generalized multiple scale reproducing kernel particle methods abstract an approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed the study is divided into three major topics the groundwork is set by revisiting the fourier analysis of discrete systems the multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart reproducing kernel particle methods are explained an edge detection technique based on multiresolution analysis is developed this wavelet approach together with particle methods gives rise to a straightforward h adaptivity algorithm by using this framework a hermite reproducing kernel method is also proposed and its relation to wavelet methods is presented it is also shown that the new approach generalizes existing kernel methods and it can easily be degenerated into other widely used methods such as partition of unity moving least square interpolants smooth particle hydrodynamics scaling functions and wavelets and multiple scale analysis furthermore the hermite reproducing kernel particle method a particle based discrete version of the hermite reproducing kernel method is developed finally multiple scale methods based on frequency and wave number shifting techniques are presented a stability analysis is also presented for newmark time integration schemes for the low frequency equation numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods

canonical property of the miura maps between the mkp and kp hierarchies continuous and discrete apart from the case of the kp hierarchy all known miura maps between integrable hamiltonian systems had been proven to be canonical the remaining kp case is settled below as a corollary it is shown that the kp hierarchy is a factor hierarchy of the mkp one with the kernel consisting of a single scalar field a discrete mkp hierarchy and the associated miura map are constructed and the latter is shown to be canonical as well as in the continuous case this implies that one can extend the discrete kp hierarchy by a single new field into an extended discrete kp hierarchy in such a way that the extended discrete miura map mkp ekp is a canonical isomorphism

a new multiplicative consistency based method for decision making with triangular fuzzy reciprocal preference relations abstract triangular fuzzy reciprocal preference relations tfrprs are good choices to denote the decision makers fuzzy preferences consistency analysis is a very important topic to preference relations that can avoid the illogical ranking order however all previous multiplicative consistency concepts for tfrprs cannot well address this type of preference relations to address this issue this paper introduces a new multiplicative consistency concept for tfrprs which can be seen as a natural extension of tanino s multiplicative consistency concept for reciprocal preference relations based on the new consistency concept 0 1 mixed programming models for judging the multiplicative consistency of tfrprs are constructed and an approach for deriving multiplicative consistent tfrprs is provided considering the incomplete case programming models to determine missing values are constructed subsequently a multiplicative consistency based method for decision making with tfrprs is presented that can address the inconsistent and incomplete cases meanwhile illustrative examples are provided to show the concrete application of the proposed procedure and comparison analysis is also made

a parametric model for determining consensus priority vectors from fuzzy comparison matrices abstract we consider a group decision making problem where a set of alternatives have to be ranked according to fuzzy preference judgments given by multiple experts we assume that expert assessments are expressed in the form of fuzzy multiplicative preference relations or fuzzy comparison matrices in this paper we propose a general model to generate crisp priority weights of the alternatives from possibly inconsistent and conflicting fuzzy preference relations we express our model in terms of matrix approximations to address the consistency problem and we use weighted metrics to simulate the group dynamics matrix approximation techniques for deriving a common crisp consistent matrix are extended to work with fuzzy matrices by using the concept of cuts in the aggregation process the importance of each expert is taken into consideration according to the agreement of the group with the expert this results in a parametric optimization problem for which a computational formulation is given

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

second and fourth painleve hierarchies and jimbo miwa linear problems the relations between the different linear problems for painleve equations is an intriguing open problem here we consider our previously given second and fourth painleve hierarchies publ res inst math sci kyoto 37 327 347 2001 and show that they could alternatively have been derived using the linear problems of jimbo and miwa that is we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth painleve equations themselves onto those of jimbo and miwa

fuzzy hierarchical analysis the lambda max method abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the max method used by saaty in the analytical hierarchical process an example is presented where there are five criteria and three alternatives

a note on the fuzzy extension of saaty s priority theory we consider the problem of the fuzzy extension of saaty s priority theory by van laarhoven and pedrycz in this paper since this problem has not been solved yet we apply the generalised pseudoinverse approach to find a structure of the general solution we recall the arithmetic of triangular fuzzy numbers and show links with the interval analysis by moore sample calculations are included

application of the variational iteration method to the whitham broer kaup equations explicit traveling wave solutions including blow up and periodic solutions of the whitham broer kaup equations are obtained by the variational iteration method moreover the results are compared with those obtained by the adomian decomposition method revealing that the variational iteration method is superior to the adomian decomposition method

