyperbolic spaces and their global degrees of freedom are completely worked out to obtain compact universes spatial points are identified by discrete subgroups of the isometry group of the generalized thurston geometries which are related to the bianchi and the kantowski sachs nariai universes corresponding to this procedure their total degrees of freedom are shown to be categorized into those of the universal covering space and the teichmuller parameters the former are given by constructing homogeneous metrics on a simply connected manifold the teichmuller spaces are also given by explicitly constructing expressions for the discrete subgroups of the isometry group

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

an extension to fuzzy mcdm an extension to the fuzzy multiple criteria decision making mcdm model is suggested in this work where the ratings of alternatives versus criteria and the importance weights of all criteria are assessed in linguistic values represented by fuzzy numbers moreover values of alternatives under objective criteria are normalized by a suggested approach meanwhile the membership function of the final fuzzy evaluation value of each alternative can be developed in addition a riemann integral based mean of removals is suggested to rank all the final fuzzy evaluation values for final decision making so that the ranking procedure can be clearly formulated finally a numerical example demonstrates the feasibility of the proposed model

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

a high order schema for the numerical solution of the fractional ordinary differential equations in this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations fodes this technique is based on the so called block by block approach which is a common method for the integral equations in our approach the classical block by block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps while preserving the good stability property of the block by block schemes by using this new approach we are able to construct a high order schema for fodes of the order a a 0 the stability and convergence of the schema is rigorously established we prove that the numerical solution converges to the exact solution with order 3 a for 0 1 a series of numerical examples are provided to support the theoretical claims

a decision support system for multiple criteria alternative ranking using topsis and vikor in fuzzy and nonfuzzy environments abstract topsis and vikor are two well known and widely used multiple attribute decision making methods many extensions of these methods have been proposed that either use different techniques to rank alternatives or utilize fuzzy logic to handle alternatives and criteria that are unquantifiable and or incomplete in this paper we present the implementation of a web based decision support system that incorporates topsis and vikor to solve multicriteria decision making problems either in a nonfuzzy or in a fuzzy environment the aim of this paper is to present a tool that will be used by decision makers to compare various alternative solutions and understand how robust a decision will be the proposed system can be used both in single and in group decision making problems in addition we review several variations for each step of these methods and implement different techniques when applicable hence decision makers can experiment with different techniques in each method we implement ten normalization techniques three methods for the calculation of the ideal and anti ideal solutions fifteen distance metrics five fuzzy distance metrics and eight defuzzification techniques an illustrative example is presented to highlight the key features of the implemented system and the different scenarios that can be built using the proposed dss