numerical computations of coarsening in the one dimensional cahn hilliard model of phase separation time dependent solutions of the cahn hilliard equation are studied numerically in particular heteroclinic orbits which connect different equilibrium solutions at t and t are sought thus boundary value problems in space time are computed this computation requires an investigation of the stability of equilibria since projections onto the stable and unstable manifolds determine the boundary conditions at t and t this stability analysis is then followed by solution of the appropriate boundary value problem in space time the results obtained cannot be found by standard initial value simulations by specifying the two steady states at t appropriately it is possible to find orbits reflecting a given degree of coarsening over the time evolution this gives a clear picture of the dynamic coarsening admissible in the equation it also provides an understanding of orbits on the global attractor for the equation

ls svr based solving volterra integral equations abstract in this paper a novel hybrid method is presented for solving the second kind linear volterra integral equations due to the powerful regression ability of least squares support vector regression ls svr we approximate the unknown function of integral equations by using ls svr in intervals with known numerical solutions the trapezoid quadrature is used to approximate subsequent integrations in intervals with unknown numerical solutions the feasibility of the proposed method is examined on some integral equations experimental results of comparison with analytic and repeated modified trapezoid quadrature method s solutions show that the proposed algorithm could reach a very high accuracy the proposed algorithm could be a good tool for solving the second kind linear volterra integral equations

spinodal decomposition for the cahn hilliard equation in higher dimensions nonlinear dynamics this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation where n n 1 2 3 is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u 3 based on the results of 26 the nonlinear cahn hilliard equation will be discussed this equation generates a nonlinear semiflow in certain affine subspaces of h 2 in a neighborhood u e with size proportional to e n around the constant solution where lies in the spinodal region we observe the following behavior within a local inertial manifold containing there exists a finite dimensional invariant manifold which dominates the behavior of all solutions starting with initial conditions from a small ball around with probability almost 1 the dimension of is proportional to e n and the elements of exhibit a common geometric quantity which is strongly related to a characteristic wavelength proportional to e

pattern formation in a nonlinear model for animal coats abstract several models have been proposed for describing the formation of animal coat patterns we consider reaction diffusion models due to murray which rely on a turing instability for the pattern selection in this paper we describe the early stages of the pattern formation process for large domain sizes this includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one two and three dimensional base domains these results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the cahn hilliard equation we use techniques of dynamical systems viewing solutions of the reaction diffusion model in terms of nonlinear semiflows our results are applicable to any parabolic system exhibiting a turing instability

topological microstructure analysis using persistence landscapes phase separation mechanisms can produce a variety of complicated and intricate microstructures which often can be difficult to characterize in a quantitative way in recent years a number of novel topological metrics for microstructures have been proposed which measure essential connectivity information and are based on techniques from algebraic topology such metrics are inherently computable using computational homology provided the microstructures are discretized using a thresholding process however while in many cases the thresholding is straightforward noise and measurement errors can lead to misleading metric values in such situations persistence landscapes have been proposed as a natural topology metric common to all of these approaches is the enormous data reduction which passes from complicated patterns to discrete information it is therefore natural to wonder what type of information is actually retained by the topology in the present paper we demonstrate that averaged persistence landscapes can be used to recover central system information in the cahn hilliard theory of phase separation more precisely we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot considering that persistent homology only measures discrete connectivity information regardless of the size of the topological features these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated we believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations

linear discriminant analysis based on l1 norm maximization linear discriminant analysis lda is a well known dimensionality reduction technique which is widely used for many purposes however conventional lda is sensitive to outliers because its objective function is based on the distance criterion using l2 norm this paper proposes a simple but effective robust lda version based on l1 norm maximization which learns a set of local optimal projection vectors by maximizing the ratio of the l1 norm based between class dispersion and the l1 norm based within class dispersion the proposed method is theoretically proved to be feasible and robust to outliers while overcoming the singular problem of the within class scatter matrix for conventional lda experiments on artificial datasets standard classification datasets and three popular image databases demonstrate the efficacy of the proposed method

