interactive decision making for multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers abstract in this paper by considering the experts vague or fuzzy understanding of the nature of the parameters in the problem formulation process multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated through the use of the level sets of fuzzy numbers an extended pareto optimality concept called the pareto optimality is introduced to generate a candidate for the satisficing solution which is also pareto optimal the decision maker is asked to specify the degree and the reference objective values it is shown that the corresponding pareto optimal solution can be easily obtained by solving the minimax problems for which the dantzig wolfe decomposition method and ritter s partitioning procedure are applicable then a linear programming based interactive decision making method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an pareto optimal solution set is presented an illustrative numerical example is provided to demonstrate the feasibility of the proposed method

interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

nonlinear manifold learning for meshfree finite deformation thin shell analysis calculations on general point set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges the absence of a mesh makes it nontrivial to decide if two neighboring points in the three dimensional embedding are nearby or rather far apart on the manifold furthermore the topology of surfaces is generally not that of an open two dimensional set ruling out global parametrizations we propose a general and simple numerical method analogous to the mathematical theory of manifolds in which the point set surface is described by a set of overlapping charts forming a complete atlas we proceed in four steps 1 partitioning of the node set into subregions of trivial topology 2 automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods 3 parametrization of the surface using smooth meshfree here maximum entropy approximants and 4 gluing together the patch representations by means of a partition of unity each patch may be viewed as a meshfree macro element we exemplify the generality flexibility and accuracy of the proposed approach by numerically approximating the geometrically nonlinear kirchhoff love theory of thin shells we analyze standard benchmark tests as well as point set surfaces of complex geometry and topology

fuzzy mathematical programming for multi objective linear fractional programming problem the present paper deals with a solution procedure for multi objective linear fractional programming problems an equivalent multi objective linear programming form of the problem has been formulated in the proposed methodology using fuzzy set theoretic approach a procedure has been explored the proposed solution procedure has also been used to solve numerical examples

a locking free meshfree method for the simulation of shear deformable plates based on a mixed variational formulation abstract the problem of shear locking in the thin plate limit is a well known issue that must be overcome when discretising the reissner mindlin plate equations in this paper we present a shear locking free method utilising meshfree maximum entropy basis functions and rotated raviart thomas nedelec elements within a mixed variational formulation the formulation draws upon well known techniques in the finite element literature due to the inherent properties of the maximum entropy basis functions our method allows for the direct imposition of dirichlet essential boundary conditions in contrast to methods based on moving least squares basis functions we present benchmark problems that demonstrate the accuracy and performance of the proposed method

runge kutta nystrom triples abstract the embedded runge kutta nystrom process for the numerical solution of the special second order initial value problem can be extended by the addition of dense formulae for both y and y which yield solutions at points within the normal step intervals as for runge kutta methods two modes of implementation are possible and it is shown that these are equivalent under certain conditions the cases of dense formulae applied to rkn4 3 and rkn6 4 families are considered and test results are presented

continuation of periodic solutions in the waveguide array mode locked laser abstract we apply the adjoint continuation method to construct highly accurate periodic solutions that are observed to play a critical role in the multi pulsing transition of mode locked laser cavities the method allows for the construction of solution branches and the identification of their bifurcation structure supplementing the adjoint continuation method with a computation of the floquet multipliers allows for explicit determination of the stability of each branch this method reveals that when gain is increased the multi pulsing transition starts with a hopf bifurcation followed by a period doubling bifurcation and a saddle node bifurcation for limit cycles finally the system exhibits chaotic dynamics and transitions to the double pulse solutions although this method is applied specifically to the waveguide array mode locking model the multi pulsing transition is conjectured to be ubiquitous and these results agree with experimental and computational results from other models

modified moving least squares with polynomial bases for scattered data approximation one common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes to tackle such problems we present a modified robust moving least squares mls method for scattered data smoothing and approximation the error functional used in the derivation of the classical mls approximation is augmented with additional terms based on the coefficients of the polynomial base functions this allows quadratic base functions to be used with the same size of the support domain as linear base functions resulting in better approximation capability the increased robustness of the modified mls method to irregular nodal distributions makes it suitable for use across many fields the analysis is supported by several univariate and bivariate examples

