positive solutions of nth order impulsive boundary value problems with integral boundary conditions in banach spaces abstract by using the fixed point theory for strict set contraction operator this paper investigates the existence of positive solutions for a class of n th order impulsive boundary value problems with integral boundary conditions in banach spaces moreover we offer some interesting discussion of the associated boundary value problems since the technique to deal with n th order bvp is completely different from that of previous papers the expression of the associated integral equation is simpler than that of relevant papers and upper and lower bounds for these positive solutions also are given our work is new

a class of nth order impulsive integrodifferential equations in banach spaces abstract by establishing a comparison result and using the monotone iterative technique the author obtains the existence of maximal and minimal solutions of the initial value problem for a class of n th order impulsive integrodifferential equations in a banach space

point group identification algorithm in dynamic response analysis of nonlinear stochastic systems abstract the point group identification pgi algorithm is proposed to determine the representative point sets in response analysis of nonlinear stochastic dynamic systems the pgi algorithm is employed to identify point groups and their feature points in an initial point set by combining subspace clustering analysis and the graph theory further the representative point set of the random variate space is determined according to the minimum generalized f discrepancy the dynamic responses obtained by incorporating the algorithm pgi into the probability density evolution method pdem are compared with those by the monte carlo simulation method the investigations indicate that the proposed method can reduce the number of the representative points lower the generalized f discrepancy of the representative point set and also ensure the accuracy of stochastic structural dynamic analysis

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

asymmetric cubature formulae with few points in high dimension for symmetric measures let mu be a positive measure on mathbb r d invariant under the group of reflections and permutations and let m be a natural number we describe a method to construct cubature formulae of degree m with respect to mu with n positive weights and n points in the support of mu and such that n grows at most like dm with the dimension d we apply this method to classical measures to explicitly construct cubature formulae of degree 5 with the number of points growing at most like d3

the probability density evolution method for dynamic response analysis of non linear stochastic structures the probability density evolution method pdem for dynamic responses analysis of non linear stochastic structures is proposed in the method the dynamic response of non linear stochastic structures is firstly expressed in a formal solution which is a function of the random parameters in this sense the dynamic responses are mutually uncoupled a state equation is then constructed in the augmented state space based on the principle of preservation of probability a one dimensional partial differential equation in terms of the joint probability density function is set up the numerical solving algorithm where the newmark beta time integration algorithm and the finite difference method with lax wendroff difference scheme are brought together is studied in the numerical examples free vibration of a single degree of freedom non linear conservative system and dynamic responses of an 8 storey shear structure with bilinear hysteretic restoring forces subjected to harmonic excitation and seismic excitation respectively are investigated the investigations indicate that the probability density functions of dynamic responses of non linear stochastic structures are usually irregular and far from the well known distribution types they exhibit obvious evolution characteristics the comparisons with the analytical solution and monte carlo simulation method demonstrate that the proposed pdem is of fair accuracy and efficiency copyright 2005 john wiley sons ltd

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

multiple positive solutions for first order nonlinear integro differential equations in banach spaces by using the fixed point index theory the author obtains the existence of two positive solutions for a boundary value problem of first order nonlinear integro differential equations on an infinite interval in a real banach space

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

multiple positive solutions for nth order impulsive integro differential equations in banach spaces abstract in this paper the author obtains the existence of multiple positive solutions for a boundary value problem of n th order nonlinear impulsive integro differential equations on an infinite interval in a banach space by means of fixed point index theory of completely continuous operators

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

adaptive mesh strategies for the spectral element method abstract an adaptive spectral element method has been developed for the efficient solution of time dependent partial differential equations adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the one dimensional viscous burgers equation and the two dimensional navier stokes equations for driven flow in a cavity sharp gradients singularities and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used the adaptive formulation presents significant increases in efficiency flexibility and general capabilities for high order spectral methods

improving point selection in cubature by a new discrepancy reasonable point set selection is of paramount importance to the accuracy of high dimensional integrals that will be encountered in various disciplines in the present paper to improve the point selection and to overcome the computational complexity of evaluating classical discrepancies the concept of extended f discrepancy ef discrepancy and generalized f discrepancy gf discrepancy of a point set is introduced and justified by comparative studies with other existing discrepancies meanwhile the extensions of the koksma hlawka inequality for ef discrepancy are proved and a conjecture for gf discrepancy is put forward and discussed this gf discrepancy is then employed as the objective function when selecting the optimal rotation angles in the rotation transform of the quasi symmetric point method q spm meanwhile it is also proved that the rotation transform will keep the degree of algebraic accuracy a genetic algorithm is adopted to solve the optimization problem several numerical examples a