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

squared eigenfunction symmetries for soliton equations part i abstract nonlinear integrable evolution equations in 1 1 dimensions arise from constraints of the 2 1 dimensional hierarchies associated with the kadomtsev petviashvili kp equation the modified kp equation and the dym equation respectively links of backlund type and of reciprocal type are known to exist between the 2 1 dimensional systems the corresponding links between the constrained flows are discussed in a general framework in particular squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered the links between the soliton hierarchies are extended to these symmetries

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

the piecewise polynomial partition of unity functions for the generalized finite element methods a partition of unity pu function is an essential component of the generalized finite element method gfem the popular shepard pu functions which are rational functions are easy to construct but have difficulties in dealing with essential boundary conditions and require lengthy computing time for reasonable accuracy in numerical integration in this paper we introduce two simple pu functions the first is a highly regular piecewise polynomial consisting of two distinct polynomials that is effective for uniformly partitioned patches the second is a highly regular piecewise polynomial consisting of three distinct polynomials which is for arbitrary partitioned patches

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

classical liouville completely integrable systems associated with the solutions of boussinesq burgers hierarchy two new finite dimensional completely integrable systems in the liouville sense are obtained the solutions of boussinesq burgers hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems

reproducing kernel element method part iv globally compatible cn n 1 triangular hierarchy in this part of the work a globally compatible c n x triangular element hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary two dimensional domains in principle the smoothness of the globally conforming element can be made arbitrarily high np 1 the triangle interpolation field can interpolate the derivatives of an unknown function up to arbitrary mth order i m and it can reproduce complete kth order polynomials with k p m this is the first interpolation hierarchical structure that has ever been constructed with both minimal degrees of freedom and higher order smoothness and continuity over discretizations of a multiple dimensional domain the performance of the newly constructed compatible element is evaluated in solving several kirchhoff plate problems

the method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities babuska and oh have introduced a new approach called the method of auxiliary mapping mam to deal with elliptic boundary value problems with singularities they showed that for the laplace equation with corner singularities in the context of the p version of the finite element method mam yielded an exponential rate of convergence at virtually no extra cost in this paper those results are extended by showing the exponential convergence of mam for homogeneous laplace equations with boundary singularities and helmholtz equations with both corner and boundary singularities in addition a convergence result is developed for mam as applied to the h p version of the finite element method to clarify the power of mam a series of benchmark runs are made for three examples using our implementation mapfem comparisons are made with two h version finite element codes pltmg6 and fesop both of which use adaptive meshes the finite difference code ellpack and the recent singular element code isbfm the examples include the well known motz problem and both homogeneous and nonhomogeneous helmholtz equations

the integrable hierarchy constructed from a pair of kdv type hierarchies and its associated w algebra for any two arbitrary positive integers n and m using them th kdv hierarchy and the n m th kdv hierarchy as building blocks we are able to construct another integrable hierarchy referred to as the n m th kdv hierarchy thew algebra associated to the second hamiltonian structure of the n m th kdv hierarchy calledw n m algebra is isomorphic via a miura map to the direct sum of aw m algebra aw n m algebra and an additionalu 1 current algebra in turn from the latter we can always construct a representation of aw algebra

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

nonstandard drinfeld sokolov reduction subject to some conditions the input data for the drinfeld sokolov construction of kdv type hierarchies is a quadruplet where the are gradations of a loop algebra and is a semisimple element of the nonzero grade a new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented the proposal relies on splitting the grade zero part of into a vector space direct sum of two subalgebras this permits one to interpret certain gelfand dickey type systems associated with a nonstandard splitting of the algebra of pseudodifferential operators in the drinfeld sokolov framework

string theory and water waves we uncover a remarkable role that an infinite hierarchy of nonlinear differential equations plays in organizing and connecting certain string theories non perturbatively we are able to embed the type 0a and 0b a a minimal string theories into this single framework the string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy we observe that there are several other string like limits of the system and conjecture that some of them are type iia and iib a d minimal string backgrounds we explain how these and several string like special points arise and are connected in some cases the framework endows the theories with a non perturbative definition for the first time notably we discover that the painlev iv equation plays a key role in organizing the string theory physics joining its siblings painlev i and ii whose roles have previously been identified in this minimal string context

mesh based construction of flat top partition of unity functions a novel idea to construct flat top partition of unity functions in a closed form on a general structured or unstructured finite element mesh is presented an efficient and practical construction method of a flat top partition of unity function is important in the generalized finite element method gfem details on how to construct flat top partition of unity functions on a provided mesh are given the generalized finite element approximation with the use of the new flat top partition of unity function is presented with various numerical examples that demonstrate the effectiveness of proposed flat top partition of unity functions

effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for gfem xfem stress approximations around brittle cracks abstract the convergence rates of the conventional generalized extended finite element method gfem xfem in crack modeling are similar to the convergence rates of the finite element method fem laborde et al 2005 and bechet et al 2005 unless the crack tip enrichment functions are applied in a subdomain with fixed dimension independently of the mesh parameter h and also demanding some special care for blending along the transition zone chahine et al 2006 and taracon et al 2009 thus to improve convergence rates more degrees of freedom dof are generated due to the larger quantity of enriched nodes this work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh based smooth partitions of unity pou such pou with higher regularity can be built through the so called c k gfem framework following duarte et al 2006 based on a moving least square of degree zero and considering mesh based smooth weighting functions associated with arbitrary polygonal clouds the purpose herein is to investigate some possible advantages of mesh based smooth pou for modeling discontinuities and singularities in two dimensional problems of linear elastic fracture mechanics in such a fashion that the discretization error associated to stress discontinuities inherent in standard c 0 continuous gfem xfem approximations is eliminated the procedure shares features similar to the standard fem regarding domain partition and numerical integration but as neither the pou nor the enrichment functions are defined in natural domains the integrations are performed using only global coordinates the approximation capabilities of c k gfem discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the h and p versions considering global measures in terms of strain energy and l 2 norm of displacements the effects on stability are also verified by analyzing the evolution of the condition number the effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved the smoothness of the pou the different types of enrichment functions and the pattern of enrichment the performance of the smooth approximations is compared to the c 0 counterparts built using conventional c 0 fem based pou it is shown that smoothness in the presence of extrinsically applied singular enrichments is important as the information provided by such enrichment is better captured in addition there are no stress jumps around singularities reducing error propagation beyond the neighborhood of the singularity

reproducing kernel element method part ii globally conforming im cn hierarchies in this part of the work a minimal degrees of freedom arbitrary smooth globally compatible i m c n interpolation hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary multiple dimensional domains this is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher order smoothness or continuity over multi dimensional domain the proposed hierarchical structure possesses the generalized kronecker property i e o a w b i ox a x j dijdab jaj jbj 6 m

the kaup broer system and trilinear odes of painleve type the kaup broer hierarchy can be considered as a certain reduction of the kp hierarchy upon introducing the function the kaup broer system can be formulated in trilinear form giving rise to wronskian solutions in this paper several odes of the painleve type obtained through reductions of the kaup broer system are considered those odes take a trilinear form which allows a linearization

non perturbative string theory from water waves we use a combination of a t hooft limit and numerical methods to find non perturbative solutions of exactly solvable string theories showing that perturbative solutions in different asymptotic regimes are connected by smooth interpolating functions our earlier perturbative work showed that a large class of minimal string theories arise as special limits of a painleve iv hierarchy of string equations that can be derived by a similarity reduction of the dispersive water wave hierarchy of differential equations the hierarchy of string equations contains new perturbative solutions some of which were conjectured to be the type iia and iib string theories coupled to 4 4k 2 superconformal minimal models of type a d our present paper shows that these new theories have smooth non perturbative extensions we also find evidence for putative new string theories that were not apparent in the perturbative analysis

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

fuzzy hierarchical analysis revisited abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by saaty in the analytic hierarchy process we test our new procedure in two cases where there are formulas for the crisp weights an example is presented where there are five criteria and three alternatives

regular articlesquared eigenfunction symmetries for soliton equations part ii nonlinear integrable evolution equations in 1 1 dimensions arise from constraints of the 2 1 dimensional hierarchies associated with the kadomtsev petviashvili kp equation the modified kp equation and the dym equation respectively links of backlund type and of reciprocal type are known to exist between the 2 1 dimensional systems the corresponding links between the constrained flows are discussed in a general framework in particular squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered the links between the soliton hierarchies are extended to these symmetries

multi lagrangians for integrable systems we propose a general scheme to construct multiple lagrangians for completely integrable nonlinear evolution equations that admit multi hamiltonian structure the recursion operator plays a fundamental role in this construction we use a conserved quantity higher lower than the hamiltonian in the potential part of the new lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map this leads to some remarkable new developments we show that nonlinear evolutionary systems that admit n fold first order local hamiltonian structure can be cast into variational form with 2n 1 lagrangians which will be local functionals of clebsch potentials this number increases to 3n 2 when the miura transformation is invertible furthermore we construct a new lagrangian for polytropic gas dynamics in 1 1 dimensions which is a free local functional of the physical field variables namely density and velocity thus dispensing with the necessity of introducing clebsch potentials entirely this is a consequence of bi hamiltonian structure with a compatible pair of first and third order hamiltonian operators derived from sheftel s recursion operator

