optimality conditions for linear programming problems with fuzzy coefficients the optimality conditions for linear programming problems with fuzzy coefficients are derived in this paper two solution concepts are proposed by considering the orderings on the set of all fuzzy numbers the solution concepts proposed in this paper will follow from the similar solution concept called the nondominated solution in the multiobjective programming problem under these settings the optimality conditions will be naturally elicited

multiple objective programming problems with possibilistic coefficients abstract we consider the problem of incorporating possibilistic data into a multiple objective linear programming framework a solution concept that is attractive from the standpoints of feasibility and efficiency is specified a necessary and sufficient condition for such a solution is established and ways to single out some satisfying alternative in this context are described a numerical example is also included for the sake of illustration throughout the paper we indicate topics for further research

maximal l p regularity for the laplacian on lipschitz domains we consider the laplacian with dirichlet or neumann boundary conditions on bounded lipschitz domains both with the following two domains of definition d 1 delta u in w 1 p omega delta u in l p omega bu 0 or d 2 delta u in w 2 p omega bu 0 where b is the boundary operator we prove that under certain restrictions on the range of p these operators generate positive analytic contraction semigroups on lp which implies maximal regularity for the corresponding cauchy problems in particular if is bounded and convex and 1 p leq 2 the laplacian with domain d2 has the maximal regularity property as in the case of smooth domains in the last part we construct an example that proves that in general the dirichlet laplacian with domain d1 is not even a closed operator

the high order compact numerical algorithms for the two dimensional fractional sub diffusion equation in this paper performing the average operators on the space variables a numerical scheme with third order temporal convergence for the two dimensional fractional sub diffusion equation is considered for which the unconditional stability and convergence in l1 l norm are strictly analyzed for 0 0 9569347 by using the discrete energy method therewith adding small perturbation terms we construct a compact alternating direction implicit difference scheme for the two dimensional case finally some numerical results have been given to show the computational efficiency and numerical accuracy of both schemes for all 0 1

global in time behavior of weak solutions to reaction diffusion systems with inhomogeneous dirichlet boundary condition we study reaction diffusion systems describing in particular the evolution of concentrations in general reversible chemical reactions we concentrate on inhomogeneous dirichlet boundary conditions we first prove global existence of very weak solutions then we prove that these although rather weak solutions converge exponentially in l 1 norm toward the homogeneous equilibrium these results are proven by means of l 2 duality arguments and through estimates provided by the nonincreasing entropy

linear programming with a possibilistic objective function abstract this paper proposes some ways for dealing with a linear program when the coefficients of the objective function are subject to possibilistic imprecision i e they are characterized by fuzzy restrictions emphasis is placed upon a passive approach that yields a satisfying solution via an appropriate semi infinite program and an active one that allows to reach a solution with a high possibility level of optimality extensions to the possibilistic constraints case and to the case of multiple objective programming problems with possibilistic coefficients are also hinted we end up with some concluding remarks and indicate axes for further developments

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

data assimilation for linear parabolic equations minimax projection method in this paper we propose a state estimation method for linear parabolic partial differential equations pde that accounts for errors in the model truncation and observations it is based on an extension of the galerkin projection method the extended method models projection coefficients representing the state of the pde in some basis by means of a differential algebraic equation dae the original estimation problem for the pde is then recast as a state estimation problem for the constructed dae using a linear continuous minimax filter we construct a numerical time integrator that preserves the monotonic decay of a nonstationary lyapunov function along the solution to conclude we demonstrate the efficacy of the proposed method by applying it to the tracking of a discharged pollutant slick in a two dimensional fluid

global regularity for divergence form elliptic equations on quasiconvex domains abstract in this paper we introduce a notion of quasiconvex domain and show that the global w 1 p regularity holds on such domains for a wide class of divergence form elliptic equations the modified vitali covering lemma compactness method and the maximal function technique are the main analytical tools

a new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications in the present work first a new fractional numerical differentiation formula called the l1 2 formula to approximate the caputo fractional derivative of order 0 1 is developed it is established by means of the quadratic interpolation approximation using three points t j 2 f t j 2 t j 1 f t j 1 and t j f t j for the integrand f t on each small interval t j 1 t j j 2 while the linear interpolation approximation is applied on the first small interval t 0 t 1 as a result the new formula can be formally viewed as a modification of the classical l1 formula which is obtained by the piecewise linear approximation for f t both the computational efficiency and numerical accuracy of the new formula are superior to that of the l1 formula the coefficients and truncation errors of this formula are discussed in detail two test examples show the numerical accuracy of l1 2 formula second by the new formula two improved finite difference schemes with high order accuracy in time for solving the time fractional sub diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed respectively in addition the application of the new formula into solving fractional ordinary differential equations is also presented several numerical examples are computed the comparison with the corresponding results of finite difference methods by the l1 formula demonstrates that the new l1 2 formula is much more effective and more accurate than the l1 formula when solving time fractional differential equations numerically