quadratic maximum entropy serendipity shape functions for arbitrary planar polygons in this paper we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons drawing on the work of bompadre et al 2012 1 and hormann and sukumar 2008 2 we adopt a relative entropy measure for signed positive or negative shape functions with nodal prior weight functions that have the appropriate zero set on the boundary of the polygon we maximize the objective functional subject to the constraints for quadratic completeness proposed by rand et al 2013 3 along an edge of a polygon the approximation is identical to univariate bernstein polynomials the choice of the nodal prior weight function ensures that the shape functions satisfy a weak kronecker delta property on each edge the shape functions are well defined for arbitrary planar polygons without self intersections on using a modified numerical integration scheme we show that the quadratic patch test is passed on polygonal meshes with convex and nonconvex elements numerical tests for the poisson equation on self similar trapezoidal meshes and quasiuniform polygonal meshes are presented which reveal the sound accuracy of the method and optimal rates of convergence in the l2l2 norm and the h1h1 seminorm are established

a higher order method based on local maximum entropy approximation we present here a generalization of local maximum entropy max ent approximation for high orders of consistency i e quadratic cubic the method is based upon the application of the de boor s algorithm to the standard linear local max ent approximation the resulting approximation possesses some interesting properties such as non negativity smoothness exact interpolation on the boundary and variation diminishing no gibbs effect the resulting structure has many similarities with b spline surfaces but without the tensor product structure typical of that approximation examples are provided of its use in the framework of a galerkin method showing the potential of the proposed method in solving boundary value problems copyright 2010 john wiley sons ltd

a fifth order interpolant for the dornand and prince runge kutta method abstract a family of fifth order interpolants for the fifth order solution provided by the dormand and prince runge kutta pair rk5 4 7m which requires two additional function evaluations per step is presented an optimal interpolant in this family has been determined by choosing the parameters to minimize the leading coefficients of the local truncation error of the continuous solution some numerical experiments with the nonstiff detest problems show that the proposed optimal method has a good interpolatory behavior

fuzzy approaches for multiple criteria linear fractional optimization a comment abstract in this note a fallacy of the second solution procedure for fuzzy multiple criteria linear fractional optimization described by luhandjula 1984 is noted

sensitivity analysis in fuzzy multiobjective linear fractional programming problem in this paper we study measurement of sensitivity for changes of violations in the aspiration level for the fuzzy multiobjective linear fractional programming problem

interactive fuzzy programming for two level linear fractional programming problems this paper presents interactive fuzzy programming for two level linear fractional programming problems with the essentially cooperative decision makers in our interactive method after determining the fuzzy goals of the decision makers at both levels a satisfactory solution is efficiently derived by updating the satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels in an interactive process optimal solutions to the formulated programming problems are obtained by combined use of the bisection method and the phase one of linear programming and the variable transformation by charnes and cooper an illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method

an adaptive meshfree method for phase field models of biomembranes part i we present an adaptive meshfree method for phase field models of biomembranes max ent smooth approximants can deal with the high order phase field equations the method resolves adaptively the models at an affordable computational cost convergence to sharp interface limit is shown with small regularization parameters we present an adaptive meshfree method to approximate phase field models of biomembranes in such models the helfrich curvature elastic energy the surface area and the enclosed volume of a vesicle are written as functionals of a continuous phase field which describes the interface in a smeared manner such functionals involve up to second order spatial derivatives of the phase field leading to fourth order euler lagrange partial differential equations pde the solutions develop sharp internal layers in the vicinity of the putative interface and are nearly constant elsewhere thanks to the smoothness of the local maximum entropy max ent meshfree basis functions we approximate numerically this high order phase field model with a direct ritz galerkin method the flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions thus the proposed approach is more efficient than common tensor product methods e g finite differences or spectral methods and simpler than unstructured c 0 finite element methods applicable by reformulating the model as a system of second order pde the proposed method implemented here under the assumption of axisymmetry allows us to show numerical evidence of convergence of the phase field solutions to the sharp interface limit as the regularization parameter approaches zero in a companion paper we present a lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid

