spline collocation method for integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions to initial or boundary value problems of fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms

cross domain soft partition clustering with diversity measure and knowledge reference conventional soft partition clustering approaches such as fuzzy c means fcm maximum entropy clustering mec and fuzzy clustering by quadratic regularization fc qr are usually incompetent in those situations where the data are quite insufficient or much polluted by underlying noise or outliers in order to address this challenge the quadratic weights and gini simpson diversity based fuzzy clustering model qwgsd fc is first proposed as a basis of our work based on qwgsd fc and inspired by transfer learning two types of cross domain soft partition clustering frameworks and their corresponding algorithms referred to as type i type ii knowledge transfer oriented c means ti kt cm and tii kt cm are subsequently presented respectively the primary contributions of our work are four fold 1 the delicate qwgsd fc model inherits the most merits of fcm mec and fc qr with the weight factors in the form of quadratic memberships similar to fcm it can more effectively calculate the total intra cluster deviation than the linear form recruited in mec and fc qr meanwhile via gini simpson diversity index like shannon entropy in mec and equivalent to the quadratic regularization in fc qr qwgsd fc is prone to achieving the unbiased probability assignments 2 owing to the reference knowledge from the source domain both ti kt cm and tii kt cm demonstrate high clustering effectiveness as well as strong parameter robustness in the target domain 3 ti kt cm refers merely to the historical cluster centroids whereas tii kt cm simultaneously uses the historical cluster centroids and their associated fuzzy memberships as the reference this indicates that tii kt cm features more comprehensive knowledge learning capability than ti kt cm and tii kt cm consequently exhibits more perfect cross domain clustering performance and 4 neither the historical cluster centroids nor the historical cluster centroid based fuzzy memberships involved in ti kt cm or tii kt cm can be inversely mapped into the raw data this means that both ti kt cm and tii kt cm can work without disclosing the original data in the source domain i e they are of good privacy protection for the source domain in addition the convergence analyses regarding both ti kt cm and tii kt cm are conducted in our research the experimental studies thoroughly evaluated and demonstrated our contributions on both synthetic and real life data scenarios the novel qwgsd fc model is designed for soft partition clustering two cross domain clustering methods ti tii kt cm are subsequently proposed ti tii kt cm are superior to or competitive with other existing methods the convergences of both ti kt cm and tii kt cm are proved

numerical solution of linear volterra integral equations of the second kind with sharp gradients collocation methods are a well developed approach for the numerical solution of smooth and weakly singular volterra integral equations in this paper we extend these methods through the use of partitioned quadrature based on the qualocation framework to allow the efficient numerical solution of linear scalar volterra integral equations of the second kind with smooth kernels containing sharp gradients in this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel we illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations while keeping the solution approximation fixed the computational performance improvement introduced by our new method is examined through several test examples the final example we consider is the original problem that motivated this work the problem of calculating the probability density associated with a continuous time random walk in three dimensions that may be killed at a fixed lattice site to demonstrate how separating the solution approximation from quadrature approximation may improve computational performance we also compare our new method to several existing gregory sinc and global spectral methods where quadrature approximation and solution approximation are coupled

second order sensitivity derivatives in structural analysis haug s 1981 expressions for the calculation of static constraint function second derivatives have resulted in an order of magnitude saving over conventional methods for the calculation of such derivatives a similar but simplified procedure is presented which results in a reduction of computational cost by an additional factor of two the results are presented in a notation that is different from that of haug but more common in other work on structural synthesis and analysis

on the implementation of discontinuous galerkin methods for volterra integral equations with highly oscillatory bessel kernels in this paper we consider implementation of discontinuous galerkin methods for weakly singular volterra integral equations of first and second kinds with highly oscillatory bessel kernels which cost the same operations independent of large values of frequencies preliminary numerical results show that the proposed method is efficient for verifying accuracy of approximate solutions