multiplicity of solutions to two point boundary value problems for second order impulsive differential equations abstract in this paper we obtain some sufficient condition for the existence of at least three solutions to two point boundary value problems for second order impulsive differential equations by using leray schauder degree theory we assume the existence of two pairs of strict lower and upper solutions and the nonlinear term f satisfies a nagumo s growth condition with respect to the first order derivative

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

structure of some categories of representations of infinite dimensional lie algebras the highest weight representations of a finite dimensional complex semisimple lie algebra 9 have been studied extensively and a number of interesting results have been obtained in the last few years an extensive bibliography is given in 7 in 2 3 a category cy of representations of g was introduced which turns out to be the proper setting for considering various questions regarding highest weight representations the aim of this paper is to extend the study of this notion to a kac moody algebra or more generally to a contragredient lie algebra g a corresponding to a square matrix a over c the study of such algebras was begun independently in 8 14 and subsequently a number of interesting results were proved these results show that many important properties from the finite dimensional setup i e from the context of 9 can be proved in the general setup involving g a in particular the notion of category was extended in 9 to the case of kac moody algebras and used in the proof of combinatorial identities

positive solutions for nth order nonlinear impulsive singular integro differential equations on infinite intervals in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

positive solutions of an infinite boundary value problem for nth order nonlinear impulsive singular integro differential equations in banach spaces abstract this paper obtains the existence of positive solutions for an infinite boundary value problem of nth order nonlinear impulsive singular integro differential equations on the half line in a banach space by means of fixed point theory

existence of positive solutions for nth order nonlinear impulsive singular integro differential equations in banach spaces abstract this paper obtains the existence of positive solutions for a boundary value problem of n th order nonlinear impulsive singular integro differential equations on an infinite interval in a banach space by means of the fixed point theory of completely continuous operators

multiple positive solutions of a m point boundary value problem for 2nth order singular integro differential equations in banach spaces abstract in this paper we obtain the existence of multiple positive solutions of a m point boundary value problem for 2 n th order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

a unique positive solution for nth order nonlinear impulsive singular integro differential equations on unbounded domains in banach spaces in this paper the fixed point theory combined with a monotone iterative technique is used to investigate the unique positive solution of a boundary problem for nth order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in a banach space the conditions for the existence of a unique positive solution are established in addition an explicit iterative sequence for approximating the solution of the boundary value problem is derived together with an error estimate furthermore the conditions of the theorems can be easily verified

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

the existence of solutions of infinite boundary value problems for first order impulsive differential systems in banach spaces in this paper the existence of solutions of infinite boundary value problems for first order impulsive differential systems is obtained by means of the schauder fixed point theorem in a banach space

multiple positive solutions of a boundary value problem for a class of 2 nth order m point singular integro differential equations in banach spaces in this paper we obtain the existence of multiple positive solutions of a boundary value problem for 2nth order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

the multi element probabilistic collocation method me pcm error analysis and applications stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters in this work we present and examine the multi element probabilistic collocation method me pcm which is a generalized form of the probabilistic collocation method in the me pcm the parametric space is discretized and a collocation cubature grid is prescribed on each element both full and sparse tensor product grids based on gauss and clenshaw curtis quadrature rules are considered we prove analytically and observe in numerical tests that as the parameter space mesh is refined the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness in addition the l 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided numerical examples demonstrating adaptive me pcm are shown including low regularity problems and long time integration we test the me pcm on two dimensional navier stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions while the convergence rate of me pcm deteriorates in 50 dimensions the error in the mean and variance is two orders of magnitude lower than the error obtained with the monte carlo method using only a small number of samples e g 100 the computational cost of me pcm is found to be favorable when compared to the cost of other methods including stochastic galerkin monte carlo and quasi random sequence methods