multiple objective linear fractional programming this paper presents a simplex based solution procedure for the multiple objective linear fractional programming problem by 1 departing slightly from the traditional notion of efficiency and 2 augmenting the feasible region as in goal programming the solution procedure solves for all weakly efficient vertices of the augmented feasible region the article discusses the difficulties that must be addressed in multiple objective linear fractional programming and motivates the solution algorithm that is developed

deriving the continuity of maximum entropy basis functions via variational analysis in this paper we prove the continuity of maximum entropy basis functions using variational analysis techniques the use of information theoretic variational principles to derive basis functions is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints for a set of distinct nodes x i i 1 n in mathbb r d the convex approximation of a function u x is u h x sum i 1 n p i x u i where p i i 1 n are nonnegative basis functions and u h x must reproduce affine functions sum i 1 n p i x 1 sum i 1 n p i x x i x given these constraints we compute p i x by minimizing the relative entropy functional kullback leibler distance d p m sum i 1 n p i x ln p i x m i x where m i x is a known prior weight function distribution to prove the continuity of the basis functions we appeal to the theory of epiconvergence

a parametric approach to description of linguistic values of variables and hedges abstract an experimental approach is described to obtaining membership functions which describe the values of linhuistic estimates of variables and mathematical tools for description of hedges and computation of composite linguistic estimates applied aspects of using the proposed approach are discussed

blending isogeometric analysis and local maximum entropy meshfree approximants abstract we present a method to blend local maximum entropy lme meshfree approximants and isogeometric analysis the coupling strategy exploits the optimization program behind lme approximation treats isogeometric and lme basis functions on an equal footing in the reproducibility constraints but views the former as data in the constrained minimization the resulting scheme exploits the best features and overcomes the main drawbacks of each of these approximants indeed it preserves the high fidelity boundary representation exact cad geometry of isogeometric analysis out of reach for bare meshfree methods and easily handles volume discretization and unstructured grids with possibly local refinement while maintaining the smoothness and non negativity of the basis functions we implement the method with b splines in two dimensions but the procedure carries over to higher spatial dimensions or to other non negative approximants such as nurbs or subdivision schemes the performance of the method is illustrated with the heat equation and linear and nonlinear elasticity the ability of the proposed method to impose directly essential boundary conditions in non convex domains and to deal with unstructured grids and local refinement in domains of complex geometry and topology is highlighted by the numerical examples

a contrast of direct and transformed nystro m pairs abstract explicit nystrom pairs provide an efficient way to solve the general second order initial value problem when stability requirements do not determine the stepsize the pairs can be derived by explicitly solving the order conditions for nystrom pairs or by applying a transformation to runge kutta pairs the two approaches lead to different classes of pairs we refer to the two types of pairs as direct and transformed pairs respectively even though transformed pairs are easier to derive existing nystrom pairs are direct pairs in this paper we compare the properties of classes of direct and transformed p 1 p pairs p 4 5 6 the properties are the efficiency of the pair the accuracy of the local error estimate and stepsize prediction and the size of the stability region we also investigate the existence of an interpolation scheme for the 3 4 pairs

taylor series approach to fuzzy multiobjective linear fractional programming this paper presents the use of a taylor series for fuzzy multiobjective linear fractional programming problems fmolfp the taylor series is a series expansion that a representation of a function in the proposed approach membership functions associated with each objective of fuzzy multiobjective linear fractional programming problem transformed by using a taylor series are unified thus the problem is reduced to a single objective practical applications and numerical examples are used in order to show the efficiency and superiority of the proposed approach

neural networks with multidimensional transfer functions we present a new type of neural network nn where the data for the input layer are the value x spl epsiv r the vector y spl epsiv r sup m associated to an initial value problem ivp with y x f y x and a steplength h then the stages of a runge kutta rk method with trainable coefficients are used as hidden layers for the integration of the ivp using f as transfer function we take as output two estimations y y spl circ of ivp at the point x h training the rk method at some test problems and counting the cost of the method under the coefficients used we may achieve coefficients that help the method to perform better at a wider class of problems

numerical approaches for solution of differential equations on manifolds numerical approaches for the solution of vector fields differential equations defined on a manifold have attracted wide attention over the past few years this paper first reviews the various numerical approaches available in the literature for the solution of vector fields namely parameterization approach constraint stabilization approach and perturbation approach pa in the process the paper also makes the following useful contributions an expanded analysis and a new perturbation scheme for the pa and a new way of choosing integration error tolerances for the parameterization approach a comparison of all the approaches is carried out both by means of a crude cost analysis as well as by studying their numerical performance on examples of vector fields arising from constrained mechanical systems cms based on this comparison recommendations are made for a proper choice of a suitable approach overall the pa performs better than the other approaches