a spline product quasi interpolation method for weakly singular fredholm integral equations a discrete method of accuracy o h m is constructed and justified for a class of fredholm integral equations of the second kind with kernels that may have weak diagonal and boundary singularities the method is based on i improving the boundary behavior of the kernel with the help of a change of variables and ii the product integration using quasi interpolation by smooth splines of order m

convergence analysis of the multistep legendre pseudo spectral method for volterra integral equations with vanishing delays in this paper we extend the single step pseudo spectral method for second kind volterra integral equations with vanishing variable delays to the multistep pseudo spectral method we also analyze the convergence of the hp version of the multistep pseudo spectral method under the l2 norm and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner in addition it is very suitable for long time calculations and large step size situations numerical results show good agreement with the theoretical analysis

asymptotic stability of multistep methods for nonlinear delay differential equations this paper is concerned with the numerical solution of delay differential equations the emphasis is on the nonlinear stability of multistep methods it is shown that every a stable linear multistep method with piecewise constant or linear interpolation can preserve the asymptotic stability of a class of nonlinear systems which is an extension of the well known gp stability result to the nonlinear case as a comparison it is also proved that strict stability at infinity is necessary to the asymptotic stability of one leg methods for non autonomous equations

rough clustering utilizing the principle of indifference abstract clustering is one of the most widely used method in data mining with applications in virtually any domain its main objective is to group similar objects into the same cluster while dissimilar objects should belong to different clusters in particular k means clustering as member of the partitioning clustering family has obtained great popularity the classic hard k means assigns an object unambiguously to one and only one cluster to address uncertainty soft clustering has been introduced using concepts like fuzziness possibility or roughness a decade ago lingras and west introduced a k means approach based on the interval interpretation of rough sets theory in the past years their rough k means has gained increasing attention in our paper we propose a refined rough k means algorithm that utilizes laplace s principle of indifference to calculate the means as we will discuss this provides a sounder justification for the impacts of the objects in the approximations in comparison to established rough k means algorithms furthermore the weighting in the mean function is based on individual objects rather than on aggregated sub means in experiments we compare the refined algorithm to related approaches

numerical solution of a class of fractional order integro differential algebraic equations using muntz jacobi tau method abstract this paper presents a spectral tau method based on muntz jacobi basis functions for approximating the solutions of fractional order volterra integro differential algebraic equations we examine the solvability of the considered equation and we show that the exact solutions suffer from a singularity at the origin numerical solvability and convergence analysis of the proposed approximate approach are also studied and the error bounds are obtained in a weighted l 2 norm some illustrative examples are presented to validate the theoretical results

a multistep legendre gauss spectral collocation method for nonlinear volterra integral equations we introduce a multistep legendre gauss spectral collocation method for the nonlinear volterra integral equations of the second kind this method is easy to implement and possesses high order accuracy in addition it is very suitable for long time calculations we also derive the optimal convergence of the hp version of the multistep collocation method under the l 2 norm numerical experiments confirm the theoretical expectations

shadowed c means integrating fuzzy and rough clustering a new method of partitive clustering is developed in the framework of shadowed sets the core and exclusion regions of the generated shadowed partitions result in a reduction in computations as compared to conventional fuzzy clustering unlike rough clustering here the choice of threshold parameter is fully automated the number of clusters is optimized in terms of various validity indices it is observed that shadowed clustering can efficiently handle overlapping among clusters as well as model uncertainty in class boundaries the algorithm is robust in the presence of outliers a comparative study is made with related partitive approaches experimental results on synthetic as well as real data sets demonstrate the superiority of the proposed approach