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

extremal solutions for nth order impulsive integro differential equations on the half line in banach spaces abstract by establishing a comparison result and using the upper and lower solutions the author obtains the existence of maximal and minimal solutions for a boundary value problem of n th order nonlinear impulsive integro differential equations on the half line in banach spaces

multiple positive solutions of a boundary value problem for nth order impulsive integro differential equations in a banach space abstract in this paper the author obtains the existence of multiple positive solutions for a boundary value problem of a class of nth order nonlinear impulsive integro differential equations on an infinite interval in a banach space by means of the fixed point index theory of completely continuous operators

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

strategy for selecting representative pointsviatangent spheres in the probability density evolution method a strategy of selecting efficient integration points via tangent spheres in the probability density evolution method pdem for response analysis of non linear stochastic structures is studied the pdem is capable of capturing instantaneous probability density function of the stochastic dynamic responses the strategy of selecting representative points is of importance to the accuracy and efficiency of the pdem in the present paper the centers of equivalent non overlapping tangent spheres are used as the basis to construct a representative point set an affine transformation is then conducted and a hypersphere sieving is imposed for spherically symmetric distributions construction procedures of centers of the tangent spheres are elaborated the features of the point sets via tangent spheres including the discrepancy and projection ratio are observed and compared with some other typical point sets the investigations show that the discrepancies of the point sets via tangent spheres are in the same order of magnitude as the point sets by the number theoretical method in addition it is observed that rotation transformation could greatly improve the projection ratios numerical examples show that the proposed method is accurate and efficient for situations involving up to four random variables copyright 2007 john wiley sons ltd

extremal solutions of nonlinear impulsive integrodifferential equations in banach spaces abstract in this paper we combine the monotone iterative technique with fixed point theory to investigate the extremal solutions of nonlinear impulsive integrodifferential equations in banach spaces existence criteria are obtained and several examples are worked out

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

multiple positive solutions for first order nonlinear impulsive integro differential equations in a banach space by using the fixed point index theory the author obtains the existence of two positive solutions for a boundary value problem of first order nonlinear impulsive integro differential equations on an infinite interval with an infinite number of impulsive times in a real banach space

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

triple positive solutions for boundary value problems on infinite intervals abstract this paper deals with the existence of triple positive solutions for sturm liouville boundary value problems of second order nonlinear differential equation on a half line by using a fixed point theorem in a cone due to avery peterson we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

finite elements for stochastic media problems we consider discretization methods for stochastic partial differential equations which are used to model media with stochastic properties the wick product formulation of the stochastic pde is used and it is shown how it may be discretized both spatially and in the stochastic part a stochastic variational principle serves as the guideline for the discretization procedure it is shown that both the projection onto the wiener chaos and the hermite transform lead to the same fully discrete equations the application examples show that there are other advantageous bases besides the karhunen loeve expansion and that the methods may be used in a variety of settings

infinite boundary value problems for nth order nonlinear impulsive integro differential equations in banach spaces abstract in this paper the existence of unbounded nonnegative solutions of a boundary value problem for n th order impulsive integro differential equations on an infinite interval in banach spaces is obtained by means of the monch fixed point theory in a banach space

new interpolatory quadrature formulae with chebyshev abscissae abstract we study interpolatory quadrature formulae relative to the legendre weight function on 1 1 having as nodes the zeros of any one of the four chebyshev polynomials of degree n plus one of the points 1 or 1 in particular we derive explicit formulae for the weights and examine their positivity we determine the precise degree of exactness we obtain asymptotically optimal error bounds and we examine the definiteness f these quadrature formulae in addition we establish their convergence for riemann integrable functions on 1 1 as well as for functions having a monotonic singularity at 1 or 1

dynamic refinement algorithms for spectral element methods abstract a numerical method is presented that combines high order domain decomposition and adaptive mesh refinement the spatial discretization is based on a nonconforming spectral element method meshes are generated by adaptive refinement of a quadtree using either local solution gradients or the local polynomial spectrum as a refinement criteria the method is applied to solve scalar test problems and to simulate the flow in a lid driven cavity at moderate reynolds numbers

positive solutions of an infinite boundary value problem for nth order nonlinear impulsive singular integro differential equations in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

minimal nonnegative solutions for nth order integro differential equations in banach spaces1this research work was supported by the national natural science foundation of china and the doctoral program foundation of the state education commission of china 1 in this paper the cone theory and monotone iterative technique are used to investigate the minimal nonnegative solutions of initial value problems ivps for nth order nonlinear integro differential equations of mixed type on an infinite interval in a banach space