the generalized product partition of unity for the meshless methods the partition of unity is an essential ingredient for meshless methods named by gfem pufem partition of unity fem xfem extended fem rkpm reproducing kernel particle method rppm reproducing polynomial particle method the method of hp clouds in the literature there are two popular choices for partition of unity a piecewise linear fem mesh and the shepard type partition of unity however the partition of unity pu by a fem mesh leads to the singular or nearly singular matrices and non smooth approximation functions the shepard type partition of unity requires lengthy computing time and its implementation is difficult in order to alleviate these difficulties oh et al introduced the smooth piecewise polynomial pu functions with flat top that lead to small matrix condition numbers and almost everywhere partition of unity that can handle essential boundary conditions nevertheless we could not have the smooth closed form pu functions with flat top for general polygonal patches 2d and general polyhedral patches 3d in this paper we introduce one of the most simple and efficient partition of unity called the generalized product partition of unity the product pu functions constructed by this method are the closed form smooth piecewise polynomials with flat top and could handle background meshes general polygonal patches as well as general polyhedral patches arising in practical applications of meshless methods

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

reproducing kernel element method part i theoretical formulation in this paper and its sequels we introduce and analyze a new class of methods collectively called the reproducing kernel element method rkem the central idea in the development of the new method is to combine the strengths of both finite element methods fem and meshfree methods two distinguished features of rkem are the arbitrarily high order smoothness and the interpolation property of the shape functions these properties are desirable especially in solving galerkin weak forms of higher order partial differential equations and in treating dirichlet boundary conditions so unlike the fem there is no need for special treatment with the rkem in solving high order equations compared to meshfree methods dirichlet boundary conditions do not present any difficulty in using the rkem a rigorous error analysis and convergence study of the method are presented the performance of the method is illustrated and assessed through some numerical examples

wronskian determinants the kp hierarchy and supersymmetric polynomials by using the wronskian representation of the solutions of the bilinear kp hierarchy a connection between hirota derivatives and supersymmetric polynomials is brought to light this correspondence is used in order to give an alternative construction of the hierarchy

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

almost everywhere partition of unity to deal with essential boundary conditions in meshless methods abstract due to overlapping nature of supports of partition of unity functions and the lack of the kronecker delta property of meshless shape functions it is difficult to deal with essential boundary conditions in meshless methods in this paper in order to alleviate this difficulty we introduce almost everywhere partition of unity that is a partition of unity except a few points along boundary in two dimensional case actually the gradient of partition of unity functions become infinitely large at these exceptional points however we prove that the presence of these bad points does not change the convergence rates of computed solutions comparing with the computed solutions obtained by the lagrange multiplier method the penalty method and the nitche s method we demonstrate the proposed method is more effective in dealing with essential boundary conditions in meshless methods

application of exp function method to the whitham broer kaup shallow water model using symbolic computation in this letter the exp function method is applied to the whitham broer kaup shallow water model with the help of symbolic computation several kinds of new solitary wave solutions are formally derived

backlund transformations for fourth painleve hierarchies abstract backlund transformations bts for ordinary differential equations odes and in particular for hierarchies of odes are a topic of great current interest here we give an improved method of constructing bts for hierarchies of odes this approach is then applied to fourth painleve p iv hierarchies recently found by gordoa et al publ res inst math sci kyoto 37 2001 327 347 we show how the known pattern of bts for p iv can be extended to our p iv hierarchies remarkably the bts required to do this are precisely the miura maps of the dispersive water wave hierarchy we also obtain the important result that the fourth painleve equation has only one nontrivial fundamental bt and not two such as is frequently stated

on the normalization of interval and fuzzy weights the normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis mcda under uncertainty especially in analytic hierarchy process ahp with interval or fuzzy judgements the existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised this paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods