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

a stabilized nodally integrated tetrahedral a stabilized nodally integrated linear tetrahedral is formulated and analysed it is well known that linear tetrahedral elements perform poorly in problems with plasticity nearly incompressible materials and acute bending for a variety of reasons low order tetrahedral elements are preferable to quadratic tetrahedral elements particularly for nonlinear problems but the severe locking problems of tetrahedrals have forced analysts to employ hexahedral formulations for most nonlinear problems on the other hand automatic mesh generation is often not feasible for building many 3d hexahedral meshes a stabilized nodally integrated linear tetrahedral is developed and shown to perform very well in problems with plasticity nearly incompressible materials and acute bending the formulation is analytically and numerically shown to be stable and optimally convergent for the compressible case provided sufficient smoothness of the exact solution u c2 h1 3 future work may extend the formulation to the incompressible regime and relax the regularity requirements nonetheless the results demonstrate that the method is not susceptible to locking and performs quite well in several standard linear and nonlinear benchmarks published in 2006 by john wiley sons ltd

starting bdf and adams codes at optimal order abstract we present an algorithm based on runge kutta formulas with interpolation to start bdf and adams multistep codes at optimal order and step size it first finds automatically and reliably an on scale initial step size besides being convenient for the user this results in a more robust and efficient start especially for stiff problems because a starting step is taken with an efficient rk formula of moderate order before switching to the multistep formula starting the multistep formula at optimal order and step size makes the integration more efficient and reduces difficulties due to mesh distortion we present the results of a variety of numerical experiments which demonstrate the efficacy of our technique

optimisation problems as decision problems the case of fuzzy optimisation problems abstract the importance that decision making problems and optimisation problems have today in all aspects of life is beyond all doubt despite that importance both problems tend to be thought of as following different routes when they have in fact a symbiotic relation here we consider the different decision problems that arise when different kinds of information and framework of behaviour are considered and we explore the corresponding optimisation problems that can be derived for searching the best possible decision we explore the case where fuzzy mathematical programming problems are obtained as well as other new ones in the fuzzy context

a goal programming procedure for fuzzy multiobjective linear fractional programming problem abstract this paper presents a goal programming gp procedure for fuzzy multiobjective linear fractional programming fmolfp problems in the proposed approach which is motivated by mohamed fuzzy sets and systems 89 1997 215 gp model for achievement of the highest membership value of each of fuzzy goals defined for the fractional objectives is formulated in the solution process the method of variable change on the under and over deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming lgp methodology the approach is illustrated by two numerical examples

an efficient runge kutta 4 5 pair abstract a pair of explicit runge kutta formulas of orders 4 and 5 is derived it is significantly more efficient than the fehlberg and dormand prince pairs and by standard measures it is of at least as high quality there are two independent estimates of the local error the local error of the interpolant is to leading order a problem independent function of the local error at the end of the step

goal programming with linear fractional criteria abstract in this paper we present an extension of goal programming to include linear fractional criteria the extension forms a natural link between goal programming gp and multiple objective linear fractional programming molfp

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

a new embedded pair of runge kutta formulas of orders 5 and 6 a new pair of embedded runge kutta rk formulas of orders 5 and 6 is presented it is derived from a family of rk methods depending on eight parameters by using certain measures of accuracy and stability numerical tests comparing its efficiency to other formulas of the same order in current use are presented with an extra function evaluation per step a c 1 continuous interpolant of order 5 can be obtained

some practical runge kutta formulas a new selection is made of the most practical of the many explicit runge kutta formulas of order 4 which have been proposed a new formula is considered formulas are modified to improve their quality and efficiency in agreement with improved understanding of the issues and formulas are derived which permit interpolation it is possible to do a lot better than the pair of fehlberg currently regarded as best

additive runge kutta methods for stiff ordinary differential equations certain pairs of runge kutta methods may be used additively to solve a system of n differential equations x j t x g t x pairs of methods of order p 4 where one method is semiexplicit and a stable and the other method is explicit are obtained these methods require the lu factorization of one n x n matrix and p evaluations of g in each step it is shown that such methods have a stability property which is similar to a stability property of perturbed linear differential equations 1 introduction in a recent article 2 the authors showed that certain pairs of methods may be used in an additive fashion to solve an initial value problem for a system of n differential equations x f t x x a xo a t b