unexpectedly linear behavior for the cahn hilliard equation this paper gives theoretical results on spinodal decomposition for the cahn hillard equation we prove a mechanism which explains why most solutions for the cahn hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r namely most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected the cahn hilliard equation depends on a small parameter epsilon modeling the atomic scale interaction length we quantify the behavior of solutions as epsilon rightarrow 0 specifically we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance o epsilon 2 n 2 up to a ball of radius r in the h 2 omega norm where r is proportional to epsilon 1 rho n 4 as epsilon to 0 here n in 1 2 3 denotes the

the theoretical foundation of the adomian method abstract in this paper we work out the theoretical basis of the adomian method this powerful method that g adomian has developed in the beginning of the 1980 s supplies analytical approximations to the solution of many kinds of equations thanks to our decomposition theory we justify the practical method and we generalize the convergence results obtained by y cherruault afterward we explain how to solve rigorously equations with linear nonlinear or composed operator and also equations with several terms or with a second member

an efficient method for the numerical solution of hammerstein mixed vf integral equations on 2d irregular domains abstract in this work we adapt the implicity linear collocation method for solving nonlinear two dimensional mixed volterra fredholm integral equations of hammerstein type by using imq rbfs on a non rectangular domain the proposed method is mesh free and it is independent of the geometry of domain convergence analysis of the proposed method together with some benchmark examples is provided which support its reliability and numerical stability

a numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions abstract in this paper we use hat basis functions to solve the system of fredholm integral equations sfies and the system of volterra integral equations svies of the second kind this method converts the system of integral equations into a linear or nonlinear system of algebraic equations also we consider the order of convergence of the method and show that it is o h 2 application of the method on some examples show its accuracy and efficiency

maximum norms of random sums and transient pattern formation many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes in this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the cahn hilliard equation this model depends on a small parameter and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero recent results on spinodal decomposition have produced such lower bounds unfortunately for higher dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments the bounds exhibit a dependence on the dimension of the domain which from a theoretical point of view seemed unavoidable but which could not be observed in practice in this paper we resolve this apparent paradox by employing probabilistic methods we can improve the lower bounds for certain domains and remove the dimension dependence we thereby obtain optimal results which close the gap between analytical methods and numerical observations and provide more insight into the nature of the decomposition process we also indicate how our results can be adapted to other situations

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

a modified homotopy perturbation method for solving the nonlinear mixed volterra fredholm integral equation the purpose of this paper is to obtain the approximation solution of the strongly nonlinear mixed volterra fredholm integral equation vfie for some strongly nonlinear problems the traditional homotopy perturbation method is divergent so we propose a modified homotopy perturbation method which is still convergent when solving the strongly nonlinear mixed vfie by means of this method an algorithm is successfully established for solving the strongly nonlinear mixed vfie and the convergence of the algorithm is proved strictly finally several examples are presented to illustrate the application of the algorithm

a new computational method for volterra fredholm integral equations abstract the main purpose of this article is to demonstrate the use of the adomian decomposition method for mixed nonlinear volterra fredholm integral equations a bound is also given for the adomian decomposition series finally numerical examples are presented to illustrate the implementation and accuracy of the decomposition method

asymptotic expansion for the trapezoidal nystro m method of linear volterra fredholm equations abstract integral equations of mixed volterra fredholm type arise in various physical and biological problems in the present paper we study the particular trapezoidal nystrom method and we obtain the asymptotic error expansion for the trapezoidal nystrom solution of the linear volterra fredholm equation so the richardson extrapolation can be applied this will increase the accuracy of the numerical solution greatly some numerical examples are given to illustrate this theory

principal component extraction using recursive least squares learning a new neural network based approach is introduced for recursive computation of the principal components of a stationary vector stochastic process the neurons of a single layer network are sequentially trained using a recursive least squares squares rls type algorithm to extract the principal components of the input process the optimality criterion is based on retaining the maximum information contained in the input sequence so as to be able to reconstruct the network inputs from the corresponding outputs with minimum mean squared error the proof of the convergence of the weight vectors to the principal eigenvectors is also established a simulation example is given to show the accuracy and speed advantages of this algorithm in comparison with the existing methods finally the application of this learning algorithm to image data reduction and filtering of images degraded by additive and or multiplicative noise is considered