meshfree particle methods for thin plates abstract in this paper we are concerned with meshfree particle methods for the solutions of the classical plate model the vertical displacement of a thin plate is governed by a fourth order elliptic equation and thus the approximation functions for numerical solutions are required to have continuous partial derivatives hence the conventional finite element method has difficulties to solve the fourth order problems meshfree methods have the advantage of constructing smooth approximation functions however most of the earlier works on meshfree methods for plate problems used either moving least squares method with penalty method or coupling fem with meshfree method to deal with essential boundary conditions in this paper by using generalized product partition of unity introduced by oh et al we introduce meshfree particle methods in which approximation functions have high order polynomial reproducing property and the kronecker delta property we also prove error estimates for the proposed meshfree methods moreover to demonstrate the effectiveness of our method results of the proposed method are compared with existing results for various shapes of plates with variety of boundary conditions and loads

a lie algebraic setting for miura maps related to an energy dependent linear problem we study the affine semi direct product vect s 1 s 1 1 and show how the coadjoint action on its extended dual leads immediately to a generalisation of the miura map and also to the existence of several compatible hamiltonian structures

reproducing kernel element method part iii generalized enrichment and applications in this part of the work a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable this is accomplished by either multiplying enrichment functions with the original global partition polynomials or increasing the order of global partition polynomials in the same mesh without refining mesh high order consistency in interpolation hierarchy with generalized kronecker delta property can be straightforwardly achieved in quadrilateral and triangular mesh in 2d by the proposed scheme comparing with the traditional finite element methods the construction proposed here has more flexibility and only needs minimal degrees of freedom the optimal element with high reproducing capacity and overall minimal degrees of freedom can be constructed by the generalized enrichment procedure two optimal elements in two dimensional space have been constructed t10p3i 4 triangular element satisfies third order consistency condition with only 10 degrees of freedom and q15p4i 4 quadrilateral element satisfies fourth order consistency condition with 15 degrees of freedom the performance of interpolation hierarchy is

enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and or corners summary the mapping method was introduced in jeong et al 2013 for highly accurate isogeometric analysis iga of elliptic boundary value problems containing singularities the mapping method is concerned with constructions of novel geometrical mappings by which push forwards of b splines from the parameter space into the physical space generate singular functions that resemble the singularities in other words the pullback of the singularity into the parameter space by the novel geometrical mapping a non uniform rational basis spline nurbs surface mapping becomes highly smooth one of the merits of iga is that it uses nurbs functions employed in designs for the finite element analysis however push forwards of rational nurbs may not be able to generate singular functions moreover the mapping method is effective for neither the k refinement nor the h refinement in this paper highly accurate stress analysis of elastic domains with cracks and or corners are achieved by enriched iga in which push forwards of nurbs via the design mapping are combined with push forwards of b splines via the novel geometrical mapping the mapping technique in a similar spirit of x fem or gfem we propose three enrichment approaches enriched iga for corners enriched iga for cracks and partition of unity iga for cracks copyright 2013 john wiley sons ltd

on the non local generalised symmetries of the benny kaup system the authors have extended the formalism of vinogradov and krasil shchik 1984 for the coupled system of benny kaup equations it is explicitly demonstrated that it is possible to generate new non local symmetries for this coupled equation if they consider d 1u integral u dx to be also in the space of independent variables

the closed form reproducing polynomial particle shape functions for meshfree particle methods abstract it has been known that reproducing kernel particle rkp shape functions with kronecker delta property are not available in simple forms thus in this paper we construct highly regular piecewise polynomial reproducing polynomial particle rpp shape functions that satisfy the kronecker delta property like rkp shape functions the rpp shape functions reproduce the complete polynomials of degree k for an integer k 0 moreover the particles associated with these rpp shape functions are either uniformly distributed in or non uniformly distributed in 0 or in a compact set a b

error analysis of the reproducing kernel particle method interest in meshfree or meshless methods has grown rapidly in recent years in solving boundary value problems bvps arising in mechanics especially in dealing with difficult problems involving large deformation moving discontinuities etc in this paper we provide a theoretical analysis of the reproducing kernel particle method rkpm which belongs to the family of meshfree methods one goal of the paper is to set up a framework for error estimates of rkpm we introduce the concept of a regular family of particle distributions and derive optimal order error estimates for rkp interpolants on a regular family of particle distributions the interpolation error estimates can be used to yield error estimates for rkp solutions of bvps

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

the two component kaup kupershmidt equation the kaup kupershmidt equation is generalized to a system of equations in the same manner as the korteweg de vries equation is generalized to the hirota satsuma equation the gelfand dikii lax and hamiltonian formulation for this generalization is given the same construction is repeated for the constrained kadomtsev pietviashvili lax operator which leads to the four component kaup kupershmidt equation the modified version of the two component kaup kupershmidt equation is presented and analysed