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

an interactive fuzzy satisficing method for multiobjective linear fractional programming problems abstract we present a new interactive fuzzy decision making method for solving multiobjective linear fractional programming problems by assuming that the decision maker dm has fuzzy goals for each of the objective functions through the interaction with the dm the fuzzy goals of the dm are quantified by eliciting the corresponding membership functions including nonlinear functions after determining the membership functions if the dm specifies reference membership values the minimax problem is solved by combined use of the bisection method and the linear programming method and the dm is supplied with the corresponding pareto optimal solution together with the trade off rates between the membership functions then by considering the current values of the membership functions as well as the trade off rates the dm responds by updating his her reference membership values in this way the compromise or satisficing solution for the dm can be derived efficiently from among a pareto optimal solution set by updating his her reference membership values on the basis of the proposed method a time sharing computer program is written and an illustrative numerical example is demonstrated along with the computer outputs

symbolic derivation of runge kutta nystrom type order conditions and methods for solving y f x y abstract in this work we study the runge kutta nystrom rkn type methods for the solution of a special third order initial value problems based on rooted trees the relative order conditions theory is presented introducing a new set of sn trees named 3 whose elements enumeration is given a mathematica package that furnishes instantly order conditions of high orders is also listed finally a new method of order 8 is constructed that outperforms by far the methods found in the literature

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

fuzzy approaches for multiple objective linear fractional optimization abstract the problem of finding a solution to a multiple objective linear fractional program arises in several real world situations in this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness after representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program techniques of fuzzy linear programming may be used to reach a satisfactory solution it is shown that under reasonable restrictions this solution is efficient pareto optimal for the original problem numerical examples are also included for illustration

computing nearly singular solutions using pseudo spectral methods in this paper we investigate the performance of pseudo spectral methods in computing nearly singular solutions of fluid dynamics equations we consider two different ways of removing the aliasing errors in a pseudo spectral method the first one is the traditional 2 3 dealiasing rule the second one is a high 36th order fourier smoothing which keeps a significant portion of the fourier modes beyond the 2 3 cut off point in the fourier spectrum for the 2 3 dealiasing method both the 1d burgers equation and the 3d incompressible euler equations are considered we demonstrate that the pseudo spectral method with the high order fourier smoothing gives a much better performance than the pseudo spectral method with the 2 3 dealiasing rule moreover we show that the high order fourier smoothing method captures about 12 15 more effective fourier modes in each dimension than the 2 3 dealiasing method for the 3d euler equations the gain in the effective fourier codes for the high order fourier smoothing method can be as large as 20 over the 2 3 dealiasing method another interesting observation is that the error produced by the high order fourier smoothing method is highly localized near the region where the solution is most singular while the 2 3 dealiasing method tends to produce oscillations in the entire domain the high order fourier smoothing method is also found be very stable dynamically no high frequency instability has been observed in the case of the 3d euler equations the energy is conserved up to at least six digits of accuracy throughout the computations 2007 elsevier inc all rights reserved

fourth and fifth order scaled rungs kutta algorithms for treating dense output new runge kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step as well as at the end of each step

an interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers abstract in this paper by considering the experts vague or fuzzy understanding of the nature of the parameters in the problem formulation process multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated using the a level sets of fuzzy numbers the corresponding nonfuzzy a multiobjective linear fractional programming problem is introduced the fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones through the introduction of extended pareto optimality concepts if the decision maker specifies the degree a and the reference membership values the corresponding extended pareto optimal solution can be obtained by solving the minimax problems for which the dantzig wolfe decomposition method and ritter s partitioning procedure are applicable then a linear programming based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended pareto optimal solution set is presented an illustrative numerical example is provided to demonstrate the feasibility of the proposed method