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

spinodal decomposition for the cahn hilliard cook equation this paper gives theoretical results on spinodal decomposition for the stochastic cahn hilliard cook equation which is a cahn hilliard equation perturbed by additive stochastic noise we prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation leading to the formation of complex patterns of a characteristic size

monte carlo simulations for spinodal decomposition this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation namely we are interested in why most solutions to the cahn hilliard equation which start near a homogeneous equilibrium u 0 in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r in a hilbert space centered at u 0 there are two mathematical explanations for spinodal decomposition due to grant and to maier paape and wanner in this paper we numerically compare these two mathematical approaches in fact we are able to synthesize the understanding we gain from our numerics with the approach of maier paape and wanner leading to a better understanding of the underlying mechanism for this behavior with this new approach we can explain spinodal decomposition for a longer time and larger radius than either of the previous two approaches a rigorous mathematical explanation is contained in a separate paper our approach is to use monte carlo simulations to examine the dependence of r the radius to which spinodal decomposition occurs as a function of the parameter e of the governing equation we give a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points we observe and can show rigorously that the behavior of most solutions originating near the equilibrium is determined completely by the linearization for an unexpectedly long time we explain the mechanism for this unexpectedly linear behavior and show that for some exceptional solutions this cannot be observed we also describe the dynamics of these exceptional solutions

continuous time collocation methods for volterra fredholm integral equations integral equations of mixed volterra fredholm type arise in various physical and biological problems in the present paper we study continuous time collocation time discretization and their global and discrete convergence properties

numerical solution of volterra fredholm integral equations via modification of hat functions abstract in this paper a new numerical approach is developed for solving linear and nonlinear volterra fredholm integral equations the fundamental structure of the presented method is based on the modification of hat functions the properties of modification of hat functions mhfs are first presented after implementation of our scheme the solution of the main problem would be transformed into the solution of a system of linear or nonlinear algebraic equations also an error analysis is provided under several mild conditions in addition examples are presented to illustrate the pertinent features of the method and the results are discussed

a hybrid legendre block pulse method for mixed volterra fredholm integral equations abstract mixed volterra fredholm integral equations can arise from mathematical modelling of the spatio temporal development of an epidemic in this paper a hybrid legendre block pulse method is used to provide solutions to a general integral equation of this type the main idea of this approach is to reduce a volterra fredholm integral equation to a system of algebraic equations an order convergence analysis is provided the accuracy of the proposed method is then confirmed by solving some problems the first being a nonlinear epidemic problem

spinodal decomposition for the cahn hilliard equation in higher dimensions part i probability and wavelength estimate this paper is the first in a series of two papers addressing the phenomenon of spinodal decomposition for the cahn hilliard equation where is a bounded domain with sufficiently smooth boundary and f is cubic like for example f u u u3 we will present the main ideas of our approach and explain in what way our method differs from known results in one space dimension due to grant 26 furthermore we derive certain probability and wavelength estimates the probability estimate is needed to understand why in a neighborhood of a homogeneous equilibrium u0 of the cahn hilliard equation with mass in the spinodal region a strongly unstable manifold has dominating effects this is demonstrated for the linearized equation but will be essential for the nonlinear setting in the second paper 37 as well moreover we introduce the notion of a characteristic wavelength for the strongly unstable directions

numerical solution for three dimensional nonlinear mixed volterra fredholm integral equations via three dimensional block pulse functions we are concerned here with a three dimensional nonlinear mixed volterra fredholm integral equations of the second kind which include many key integral that appear in the theory of nonlinear parabolic boundary value problems the existence of a unique solution will be proved a new numerical method for solving these type of equations will be presented the method is based upon three dimensional block pulse functions approximation in addition convergence analysis of the method is discussed illustrative examples are included to demonstrate the validity and applicability of the technique

solving second kind integral equations by galerkin methods with hybrid legendre and block pulse functions in this paper we use a combination of legendre and block pulse functions on the interval 0 1 to solve the linear integral equation of the second kind we convert the integral equation to a system of linear equations we show that our estimates have a good degree of accuracy

asymptotic error expansion for the nystro m method for a nonlinear volterra fredholm integral equation in this paper we study the numerical solution of general nonlinear mixed volterra fredholm integral equations by the trapezoidal nystrom method it focuses on the derivation and analysis of asymptotic error expansion of the nystrom solution providing the basis for richardson s extrapolation this will greatly increase the accuracy of the numerical solution some numerical results are given to illustrate this theory