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

on one approach to solve the linear boundary value problems for fredholm integro differential equations necessary and sufficient conditions for the well posedness of linear boundary value problems for fredholm integro differential equations are established algorithms for finding their solutions are offered

analysis of collocation methods for generalized auto convolution volterra integral equations we first study the existence uniqueness and regularity properties of solutions to a generalized version of the auto convolution volterra integral equation of the second kind these results are then used to establish the optimal global and local super convergence properties of piecewise polynomial collocation solutions for such integral equations the theoretical results are illustrated by extensive numerical examples

rough clustering using generalized fuzzy clustering algorithm in this paper we present a rough k means clustering algorithm based on minimizing the dissimilarity which is defined in terms of the squared euclidean distances between data points and their closest cluster centers this approach is referred to as generalized rough fuzzy k means grfkm algorithm the proposed method solves the divergence problem of available approaches where the cluster centers may not be converged to their final positions and reduces the number of user defined parameters the presented method is shown to be converged experimentally compared to available rough k means clustering algorithms the proposed method provides less computing time unlike available approaches the convergence of the proposed method is independent of the used threshold value moreover it yields better clustering results than rfkm for the handwritten digits data set landsat satellite data set and synthetic data set in terms of validity indices compared to mrkm and rfkm grfkm can reduce the value of xie beni index using the handwritten digits data set where a lower xie beni index value implies the better clustering quality the proposed method can be applied to handle real life situations needing reasoning with uncertainty

blow up behavior of collocation solutions to hammerstein type volterra integral equations we analyze the blow up behavior of one parameter collocation solutions for hammerstein type volterra integral equations vies whose solutions may blow up in finite time to approximate such solutions and the corresponding blow up time we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow up behavior is the same as the one for the exact solution based on the local convergence of the collocation methods for vies we present the convergence analysis for the numerical blow up time numerical experiments illustrate the analysis

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical examples

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

numerical solution of linear fredholm integral equations via two dimensional modification of hat functions abstract in this work we approximate the solution of the linear fredholm integral equations by means of a new two dimensional modification of hat functions 2d mhfs and a new operational matrix of integration by this idea the basic equations will be changed into the associated systems of algebraic equations also an error analysis is provided under several mild conditions the method is computationally attractive and some numerical examples are provided to illustrate its high accuracy

silhouettes a graphical aid to the interpretation and validation of cluster analysis a new graphical display is proposed for partitioning techniques each cluster is represented by a so called silhouette which is based on the comparison of its tightness and separation this silhouette shows which objects lie well within their cluster and which ones are merely somewhere in between clusters the entire clustering is displayed by combining the silhouettes into a single plot allowing an appreciation of the relative quality of the clusters and an overview of the data configuration the average silhouette width provides an evaluation of clustering validity and might be used to select an appropriate number of clusters

mean square stability analysis for nonlinear stochastic pantograph equations by transformation approach abstract in this paper we deal with the mean square stability of nonlinear stochastic pantograph equations an equivalent form of stochastic delay differential equations with constant delay is introduced by the transformation approach an optimal stability condition is presented since it is independent of the proportion and coincides with the necessary condition for linear pantograph equations with real coefficients

recent advances in the numerical analysis of volterra functional differential equations with variable delays the numerical analysis of volterra functional integro differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving classical delay differential equations with non vanishing delays in this paper i shall describe recent results in the analysis of optimal global and local superconvergence orders in collocation methods for such evolutionary problems following a brief survey of results for equations containing volterra integral operators with non vanishing delays the discussion will focus on pantograph type volterra integro differential equations with linear and nonlinear vanishing delays the paper concludes with a section on open problems these include the asymptotic stability of collocation solutions u h on uniform meshes for pantograph type functional equations and the analysis of collocation methods for pantograph type functional equations with advanced arguments

relative entropy collaborative fuzzy clustering method the main task of clustering methods especially fuzzy methods is to find whether natural grouping exists in data and to impose identity on them in some situations data are stored in several data sites and to discover the global structures clustering methods have to be aware of dependencies in all data sites collaborative fuzzy clustering methods have been proposed and widely studied to answer such need in this paper a novel collaborative fuzzy clustering method is proposed in this method relative entropy concept is used as the communication method a new approach is applied to calculate the interaction coefficient between data sites and horizontal and vertical modes of the proposed method are discussed performance of the proposed method is evaluated using several experiments and the results show that it has the highest quality of collaboration and could classify data more efficiently in many clustering methods it is assumed that the data are stored in one data site ways of communication between data sites effects the clustering results a new communication method based on relative entropy is proposed it minimizes the divergence of membership degrees of kth data in ith cluster in all data sites recfc and its two modes could efficiently classify data stored in several data sites