high phase lag order runge kutta and nystrom pairs we exploit the freedom in the selection of the free parameters of one family of eighth algebraic order runge kutta rk pairs and of three families of fourth sixth and eighth order rk nystrom rkn pairs with the purpose of obtaining specific pairs of the highest possible phase lag order which are also characterized by minimized principal truncation error coefficients we present a method for the analytic derivation of the dissipation order conditions for rk methods and the phase lag and dissipation order conditions for nystrom methods the rk pairs we study here are based on a one parameter generalization of some older families of pairs an algorithm and specific optimized 8 6 nystrom pairs are also provided for a class of initial value problems whose solution is known to be described by free oscillations or free oscillations of low frequency with forced oscillations of high frequency superimposed over long integration intervals these new pairs seem to offer some advantages with respect to some older pairs the latter are of the same algebraic orders as the new ones but are characterized by the minimal phase lag order according to their algebraic order and number of stages

on the optimum support size in meshfree methods a variational adaptivity approach with maximum entropy approximants we present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle the method is based on a variational approach and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters here those defining the support size of each basis function we consider local maximum entropy approximation schemes which exhibit smooth basis functions with respect to both space and the discretization parameters the node location and the locality parameters we illustrate by the poisson linear and non linear elasticity problems the effectivity of the method which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions copyright 2009 john wiley sons ltd

a family of parallel runge kutta pairs abstract increasing availability of parallel computers has recently spurred a substantial amount of research concerned with designing explicit runge kutta methods to be implemented on such computers here we discuss a family of methods that require fewer processors than methods presently available do still achieving a similar speed up in particular 5 6 and 6 7 pairs are derived that require a minimum number of function evaluations on two and three processors respectively

multiple objective linear fractional programming a fuzzy set theoretic approach abstract a method for optimizing a multiobjective linear fractional programming problem is developed which yields always an efficient solution

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters in this paper we present interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters using the level sets of fuzzy parameters the corresponding nonfuzzy two level linear fractional programming problem is introduced in our interactive method after determining fuzzy goals of decision makers at both levels a satisfactory solution is derived efficiently by updating a minimal satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels the satisfactory solution well balanced between both levels is easily computed by combined use of the bisection method the phase one of the simplex method and the variable transformation method by charnes and cooper an illustrative numerical example for two level linear fractional programming problems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method

information flux method a meshfree maximum entropy petrov galerkin method including stabilised finite element methods an information flux method incorporating a novel approach to stable methods is proposed the method may be considered as a meshfree petrov galerkin approximation scheme with basis functions based on the principle of maximum entropy the two goals of accuracy and stability are distinctly assigned to solution and weighting functions respectively it is emphasised that stability can be ensured if the weighting functions are chosen such that they resemble the information flux of the underlying physical problem in this study the proposed method is applied to convection dominated convection diffusion problems a seamless transition of the proposed method to stabilised finite element methods is demonstrated for increasing locality of the basis functions in one dimension for lower localities a superior convergence behaviour can be shown compared to stabilised finite elements the method is presented and discussed for the general multi dimensional case with numerical results shown for one and two dimensional problems

linear programming under randomness and fuzziness this paper is concerned with the use of fuzzy sets theory well suited for nuanced human behavior to extend the applicability of linear programming in a way to let both randomness and fuzziness uncertainties be taken into account a fuzzy stochastic linear programming problem is considered and ways to handle it suggested the basic idea behind our approaches is to translate the original problem in deterministic terms via tools provided by fuzzy sets theory to get a classical one some conditions for the resulting problems to be solved without extra computational effort are stressed and illustrative examples given

a goal programming procedure for solving problems with multiple fuzzy goals using dynamic programming abstract this paper describes how the preemptive priority based goal programming gp can be used to solve a class of fuzzy programming fp problems with the characteristics of dynamic programming dp in the proposed approach the membership functions of the objective goals of a problem with fuzzy aspiration levels are defined first then under the framework of preemptive priority based gp a multi stage dp model of the problem for achievement of the highest degree unity of each of the membership functions is developed in the decision process the goal satisficing philosophy of gp is used recursively to arrive at the most satisfactory solution two numerical examples are provided to illustrate the approach