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

second phase spinodal decomposition for the cahn hilliard cook equation we consider the dynamics of a nonlinear partial differential equation perturbed by additive noise assuming that the underlying deterministic equation has an unstable equilibrium we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium more precisely we show that most trajectories starting at the unstable equilibrium are driven away in two stages after passing through a cylindrical region most trajectories diverge from the deterministic equilibrium through a cone shaped region which is centered around a finite dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation and on which the influence of the nonlinearity is surprisingly small this abstract result is then applied to explain spinodal decomposition in the stochastic cahn hilliard cook equation on a domain g this equation depends on a small interaction parameter e 0 and one is generally interested in asymptotic results as e 0 specifically we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach e 2 dim g 2 with respect to the h 2 g norm

solution of physical problems by decomposition recent generalizations are discussed and results are presented for the theory and applications of the decomposition method application is made to the duffing equation with an error of 0 0001 in only four terms and less than 10 16 in thirteen terms of the decomposition series application is also made to a dissipative wave equation a matrix riccati equation and advection diffusion nonlinear transport

an optimal orthonormal system for discriminant analysis abstract this paper proposes a new discriminant analysis with orthonormal coordinate axes of the feature space in general the number of coordinate axes of the feature space in the traditional discriminant analysis depends on the number of pattern classes therefore the discriminatory capability of the feature space is limited considerably the new discriminant analysis solves this problem completely in addition it is more powerful than the traditional one in so far as the discriminatory power and the mean error probability for coordinate axes are concerned this is also shown by a numerical example

using runge kutta method for numerical solution of the system of volterra integral equation in this paper we introduce some of general methods of runge kutta they are using for numerical solution of volterra integral equations of the second kind h brunner p j vander houwen the numerical solution of volterra equation cwi monograph 3 north holland amsterdam 1986 p w sharp j h verner extended explicit bel tyukov pairs of orders 4 and volterra integral equations of the second kind appl numer math from author s homepage we first introduce pouzet volterra runge kutta methods implicit and explicit forms pvrk and then explain about some extended explicit bel tyukov pairs ebvrk each pair uses six stages and consists of on order 3 formula completely embedded in order 4 formula in numerical examples this method uses for solving a system of volterra integral equations and comparison with iterative methods as comput math appl 37 1999 1 nonlinear stud 6 1 1999

numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets in this work we present a computational method for solving nonlinear fredholm integral equations of the second kind which is based on the use of haar wavelets error analysis is worked out that shows efficiency of the method finally we also give some numerical examples

representation of exact solution for the nonlinear volterra fredholm integral equations abstract in this paper the representation of the exact solution for the nonlinear volterra fredholm integral equations will be obtained in the reproducing kernel space the exact solution is given by the form of series its approximate solution is obtained by truncating the series and a new numerical approximate method is obtained the error of the approximate solution is monotone deceasing in the sense of w 2 1 a b the intrinsic merit of the method given in the paper lies in its speedy convergence in addition it must be pointed out that additional conditions as linearization or unjustified assumptions are not required for nonlinear cases

two point iterative methods for solving nonlinear equations in this study a new root finding technique for solving nonlinear equations is proposed then two new more algorithms are derived from this new technique by employing the adomian decomposition method adm these three algorithms require two starting values that do not necessarily bracketing the root of a given nonlinear equation however when the starting values are closed enough or bracketed the root then the proposed methods converge to the root faster than the secant method another advantage over all iterative methods is that the proposed methods usually converge to two distinct roots when the handled nonlinear equation has more than one root that is the odd iterations of the new techniques converge to a root and the even iterations converge to another root some numerical examples including a sine polynomial equation are solved by using the proposed methods and compared with the secant method perfect agreements were found