operational tau method for singular system of volterra integro differential equations the legendre spectral tau matrix formulation is proposed to approximate solution of singular system of volterra integro differential equations the existence and uniqueness solution of this system are investigated by means of the smoothing property of a volterra integral operator and some projectors the l 2 convergence of the numerical method is analyzed it is proved theoretically and demonstrated numerically that the proposed method converges exponentially finally two numerical examples illustrate the theoretical results

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

numerical solution of integro differential equations arising from singular boundary value problems abstract we consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain the proposed approaches are complex and require a considerable computational effort in the present paper we describe an alternative approach based on the reduction of the original problem to an integro differential equation though the problem is posed on the half line we just need to approximate the solution on a finite interval by analyzing the behavior of the numerical approximation on this interval we identify the solution that satisfies the prescribed boundary condition although the numerical algorithm is much simpler than the ones proposed before it provides accurate approximations we illustrate the proposed methods with some numerical examples

stability and error analysis of one leg methods for nonlinear delay differential equations this paper is concerned with the numerical solution of delay dierential equations ddes we focus on the stability behaviour and error analysis of one leg methods with respect to nonlinear ddes the new concepts of gr stability gar stability and weak gar stability are introduced it is proved that a strongly a stable one leg method with linear interpolation is gar stable and that an a stable one leg method with linear interpolation is gr stable weakly gar stable and d convergent of order s if it is consistent of order s in the classical sense c 1999 elsevier science b v all rights reserved ams classication 65l06 65l20

asymptotic stability analysis of runge kutta methods for nonlinear systems of delay differential equations we consider systems of delay differential equations ddes of the form y t f t y t y t tau t with the initial condition y t phi t t leq t 0 recently torelli 10 introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of ddes in some inner product norm namely rn stability which is the straighforward generalization of the wellknown concept of bn stability of numerical methods with respect to dissipative systems of odes dissipativity means that the solutions y t and z t corresponding to different initial functions phi t and psi t respectively satisfy the inequality y t z t leq max x leq t 0 phi x psi x t geq t 0 and is guaranteed by suitable conditions on the lipschitz constants of the right hand side function f t y x a numerical method is said to be rn stable if it preserves this contractivity property after showing that under slightly more stringent hypotheses on the lipschitz constants and on the delay function tau t the solutions y t and z t are such that lim t rightarrow infty y t z t 0 in this paper we prove that rn stable continuous runge kutta methods preserve also this asymptotic stability property

piecewise polynomial collocation for linear boundary value problems of fractional differential equations we consider a class of boundary value problems for linear multi term fractional differential equations which involve caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed in particular we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by two numerical examples

bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations a new operational matrix for integration of bernoulli polynomials is introduced by using this new operational matrix of integration and the so called collocation method linear volterra and nonlinear volterra fredholm hammerstein integral equations are reduced to systems of algebraic equations with unknown bernoulli coefficients some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the technique a new operational matrix for integration of bernoulli polynomials is introduced a collocation method is used to reduce equations to systems of algebraic equations some error bounds and convergence results are provided for the proposed method the efficiency and applicability of the technique are checked by some examples

laplace transforms for approximation of highly oscillatory volterra integral equations of the first kind abstract this paper focuses on laplace and inverse laplace transforms for approximation of volterra integral equations of the first kind with highly oscillatory bessel kernels where the explicit formulae for the solution of the first kind integral equations are derived from which the integral equations can also be efficiently calculated by the clenshaw curtis filon type methods furthermore by applying the asymptotics of the solution some simpler formulas for approximating the solution for large values of the parameters are deduced preliminary numerical results are presented based on the approximate formulae and the explicit formulae which are compared with the convolution quadrature and numerical inverse laplace transform methods all these methods share that the costs the same independent of large values of frequencies

existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous volterra integral equations we analyse collocation methods for nonlinear homogeneous volterra hammerstein integral equations with non lipschitz nonlinearity we present different kinds of existence and uniqueness of nontrivial collocation solutions and give conditions for such existence and uniqueness in some cases finally we illustrate these methods with an example of a collocation problem and give some examples of collocation problems that do not fit in the cases studied previously

induction of shadowed sets based on the gradual grade of fuzziness the existing methods of determining an cut of a fuzzy set to construct its underlying shadowed set do not fully comply with the concept of shadowed sets namely a retention of the total amount of fuzziness and its localized redistribution throughout a universe of discourse moreover no closed formula to calculate the corresponding cut is available this paper proposes analytical formulas to calculate threshold values required in the construction of shadowed sets we introduce a new algorithm to design a shadowed set from a given fuzzy set the proposed algorithm which adheres to the main premise of shadowed sets of capturing the essence of fuzzy sets helps localize fuzziness present in a given fuzzy set we represent the fuzziness of a fuzzy set as a gradual number through defuzzification of the gradual number of fuzziness we determine the required threshold i e some cut used in the formation of the shadowed set we show that the shadowed set obtained in this way comes with a measure of fuzziness that is equal to the one characterizing the original fuzzy set

asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations this paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type we use a stable linear multistep methods to compute the numerical solution the asymptotic stability of the a stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated and it is shown that the a stable linear multistep methods with linear interpolation are gas stable we validate our conclusions by numerical experiments

a mathematical treatment of the fluorescence capillary fill device a mathematical model in the form of two coupled diffusion equations is provided for a competitive chemical reaction between an antigen and a labeled antigen for antibody sites on a cell wall boundary conditions are such that the problem is both nonlinear and nonlocal this is then recharacterized first as a pair of coupled singular integro differential equations and then as a system of four volterra integral equations the latter permits a proof of existence and uniqueness of the solution of the original problem small and large time asymptotic solutions are derived and from the first characterization a regular perturbation solution is obtained numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin

on sinc discretization for systems of volterra integral algebraic equations abstract integral algebraic equations iaes are coupled systems of volterra integral equations of the first and second kind which naturally arise in many applications in mathematical physics in this paper we solve the iaes of index 1 by sinc collocation discretization and prove that the discrete solutions converge to the true solutions of the iaes exponentially the discrete solutions are determined by linear systems which can be effectively solved by suitable iteration methods numerical examples are given to illustrate the effective performance of our method

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

numerical solutions for fredholm integral equations of the second kind with weakly singular kernel using spectral collocation method abstract we investigate the spectral collocation for fredholm integral equations of the second kind with weakly singular kernel the jacobi gauss quadrature formula is used to approximate the integral operator in the numerical implementation we obtain the convergence rates for the approximated solution of weakly singular fredholm integral equations which show that the errors of the approximate solution decay exponentially in l norm and weighted l2 norm some numerical examples are given to illustrate the theoretical results

delay dependent stability of high order runge kutta methods this paper is concerned with the study of the delay dependent stability of runge kutta methods for delay differential equations first a new sufficient and necessary condition is given for the asymptotic stability of analytical solution then based on this condition we establish a relationship between 0 stability and the boundary locus of the stability region of numerical methods for ordinary differential equations consequently a class of high order runge kutta methods are proved to be 0 stable in particular the 0 stability of the radau iia methods is proved

the stability of difference formulas for delay differential equations a new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y t py t qy t 8 where p and q are complex constants and the delay 8 is a positive constant the technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py t term with the region corresponding to the delay term qy t 8 a sufficient condition for stability is that these regions be disjoint the technique is used to show that for each a stable a a stable or stiffly stable linear multistep formula for ordinary differential equations there is a correspond ing linear multistep formula for delay differential equations with analogous stability properties the analogy does not extend however to a stable one step formulas

positive almost periodic solutions of nonlinear discrete systems with finite delay the existence of positive almost periodic solutions of nonlinear discrete systems with finite delay is obtained by using a variant of hilbert s projective metric on a subset of the space of almost periodic sequences