global error estimation with runge kutta triples abstract the applications of runge kutta rk interpolation to global error estimation using the zadunaisky and related techniques are considered it is shown that the pseudo problem can be based on dense output values within any one step and reliable global error estimates can be obtained at the integration mesh points by using special rk formulae some special formulae of orders 2 6 are presented together with numerical results

applications of fuzzy set theory to mathematical programming mathematical programming is one of the areas to which fuzzy set theory has been applied extensively primarily based on bellman and zadeh s model of decision in fuzzy environments models have been suggested which allow flexibility in constraints and fuzziness in the objective function in traditional linear and nonlinear programming in integer and fractional programming and in dynamic programming these models in turn have been used to offer computationally efficient approaches for solving vector maximum problems this paper surveys major models and theories in this area and offers some indication on future developments which can be expected

meshfree co rotational formulation for two dimensional continua in this paper a meshfree co rotational formulation for two dimensional continua is proposed in a co rotational formulation the motion of a body is separated into rigid motion and strain producing deformation traditionally this has been done in the setting of finite elements for beams and shell type elements in the present work every node in a meshfree discretized domain has its own co rotating coordinate system three key ingredients are established in order to apply the co rotational formulation i the relationship between global and local variables ii the angle of rotation of a typical co rotating coordinate system and iii a variationally consistent tangent stiffness matrix an algorithm for the co rotational formulation based on load control is provided maximum entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations numerical examples are presented to demonstrate the validity of the meshfree co rotational formulation copyright 2009 john wiley sons ltd

cell based maximum entropy approximants in this paper we devise cell based maximum entropy max ent basis functions that are used in a galerkin method for the solution of partial differential equations the motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes in the variational scheme to obtain max ent basis functions the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in r2 more precisely we take powers of the composition of r functions via boolean operations the basis functions so constructed are nonnegative smooth linearly complete and compactly supported in a neighbor ring of segments that enclose each node the smoothness is controlled by two positive integer parameters the normalization order of the approximation of the distance function and the power to which it is raised the properties and mathematical foundations of the new compactly supported approximants are described and its use to solve two dimensional elliptic boundary value problems poisson equation and linear elasticity is demonstrated the sound accuracy and the optimal rates of convergence of the method in sobolev norms are established

compensatory operators in fuzzy linear programming with multiple objectives previously h j zimmermann 11 presented a fuzzy approach to solve the linear vectormaximum problem in the mentioned approach the author uses the minimum or the product operators to translate the semantic meaning of the and the aim of this paper is to reconsider this approach by using operators which allow some degree of compensation between aggregated membership functions firstly the c operator already defined in 12 is used to combine our fuzzy objectives it is shown that solutions obtained by this way are always efficient unfortunately the substitute problem is not generally computationally feasible we suggest then the min bounded sum operator which is a compensatory one has some desired properties and is supported by empirical arguments it allows a considerable decrease of the computational burden in the substitute problem and leads to a solution which is attractive from the stand point of efficiency

optimal transportation meshfree approximation schemes for fluid and plastic flows we develop an optimal transportation meshfree otm method for simulating general solid and fluid flows including fluid structure interaction the method combines concepts from optimal transportation theory with material point sampling and max ent meshfree interpolation the proposed otm method generalizes the benamou brenier differential formulation of optimal mass transportation problems to problems including arbitrary geometries and constitutive behavior the otm method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities the otm method exactly conserves linear and angular momentum and its convergence characteristics are verified in standard benchmark problems we illustrate the range and scope of the method by means of two examples of application the bouncing of a gas filled balloon off a rigid wall and the classical taylor anvil benchmark test extended to the hypervelocity range

conditions for bounded solutions of non markovian quantum master equations for the most general quantum master equations also called nakajima zwanzig or non markovian equations we define suitable boundedness conditions on integral kernels and inhomogeneity terms in order to derive with mathematical rigor an upper bound on solutions as required by the von neumann conditions such equations are of importance for quantum dynamics of open systems with arbitrary couplings to environment and arbitrary entangled initial states the derivation is based on an equivalent coherence vector representation in finite dimension n leading to coupled volterra integro differential equations of second kind and convolution type in an n2 1 dimensional real vector space as examples analytical and numerical model solutions are worked out for 2 level systems in order to test suitable trial functions for input quantities all this is motivated by the fact that exact solutions can hardly be found but appropriate trial functions may provide a reasonable semiphenomenological description of complicated quantum dynamics