on self organizing algorithms and networks for class separability features we describe self organizing learning algorithms and associated neural networks to extract features that are effective for preserving class separability as a first step an adaptive algorithm for the computation of q sup 1 2 where q is the correlation or covariance matrix of a random vector sequence is described convergence of this algorithm with probability one is proven by using stochastic approximation theory and a single layer linear network architecture for this algorithm is described which we call the q sup 1 2 network using this network we describe feature extraction architectures for 1 unimodal and multicluster gaussian data in the multiclass case 2 multivariate linear discriminant analysis lda in the multiclass case and 3 bhattacharyya distance measure for the two class case the lda and bhattacharyya distance features are extracted by concatenating the q sup 1 2 network with a principal component analysis network and the two layer network is proven to converge with probability one every network discussed in the study considers a flow or sequence of inputs for training numerical studies on the performance of the networks for multiclass random data are presented

numerical computation of the nonlinear feedback operators for the nonquadratic time variant optimal control problems in this paper a reliable numerical algorithm for the nonlinear feedback operators for the time variant optimal control with nonquadratic criteria is constructed our approach rests mainly on the modified adomian decomposition method the convergence of the decomposition method as applied to the synthesis equations is investigated the proposed scheme avoids the complexity provided by using other techniques

a reliable algorithm for obtaining positive solutions for nonlinear boundary value problems abstract in this paper we propose an algorithm for solving the nonlinear two point boundary value problem that has at least one positive solution 1 6 for in a compatible interval our method stems mainly from combining the decomposition series solution obtained by adomian decomposition method with pade approximates the validity of the approach is verified through illustrative numerical examples

some extended explicit bel tyukov pairs for volterra integral equations of the second kind we derive and investigate a family of pairs of extended explicit bel tyukov runge kutta ebvrk formulas to treat volterra integral equations of the second kind each pair uses six stages and consists of an order 3 formula completely embedded in an order 4 formula we examine the structure of ebvrk pairs to facilitate their derivation and in particular we show that at least six stages are needed to form such pairs this design is motivated to allow a variable stepsize in order to satisfy a prescribed error tolerance with a minimum of computation some numerical examples illustrate the procedures

fast adaptive algorithms and networks for class separability features abstract in this article we introduce accelerated algorithms for linear discriminant analysis lda and feature extraction from unimodal multiclass gaussian data current adaptive methods based on the gradient descent optimization technique use a fixed or a monotonically decreasing step size in each iteration which results in a slow convergence rate here we use a variable step size optimally computed in each iteration using the steepest descent method in order to accelerate the convergence of the algorithm based on the new adaptive algorithm we present a self organizing neural network for adaptive computation of the square root of the inverse covariance matrix 1 2 and use it i in a network for optimal feature extraction from gaussian data and ii in cascaded form with a principal component analysis network for lda experimental results demonstrate fast convergence and high stability of the algorithm and justify its advantages for on line pattern recognition applications with stationary and non stationary input data

a new lda based face recognition system which can solve the small sample size problem a new lda based face recognition system is presented in this paper linear discriminant analysis lda is one of the most popular linear projection techniques for feature extraction the major drawback of applying lda is that it may encounter the small sample size problem in this paper we propose a new lda based technique which can solve the small sample size problem we also prove that the most expressive vectors derived in the null space of the within class scatter matrix using principal component analysis pca are equal to the optimal discriminant vectors derived in the original space using lda the experimental results show that the new lda process improves the performance of a face recognition system signi cantly 2000 pattern recognition society published by elsevier science ltd all rights reserved

a reliable treatment for mixed volterra fredholm integral equations the main goal of this paper is to demonstrate the use of the modified decomposition method for mixed nonlinear volterra fredholm integral equations the modified method combined with the noise terms phenomena may provide the exact solution by using two iterations only two numerical illustrations are given to show the pertinent features of the technique

he s variational iteration method for solving nonlinear mixed volterra fredholm integral equations in this paper he s variational iteration method is used to give the approximate solution of nonlinear mixed volterra fredholm integral equations the method constructs a convergent sequence of functions which approximates the exact solution with few iterations numerical results and a comparison with the exact solution are given which reveal its efficiency