numerical approximations for highly oscillatory bessel transforms and applications abstract this paper presents an efficient numerical method for approximating highly oscillatory bessel transforms based on analytic continuation we transform the integrals into the problems of integrating the forms on 0 with the integrand that does not oscillate and decays exponentially fast which can be efficiently computed by using gauss laguerre quadrature rule we then derive the error of the method depending on the frequency and the node number moreover we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations preliminary numerical results show the efficiency and accuracy of numerical approximations

a multilevel finite element method for fredholm integral eigenvalue problems in this work we proposed a multigrid finite element mfe method for solving the fredholm integral eigenvalue problems the main motivation for such studies is to compute the karhunen loeve expansions of random fields which play an important role in the applications of uncertainty quantification in our mfe framework solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh then any existing efficient integration scheme can be used for the associated integration process the error estimates are provided and the computational complexity is analyzed it is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh several numerical experiments are presented to validate the efficiency of the proposed numerical method

modified asymptotic orders of the direct filon method for a class of volterra integral equations the direct filon method provides efficient approximation for solutions of volterra integral equations with highly oscillatory kernels in this paper we study the rate of convergence of this method for a class of equations with trigonometric kernels the asymptotic order is developed by examining the property of the solution numerical experiments are included to verify the given order

blow up collocation solutions of nonlinear homogeneous volterra integral equations in this paper collocation methods are used for detecting blow up solutions of nonlinear homogeneous volterra hammerstein integral equations to do this we introduce the concept of blow up collocation solution and analyze numerically some blow up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied finally we discuss the relationships between necessary conditions for blow up of collocation solutions and exact solutions

theoretical and numerical analysis for volterra integro differential equations with ito integral under polynomially growth conditions abstract in this paper we theoretically and numerically deal with nonlinear volterra integro differential equations with ito integral under a one sided lipschitz condition and polynomially growth conditions it is proved that both the exact solutions and vector fields are bounded and satisfy a holder condition in the pth moment sense analogously the boundedness and holder condition in the pth moment sense are preserved by the semi implicit euler method for sufficiently small step size moreover by the local truncated errors we prove the strong convergence order 1 finally numerical simulations on stochastic control models and stochastic ginzburg landau equation illustrate our results

two dimensional integral algebraic systems analysis and computational methods abstract in this article we formulate sufficient conditions for the existence and uniqueness of solution to systems of two dimensional volterra integral equations in which the coefficient of the main term is a singular matrix a numerical method is introduced which can be applied to approximate the solution when the given conditions are satisfied the convergence of this method is proved and illustrated by numerical examples

an h p version of the continuous petrov galerkin finite element method for volterra integro differential equations with smooth and nonsmooth kernels we present an h p version of the continuous petrov galerkin cpg finite element method for linear volterra integro differential equations with smooth and nonsmooth kernels we establish a priori error estimates in the l 2 h 1 and l infty norms that are completely explicit with respect to the local discretization and regularity parameters for singular solutions caused by the weakly singular kernels we prove that algebraic convergence rates can be achieved for the h p version of the cpg method with quasi uniform meshes moreover we show that exponential rates of convergence can be achieved for solutions with start up singularities by using geometric time partitions and linearly increasing polynomial degrees numerical experiments are provided to illustrate the theoretical results

fuzzy entropy a more comprehensible perspective for interval shadowed sets of fuzzy sets shadowed sets proposed by pedrycz provide a three way approximation scheme for transforming the universe of a fuzzy set into three disjoint areas i e elevated area reduced area and shadow area to calculate a pair of decision making thresholds an analytic method was proposed by solving a minimization problem of the uncertainty arising from the three areas however some uncertainty will be lost in the process of constructing the shadowed set model using pedrycz s method moreover few references on how to measure the uncertainty of shadow sets exist in this article a comprehensible method for measuring the fuzzy entropy of a shadowed set i e interval fuzzy entropy is defined based on the interval fuzzy entropy a new shadowed set model namely interval shadowed sets is proposed compared with pedrycz s model the main difference is that the range of the shadow area in this model is beta alpha 0 leq beta while not 0 1 by solving a fuzzy entropy loss minimization problem a pair of optimal thresholds alpha and beta can be obtained finally the results of the instance analysis of different types of representative membership functions and many experiments show that the fuzzy entropy loss of the interval shadowed set is lower than that of the traditional shadowed set of a fuzzy set these results enrich shadowed set theory from a new perspective