fuzzy linear fractional goal programming applied to refinery operations planning abstract chemical process plant operations planning in an oil refinery has been modelled as fuzzy linear fractional multiple goal programming problem a variant of the fuzzy linear fractional goal programming model of dutta et al fuzzy sets and systems 52 1992 39 45 has been developed and used in the present study to solve the problem which has two fuzzy fractional goals and 22 crisp constraints linear membership functions have been assumed and min operator has been used as the aggregator linguistic variable has been taken as close throughout the paper following dutta et al 1992 a software has been developed in turbo c to implement the model the software takes the parameters defining the fuzziness and crisp data from the input field and solves the crisp equivalent to the fuzzy linear fractional goal programming problem the parameters defining fuzziness are subjectively chosen by the user decision maker the software finally outputs satisfying decision variables satisfying fuzzy fractional goal values optimal values of the profit numerator and the capacities of the primary unit and cracker denominators which are involved in the fuzzy fractional goals the results indicate that the present model yielded a more efficient solution compared to the crisp solution of allen british chem eng 16 1971 685 691 as far as the fractional objectives are concerned this demonstrates the efficacy of the present model however a paltry decrement of 1 3 in the profit is noticed in the present study in comparison to the crisp solution of allen 1971

on a fuzzy set approach to solving multiple objective linear fractional programming problem in 1984 luhandjula used a linguistic variable approach in order to present a procedure for solving multiple objective linear fractional programming problem molfpp in 1992 dutta et al fuzzy sets and systems 52 1 1992 39 45 modified the linguistic approach of luhandjula such as to obtain efficient solution to problem molfpp the aim of this paper is to point out certain shortcomings in the work of dutta et al and give the correct proof of theorem which validates the obtaining of the efficient solutions we notice that the method presented there as a general one does only work efficiently if certain hypotheses restrictive enough and hardly verified are satisfied the example considered by dutta et al is again used to illustrate the approach

interactive fuzzy decision making for generalized multiobjective linear fractional programming problems with fuzzy parameters abstract in this paper we focus on multiobjective linear fractional programming problems with fuzzy parameters and present a new interactive fuzzy decision making method for obtaining the satisficing solution of the decision maker dm on the basis of the linear programming method the fuzzy parameters in the description of the objective functions and the constraints which reflect the expert s ambiguous understanding of the nature of the parameters in the problem formulation process are characterized by fuzzy numbers the concept of multiobjective linear fractional programming and m pareto optimality is introduced based on the level sets of the fuzzy numbers through the interaction with the dm the fuzzy goals of the dm for each of the objective functions in multiobjective linear fractional programming are quantified by eliciting the corresponding membership functions after determining the membership functions in order to generate a candidate for the satisficing solution which is also m pareto optimal if the dm specifies the degree of the level sets and the reference membership values the minimax problem is solved by combined use of the bisection method and the linear programming method and the dm is supplied with the corresponding m pareto optimal solution together with the trade off rates among the values of the membership functions and the degree by considering the current values of the membership functions and as well as the trade off rates the dm acts on this solution by updating his her reference membership values and or the degree in this way the satisficing solution for the dm can be derived efficiently from among an m pareto optimal solution set on the basis of the proposed method a time sharing computer program is written and an illustrative numerical example is demonstrated

solution of structured non stiff odes abstract the solution of the initial value problem for a system of ordinary differential equations odes y f x y by runge kutta methods is considered the structure to be exploited is that a large portion of the cost of evaluating f x y arises in the evaluation of functions of the independent variable x only it is shown that when the structure is favorable there are methods which are considerably more efficient than those currently used in general purpose codes

runge kutta triples abstract the addition of a third runge kutta rk formula to an embedded pair forming an rk triple can yield an algorithm which gives solutions of appropriate asymptotic accuracy at points within the normal step intervals in the numerical solution of the initial value problem two modes of implementation are possible and it is shown that these can be regarded as equivalent when certain conditions are satisfied an optimized rk5 4 triple based on a particularly efficient embedded pair is presented and tested the results indicate the efficiency of the dense output technique