an efficient numerical approximation for the linear class of mixed integral equations in this paper an efficient numerical and analytical method for the linear class of mixed integral equations is presented the bivariate chebyshev collocation method for solving the linear volterra fredholm integral equations of the second kind is examined theoretically and numerically its focus lies mainly on the truncated bivariate chebyshev expansion and chebyshev collocation points

on the numerical solution of nonlinear volterra fredholm integral equations by collocation methods particular cases of nonlinear mixed volterra fredholm integral equations of the second kind arise in the mathematical modeling of the spatio temporal development of an epidemic this paper is concerned with the numerical solution of general integral equations of this type by continuous time and discrete time spline collocation methods its focus is on the derivation and analysis of methods of high order of convergence

low cost numerical solution for three dimensional linear and nonlinear integral equations via three dimensional jacobi polynomials in recent years numerical methods have been introduced to solve two dimensional volterra and fredholm integral equations in this study a numerical scheme is constructed to solve classes of linear and nonlinear three dimensional integral equations volterra fredholm and mixed volterrafredholm this operational approach is proposed to easily and directly solve these equations at low computational costs the scheme is based on the jacobi polynomials on the interval 0 1 where three variable jacobi polynomials are introduced and their operational matrices of integration and product are derived compared to other existing methods for multidimensional problems the jacobi operational method eliminates the time consuming computations and solely employs the one dimensional operational matrix to construct corresponding multidimensional operational matrices the absolute error of the proposed method is almost constant on the studied interval even at higher dimensions confirming the stability of the proposed operational jacobi method required theorems on the convergence of the method are proved in jacobi weighted sobolev space it is established that the error function vanishes as n increases the method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the blockpulse functions differential transform and degenerate kernel methods

numerical solution of the mixed volterra fredholm integro differential multi term equations of fractional order abstract this paper aims to introduce two effective numerical approaches for solving the mixed volterra fredholm integro differential equation of fractional order and multi order with initial values despite the fact that the second approach transfers the integro differential equations into a system of algebraic equations through the usage of operational matrices these problems can be transferred to a system of algebraic equations by expanding the solution s highest order derivative through the block pulse functions bpfs this is done by using the generalized operational matrices of bpfs for differentiation along with the fractional calculus properties in the first scheme in which convergence of the solution obtained has been shown in the following the accuracy and applicability of two proposed approaches will be compared by some relevant numerical examples for which the exact solution is known

numerical solution of singular volterra integral equations system of convolution type by using operational matrices abstract in this paper we use operational matrices of piecewise constant orthogonal functions on the interval 0 1 to solve singular volterra integral equations system of convolution type we first obtain laplace transform of the problem and then we find numerical inversion of laplace transform by operational matrices numerical examples show that the approximate solutions have a good degree of accuracy

algorithms and networks for accelerated convergence of adaptive lda we introduce and discuss new accelerated algorithms for linear discriminant analysis lda in unimodal multiclass gaussian data these algorithms use a variable step size optimally computed in each iteration using i the steepest descent ii conjugate direction and iii newton raphson methods in order to accelerate the convergence of the algorithm current adaptive methods based on the gradient descent optimization technique use a fixed or a monotonically decreasing step size in each iteration which results in a slow convergence rate furthermore the convergence of these algorithms depends on appropriate choices of the step sizes the new algorithms have the advantage of automatic optimal selection of the step size using the current data samples based on the new adaptive algorithms we present self organizing neural networks for adaptive computation of s 1 2 and use them in cascaded form with a pca network for lda experimental results demonstrate fast convergence and robustness of the new algorithms and justify their advantages for on line pattern recognition applications with stationary and non stationary multidimensional input data

the numerical solution of fifth order boundary value problemsby the decomposition method in this paper we present a fast and accurate numerical scheme for the solution of fifth order boundary value problems with two point boundary conditions the adomian decomposition method and a modified form of this method are applied to construct the numerical solution the new approach provides the solution in the form of a rapidly convergent series and not at grid points two numerical illustrations are given to show the pertinent features of technique