optimal convergence results of piecewise polynomial collocation solutions for integral algebraic equations of index 3 in this work we will consider piecewise polynomial collocation methods for a new class of integral algebraic equations of index 3 an analysis of the global convergence properties of the collocation methods is given and the optimal order of convergence results is established the performance of the proposed scheme is illustrated by means of two numerical examples and confirm that we can achieve the expected theoretical orders of convergence

on graded meshes for weakly singular volterra integral equations with oscillatory trigonometric kernels we compare the collocation methods on graded meshes with that on uniform meshes for the solution of the weakly singular volterra integral equation of the second kind with oscillatory trigonometric kernels due to in general unbounded derivatives at the left endpoint of the interval of integration we should approximate the solution of the integral equation by collocation methods on graded meshes however we show that this non smooth behavior of the solution has little effect on the approximate solution when the kernels of integral equation are highly oscillatory numerical examples are given to confirm the proposed results

the superconvergence of newton cotes rules for the hadamard finite part integral on an interval we study the general composite newton cotes rules for the computation of hadamard finite part integral with the second order singularity and focus on their pointwise superconvergence phenomenon i e when the singular point coincides with some a priori known point the convergence rate is higher than what is globally possible we show that the superconvergence rate of the composite newton cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points several numerical examples are provided to validate the theoretical analysis

shadowed sets in the characterization of rough fuzzy clustering in this study we develop a technique of an automatic selection of a threshold parameter which determines approximation regions in rough set based clustering the proposed approach exploits a concept of shadowed sets all patterns data to be clustered are placed into three categories assuming a certain perspective established by an optimization process as a result a lack of knowledge about global relationships among objects caused by the individual absolute distance in rough c means clustering or individual membership degree in rough fuzzy c means clustering can be circumvented subsequently relative approximation regions of each cluster are detected and described by integrating several technologies of granular computing including fuzzy sets rough sets and shadowed sets we show that the resulting characterization leads to an efficient description of information granules obtained through the process of clustering including their overlap regions outliers and boundary regions comparative experimental results reported for synthetic and real world data illustrate the essence of the proposed idea

blow up time for solutions to some nonlinear volterra integral equations the problem of the estimating of a blow up time for solutions of volterra nonlinear integral equation with convolution kernel is studied new estimates lower and upper are found and moreover the procedure for the improvement of the lower estimate is presented main results are illustrated by examples the new estimates are also compared with some earlier ones related to a shear band model

on the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals the behaviour of a solution to a fredholm integral equation of the second kind on a union of open intervals is examined the kernel of the corresponding integral operator may have diagonal singularities information about them is given through certain estimates the weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described

second order approximations in structural genesis and shape finding the introduction of second order sensitivities to behaviour approximation in structural optimization is applied and investigated numerical computation of second derivatives is considered and evaluated two approaches in shape optimization are involved namely shape finding by moving boundaries and shape genesis by distributing mass in a specified design domain a series of examples is engaged to work out the benefits of second order information in both the above fields