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

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

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

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

membership function elicitation in possibilistic programming problems in this paper we propose an elicitation method for membership functions in the setting of possibilistic programming each possibility distribution is defined by several possible domains with different possibility degrees taking advantage of the properties of possibility and necessity measures a nested target table for the objective function and for each constraint is elicited from a decision maker to aggregate all the goals an additional nested target table is established through those nested target tables membership functions can be determined it is emphasized that the information of nested target tables is much more important than that of membership values in the setting of possibilistic programming the membership functions in possibilistic programming are not assumed to be cardinal but ordinal in this sense the decision making via possibilistic programming can be regarded as a qualitative one

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

second order approximations for variable order fractional derivatives fractional calculus allows variable order of fractional operators which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and or time in this paper we derive two second order approximation formulas for the variable order fractional time derivatives involved in anomalous diffusion and wave propagation we then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub diffusion and super diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable order function finally we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super diffusion and also simulations of the burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and or time taken together our results demonstrate that variable order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long standing issues of outflow boundary conditions and monotonicity of integer order pdes

compact difference schemes for the modified anomalous fractional sub diffusion equation and the fractional diffusion wave equation in this paper compact finite difference schemes for the modified anomalous fractional sub diffusion equation and fractional diffusion wave equation are studied schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two based on the idea of weighted and shifted grunwald difference operator we establish schemes with temporal and spatial accuracy order equal to two and four respectively

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

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

a fourth order extrapolated compact difference method for time fractional convection reaction diffusion equations with spatially variable coefficients abstract this paper is concerned with numerical methods for a class of time fractional convection reaction diffusion equations the convection and reaction coefficients of the equation may be spatially variable based on the weighted and shifted grunwald letnikov formula for the time fractional derivative and a compact finite difference approximation for the spatial derivative we establish an unconditionally stable compact difference method the local truncation error and the solvability of the resulting scheme are discussed in detail the stability of the method and its convergence of third order in time and fourth order in space are rigorously proved by the discrete energy method combining this method with a richardson extrapolation we present an extrapolated compact difference method which is fourth order accurate in both time and space a rigorous proof for the convergence of the extrapolation method is given numerical results confirm our theoretical analysis and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method

stability and convergence of fully discrete galerkin fems for the nonlinear thermistor equations in a nonconvex polygon in this paper we establish the unconditional stability and optimal error estimates of a linearized backward euler galerkin finite element method fem for the time dependent nonlinear thermistor equations in a two dimensional nonconvex polygon due to the nonlinearity of the equations and the non smoothness of the solution in a nonconvex polygon the analysis is not straightforward while most previous efforts for problems in nonconvex polygons mainly focused on linear models our theoretical analysis is based on an error splitting proposed in 30 31 together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated time discrete elliptic system in a nonconvex polygon with the proved regularity we establish the stability in l infty l infty l l and the convergence in l infty l 2 l l2 for the fully discrete finite element solution without any restriction on the time step size the approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons numerical results confirm our theoretical analysis and show clearly that no time step condition is needed

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

fuzzy optimization an appraisal abstract this paper takes a general look at core ideas that make up the burgeoning body of fuzzy mathematical programming emphasizing the methodological view although fuzzy mathematical programming has enjoyed a rapidly increasing acceptance within the scientific community some technical hurdles exist to hinder a unanimity reasons for this as well as possible ways for improvement are also discussed

evaluate fuzzy optimization problems based on biobjective programming problems abstract the embedding theorem shows that the set of fuzzy numbers can be embedded into a banach space inspired by this embedding theorem we propose a solution concept of the fuzzy optimization problem by solving a biobjective optimization problem which is transformed from the original fuzzy optimization problem using the embedding theorem we then also consider the fuzzy optimization problem with fuzzy coefficients i e the coefficients are assumed to be fuzzy numbers under a setting of core values of fuzzy numbers we show that the optimal solution of its corresponding crisp optimization problem the usual optimization problem is also an optimal solution of the original fuzzy optimization problem

measurement of membership functions and their acquisition abstract three basic views of the representation of membership functions are reviewed together with fundamental measurement of linguistic terms of linguistic variables the conclusion is that such measurements are either on an ordinal or an interval scale based on whether the appropriate axioms are validated by the empirical data with an allowance for stochastic variation the conjoint measurement is introduced for the case of multi dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct independent linguistic variables it is shown that any composition of distinct independent component linguistic terms of component linguistic variables by any t norm or s norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite once the scale properties of the measurement values of particular terms are validated composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables finally four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement

resolvent estimates in lp for elliptic systems in lipschitz domains abstract we obtain the lp resolvent estimates in lipschitz domains in r n for constant coefficient elliptic systems satisfying the legendre hadamard condition where 1 p for n 3 and 2n n 3 0

a general model for fuzzy linear programming abstract in the current literature there are several models of fuzzy linear programming problems the aim of this paper is first to describe the more important problems in fuzzy linear programming and second to give a general model of fuzzy linear programming problems involving all of the above ones finally a resolution method for that general model is proposed

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

the method of layer potentials for the heat equation in lipschitz cylinders pour l equation de la chaleur dans des cylindres de lipschitz on etudie des problemes aux valeurs limites d initiales avec conditions de neumann et de dirichlet

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

relative modalities and their use in possibilistic linear programming abstract the relative modalities defined by possibility necessity impossibility and contingency of fuzzy sets in modal semantics are investigated decision procedures based on the modal concept are introduced and classified into eight classes the relation between each decision procedure and the attitude of the decision maker toward uncertainty is discussed lastly possibilistic linear programming relying on these decision procedures is formulated and a solution algorithm is proposed a numerical example is given

numerical algorithms for time fractional subdiffusion equation with second order accuracy this article aims to fill in the gap of the second order accurate schemes for the time fractional subdiffusion equation with unconditional stability two fully discrete schemes are first proposed f

fuzzy goal programming with nested priorities abstract this paper proposes a new approach to formulating fuzzy priorities in a goal programming problem the proposed methodology remedies certain shortcomings of the composite membership function approach discussed in previous works 7 10 the principal advantage of the proposed method is that it leads to a formulation in which tradeoffs between goals more closely reflect the decision maker s intentions than in other noninteractive approaches 8 9 10 14 in some of which a fixed hierarchy of goals is assumed

a diffuse interface model and semi implicit energy stable finite element method for two phase magnetohydrodynamic flows abstract in this paper we propose a diffuse interface model and finite element approximation for two phase magnetohydrodynamic mhd flows with different viscosities and electric conductivities an energy stable scheme which is based on the finite element method for the spatial discretization and first order semi implicit scheme combined with convex splitting method for the temporal discretization is proposed to solve this new model the numerical scheme is proved to be mass conservative and energy law preserving by leray schauder fixed point theorem the existence of solutions to the numerical scheme is shown the uniqueness of the numerical solutions is obtained utilizing the stability of the numerical scheme and the compactness method the existence of the weak solutions to the two phase mhd model is established as well furthermore given more regularity on the weak solution the convergence of the numerical scheme is derived finally numerical experiments are provided to verify the theoretical results and validate the proposed model

possibilistic linear programming with triangular fuzzy numbers abstract we consider a mathematical programming problem where all the parameters may be fuzzy variables specified by their possibility distribution and we define the possibility distribution of the objective function the ideas are illustrated on possibilistic linear programming

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 new difference scheme for the time fractional diffusion equation abstract in this paper we construct a new difference analog of the caputo fractional derivative called the l 2 1 formula the basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered stability of the suggested schemes and also their convergence in the grid l 2 norm with the rate equal to the order of the approximation error are proved the obtained results are supported by the numerical calculations carried out for some test problems

adaptive wavelet schemes for parabolic problems sparse matrices and numerical results a simultaneous space time variational formulation of a parabolic evolution problem is solved with an adaptive wavelet method this method is shown to converge with the best possible rate in linear complexity thanks to the use of tensor product bases there is no penalty in complexity due to the additional time dimension special wavelets are designed such that the bi infinite system matrix is sparse this sparsity largely simplifies the implementation and improves the quantitative properties of the adaptive wavelet method numerical results for an ode and the heat equation are presented

h infty calculus for the sum of non commuting operators a recent result of kalton and weis is extended to the case of non commuting operators employing the commutator condition of labbas and terreni or of da prato and grisvard under appropriate assumptions it is shown that the sum of two non commuting operators admits an h calculus the main results are then applied to a parabolic problem on a wedge domain