a computational method for system of volterra fredholm integral equations adomian decomposition method has been applied for solving volterra fredholm equation by maleknejad k maleknejad m hadizadeh a new computational method for volterra fredholm integral equations comput math appl 37 1999 1 8 in 1999 in this paper we solve the system of nonlinear volterra fredholm integral equations by adomian decomposition method some examples are solved to illustrate the accuracy of this method

numerical solution of linear fredholm integral equation by using hybrid taylor and block pulse functions in this paper we use a combination of taylor and block pulse functions on the interval 0 1 that is called hybrid functions to estimate the solution of a linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy

the modified decomposition method for analytic treatment of differential equations the modified decomposition method is applied for analytic treatment of nonlinear differential equations the modified method accelerates the rapid convergence of the series solution dramatically reduces the size of work and provides the exact solution by using few iterations only without any need to the so called adomian polynomials numerical illustrations that include nonlinear pdes nonlinear klein gordon equations and nonlinear lane emden equation are investigated to show the pertinent features of the technique

algorithms for accelerated convergence of adaptive pca we derive and discuss adaptive algorithms for principal component analysis pca that are shown to converge faster than the traditional pca algorithms due to oja and karhunen 1985 sanger 1989 and xu 1993 it is well known that traditional pca algorithms that are derived by using gradient descent on an objective function are slow to converge furthermore the convergence of these algorithms depends on appropriate choices of the gain sequences since online applications demand faster convergence and an automatic selection of gains we present new adaptive algorithms to solve these problems we first present an unconstrained objective function which can be minimized to obtain the principal components we derive adaptive algorithms from this objective function by using 1 gradient descent 2 steepest descent 3 conjugate direction and 4 newton raphson methods although gradient descent produces xu s lmser algorithm the steepest descent conjugate direction and newton raphson methods produce new adaptive algorithms for pca we also provide a discussion on the landscape of the objective function and present a global convergence proof of the adaptive gradient descent pca algorithm using stochastic approximation theory extensive experiments with stationary and nonstationary multidimensional gaussian sequences show faster convergence of the new algorithms over the traditional gradient descent methods we also compare the steepest descent adaptive algorithm with state of the art methods on stationary and nonstationary sequences

chebyshev wavelets approach for nonlinear systems of volterra integral equations in this paper a new approach for solving nonlinear systems of volterra integral equations has been proposed the method is based on chebyshev wavelets approximations the method is described and after that the error is analyzed at the end some examples are presented to illustrate the ability and simplify of the method and the results reveal the effectiveness of the technique

posteriori error estimates for the nonlinear volterra fredholm integral equations abstract we study the numerical approximation of the nonlinear volterra fredholm integral equations by combining the discrete time collocation method 1 and the new formulation of kumar and sloan 2 which converts an integral equation of the conventional hammerstein form into a conductive form for approximation by a collocation method the intrinsic merit of this alternative formulation lies in its computational savings posteriori error estimates of the method for two typical nonlinearities i e algebraic and exponential nonlinearity are obtained some remarks on the generalization of the method to higher dimensional cases are offered and finally some numerical examples are given

fast subspace tracking and neural network learning by a novel information criterion we introduce a novel information criterion nic for searching for the optimum weights of a two layer linear neural network nn the nic exhibits a single global maximum attained if and only if the weights span the desired principal subspace of a covariance matrix the other stationary points of the nic are unstable saddle points we develop an adaptive algorithm based on the nic for estimating and tracking the principal subspace of a vector sequence the nic algorithm provides a fast on line learning of the optimum weights for the two layer linear nn we establish the connections between the nic algorithm and the conventional mean square error mse based algorithms such as oja s algorithm oja 1989 lmser past apex and gha the nic algorithm has several key advantages such as faster convergence which is illustrated through analysis and simulation

numerical solution of volterra integral equations of the second kind with convolution kernel by using taylor series expansion method in this paper we present a taylor series expansion method for a class of volterra integral equations of second kind with smooth or weakly singular kernels this method use taylor series approximation method for integral equation and transform the integral equation to an nth order linear differential equation boundary conditions for differential equation produce in easy way this method gives an approximate simple and closed form solution for integral equation some numerical example use to illustrate the accuracy of method