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

on a new class of completely integrable nonlinear wave equations ii multi hamiltonian structure the multi hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed infinitely many conservation laws had earlier been obtained for these equations starting from a primary hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi hamiltonian structure are obtained and it is shown that the second hamiltonian operator can be constructed solely through a knowledge of the first hamiltonian function the recursion operator which first appears at the level of bi hamiltonian structure gives rise to an infinite sequence of conserved hamiltonians it is found that in general there exist two different infinite sequences of conserved quantities for these equations the recursion relation defining higher hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the k 1 st hamiltonian operator which depends on the kth hamiltonian function the infinite sequence of conserved hamiltonians are common to all the higher hamiltonian structures the equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri hamiltonian structure with two distinct infinite sets of conserved quantities the isothermal case of 1 is an exceptional one that requires separate treatment this corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to poisson s equation in nonlinear acoustics

a modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process this paper revisits the fuzzy logarithmic least squares method llsm in the analytic hierarchy process and points out its incorrectness in the normalization of local fuzzy weights infeasibility in deriving the local fuzzy weights of a fuzzy comparison matrix when the lower bound value of a non normalized fuzzy weight turns out to be greater than its upper bound value uncertainty of local fuzzy weights for incomplete fuzzy comparison matrices and unreality of global fuzzy weights a modified fuzzy llsm which is formulated as a constrained nonlinear optimization model is therefore suggested to tackle all these problems a numerical example is examined to show the applicability of the modified fuzzy llsm and its advantages

on a non standard algebraic description of integrable nonlinear systems abstract we present a non standard algebraic description of some classical integrable systems in 1 1 dimensions resulting from interchanging the role of space and time variables bi or multi hamiltonian hierarchies obtained this way reduce naturally to the stationary subspaces giving some insight into the recursive structure of stationary flows

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

reproducing polynomial singularity particle methods and adaptive meshless methods for two dimensional elliptic boundary value problems abstract oh et al h s oh j g kim j w jeong the closed form reproducing polynomial particle shape functions for meshfree particle methods comput methods appl mech engrg 196 2007 3435 3461 introduced the reproducing polynomial particle rpp shape functions that are piecewise polynomial and satisfy the kronecker delta property in this paper we introduce reproducing polynomial particle methods rppm that is the galerkin approximation method associated with the use of the rpp approximation space planting particles in the computation domain in a patchwise uniform manner we also introduce patchwise rppm furthermore constructing partition of unity functions with flat top adaptively we introduce adaptive patchwise rppm and compare it with reproducing singularity particle methods rspm that is the galerkin approximation method associated with the use of reproducing singularity particle rsp shape functions

new exact travelling wave solutions for the shallow long wave approximate equations based upon a generally projective riccati equation method which is a direct and unified algebraic method for constructing more general form travelling wave solutions of nonlinear partial differential equations and implemented in a computer algebraic system we consider the shallow long wave approximate equations new and more general form solutions are obtained including kink shaped solitons bell shaped solitons singular solitons and periodic solutions the properties of the new formal solitary wave solutions are shown by some figures

rational solutions of the classical boussinesq hierarchy the classical boussinesq system and its associated higher order flows in the bilinear form are considered an infinite family of homogeneous rational solutions of this hierarchy is verified by the use of the wronskian technique developed by freeman and nimmo 1983

recursion operators and bi hamiltonian formulations of the landau lifshitz equation hierarchies using a polynomial pencil of lax pairs we give a new bi hamiltonian formulation for the landau lifshitz equation hierarchy and find a new recursion operator related to it

on the miura map between the dispersionless kp and dispersionless modified kp hierarchies the miura map between the dispersionless kp and dispersionless modified kp hierarchies is investigated it is shown that the miura map is canonical with respect to their bi hamiltonian structures moreover inspired by the works of takasaki and takebe the twistor construction of solution structure for the dispersionless modified kp hierarchy is given

bifurcation of travelling wave solutions for the modified dispersive water wave equation abstract the method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation we obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits in different regions different types of travelling solutions including solitary wave solutions shock wave solutions and periodic wave solutions are simulated furthermore with a generalized projective riccati equation method several new explicit exact solutions are obtained

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures

a new derivation of painleve hierarchies abstract we give a new derivation of two painleve hierarchies this is done by extending the accelerating wave reductions of the korteweg de vries and dispersive water wave equations to their respective hierarchies we also consider the extension of this reduction of burgers equation to the burgers hierarchy