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

optimal superconvergence results for delay integro differential equations of pantograph type we analyze the optimal global and local orders of superconvergence of collocation solutions u h on uniform meshes i h for delay volterra integro differential equations with proportional delay functions given by theta t qt 0 q 1 t in 0 t in particular we show that if u h is a continuous piecewise polynomial of degree m geq 2 and if collocation is at the gauss legendre points then the optimal order of local superconvergence on i h is p ast m 2 it turns out that the same order p ast holds for nonlinear strictly increasing delay functions vanishing at t 0 however on judiciously chosen geometric meshes collocation at the gauss points yields the order 2m varepsilon n where varepsilon n rightarrow 0 as the number n of mesh points tends to infinity optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis

an engineer s approach to optimal material distribution and shape finding abstract the fundamental aim of shape optimization involves the optimal distribution of mass in space observing global and local design constraints the practical realization in applied mechanics can be performed by introducing a foam like material with variable density as discussed by bendsoe and kikuchi this paper uses for the evaluation of material properties a simple energy approach the application is extended to three dimensional domains multiple loading cases are also considered further the approach is implemented in a preoptimizer for optimal shape finding

principles for constructing three way approximations of fuzzy sets a comparative evaluation based on unsupervised learning abstract three way approximations of fuzzy sets are an important scheme of granular computing by abstracting a fuzzy set to its discrete three option alternatives which adhere to human cognitive behaviors and reduce the computational burden the key point of such three way approximations of fuzzy sets is how to choose a suitable design leading to their realization undesired three way approximations might be produced if the selected mechanism is unsuitable to data distribution in this study the principles for constructing three way approximations of fuzzy sets are summarized the following taxonomy of these principles is provided namely i uncertainty balance based principle ii prototype based principle and iii model based invoking the tradeoff between classification error and the number of data that have to be classified a number of detailed optimization models are discussed in detail to evaluate the performance of different construction principles a general unsupervised learning framework based on three way approximations of fuzzy sets is exhibited some synthetic data sets along with sixteen data sets from uci repository are involved for experiments friedman testing followed by holm bonferroni testing are exploited to test the performance significance of the proposed criteria which can provide insights and deliver guidance when choosing a principle for constructing three way approximations of fuzzy sets in the real world scenarios the research methods in this paper can also be extended to supervised and semi supervised learning areas

integral algebraic equations theory of collocation methods i our analysis of collocation solutions for general systems of linear integral algebraic equations iaes is based on the notions of the tractability index and the nu smoothing property of a volterra integral operator these are used to decouple the given iae system into the inherent system of regular second kind volterra integral equations vies and a system of first kind vies this decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions numerical examples illustrate the theoretical results

an approximation concepts approach to shape optimal design abstract in the past much of the work done in structural optimization consisted in resizing the members of fixed configuration models in that case a powerful design procedure has now emerged which is based on the coordinate use of explicit high quality approximations of the behavior constraints and dual methods of mathematical programming there is however a large class of problems for which the main degrees of freedom for the designer correspond to the shape of the structure itself the main objectives of this paper are to recall briefly a convenient geometric representation in which the boundaries of the structure are represented by bezier or b spline curves and then to discuss the choice of optimization algorithm it is shown that cost efficient methods for structural sizing may be advantageously extended to shape optimal design problems different approximation schemes are tested and a new general optimization algorithm is presented that combines mixed approximations and dual methods many large scale applications are treated to demonstrate the generality and the efficiency of the new formulation finally considerations are given about an integrated approach including cad computer codes and finite element optimization software

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

an hp version of the discontinuous galerkin time stepping method for nonlinear second order delay differential equations with vanishing delays abstract we present and analyze an h p version of the discontinuous galerkin dg time stepping method for nonlinear second order delay differential equations with vanishing delays we derive a priori error bounds in the l 2 and l norm that are fully explicit with respect to the local time steps the local approximation orders and the local regularity of the exact solution we further prove that the h p dg scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start up singularities moreover we also propose an h p dg scheme for the state dependent delay differential equations we illustrate the theoretical results with a series of numerical experiments

retarded differential equations retarded differential equations rdes are differential equations having retarded arguments they arise in many realistic models of problems in science engineering and medicine where there is a time lag or after effect numerical techniques for such problems may be regarded as extensions of dense output methods for ordinary differential equations odes but scalar rdes are inherently infinite dimensional with a richer structure than their ode counterparts we give background material develop a theoretical foundation for the basic numerics and give some results not previously published