a fast accurate approximation method with multigrid solver for two dimensional fractional sub diffusion equation a fast accurate approximation method with multigrid solver is proposed to solve a two dimensional fractional sub diffusion equation using the finite difference discretization of fractional time derivative a block lower triangular toeplitz matrix is obtained where each main diagonal block contains a two dimensional matrix for the laplacian operator our idea is to make use of the block circulant approximation via fast fourier transforms so that the resulting task is to solve a block diagonal system where each diagonal block matrix is the sum of a complex scalar times the identity matrix and a laplacian matrix we show that the accuracy of the approximation scheme is of o because of the special diagonal block structure we employ the multigrid method to solve the resulting linear systems the convergence of the multigrid method is studied numerical examples are presented to illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver an approximation method is proposed to approximate the original block lower triangular toeplitz linear system sufficient condition is given to guarantee a high accuracy of the approximation method multigrid solver is employed to solve the resulting block diagonal linear system multigrid solver is proven to be linear convergence numerical examples illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver

possibility and necessity in optimization abstract we employ possibility and necessity in fuzzy programming and in other optimization problems that traditionally use random variables the possibility approach to multiobjective programming produces the same results as fuzzy programming a detailed example is presented to illustrate the use of possibility and necessity in optimization problems that usually employ random variables to describe uncertainty

non polynomial spline approach in two dimensional fractional sub diffusion problems abstract in this paper we propose a new numerical scheme for two dimensional fractional sub diffusion problems using non polynomial spline the solvability stability and convergence of the proposed method are established using the well known discrete energy methodology it is shown that the spatial convergence order is at least 4 5 which improves the best result achieved to date we also carry out simulation to demonstrate the accuracy and efficiency of the proposed scheme and to compare with other methods

two alternating direction implicit difference schemes with the extrapolation method for the two dimensional distributed order differential equations the grunwald formula is used to solve the two dimensional distributed order differential equations two alternating direction implicit adi difference schemes are derived it is proved that the schemes are unconditionally stable and convergent with the convergence orders o h 1 2 h 2 2 2 and o h 1 4 h 2 4 4 in h 1 norm respectively where h 1 h 2 and are the step sizes in time space in x and y direction and distributed order the extrapolation method is applied to improve the approximate accuracy to the orders o 2 ln 2 h 1 2 h 2 2 2 and o 2 ln 2 h 1 4 h 2 4 4 respectively several numerical examples are given to confirm the theoretical results

modality constrained programming problems a unified approach to fuzzy mathematical programming problems in the setting of possibility theory abstract in this paper fuzzy mathematical programming problems are formulated based on the idea analogous with the chance constrained programming problem the difference in meaning between the ambiguity of the coefficients and that of the decision maker s preference is emphasized the constraints with fuzzy coefficients are treated as the restriction that should be satisfied properly rather than perfectly the objective functions with fuzzy coefficients are treated variously depending on the interpretations i e the optimization of the modalities the optimization of the fractile the minimization of the ambiguity and so forth the deterministic equivalent constraints and the deterministic equivalent problems are shown when the constraints and the objective functions are linear a numerical example is given to illustrate the proposed formulations

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

a novel high order space time spectral method for the time fractional fokker planck equation the fractional fokker planck equation is an important physical model for simulating anomalous diffusions with external forces because of the nonlocal property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations in this paper a space time spectral method is presented for the numerical solution of the time fractional fokker planck initial boundary value problem the proposed method employs the jacobi polynomials for the temporal discretization and fourier like basis functions for the spatial discretization due to the diagonalizable trait of the fourier like basis functions this leads to a reduced representation of the inner product in the galerkin analysis we prove that the time fractional fokker planck equation attains the same approximation order as the time fractional diffusion equation developed in x li and c xu siam j numer anal 47 2009 pp 2108 2131 by using the present method that indicates an e

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

some properties of extended fuzzy preference relations using modalities abstract a fuzzy preference relation is extended to six kinds of relations between fuzzy sets the properties of the six extended relations are investigated especially a sufficient condition for reducing the extended relations to four kinds is discussed when fuzzy sets are bounded and normal and some examples are given to illustrate it further a sufficient condition for the extended negative relations to coincide with the extended inverse relations is described

finite difference methods and a fourier analysis for the fractional reaction subdiffusion equation various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations fpde for example systems biology chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments however effective numerical methods and numerical analysis for fpde are still in their infancy in this paper we consider a fractional reaction subdiffusion equation fr subde in which both the motion and the reaction terms are affected by the subdiffusive character of the process using the relationship between the riemann liouville and grunwald letnikov definitions of fractional derivatives an implicit and an explicit difference methods for the fr subde are presented the stability and the convergence of the two numerical methods are investigated by a fourier analysis the solvability of the implicit finite difference method is also proved the high accuracy algorithm is structured using richardson extrapolation finally a comparison between the exact solution and the two numerical solutions is given the numerical results are in excellent agreement with our theoretical analysis