a modified decomposition abstract the maclaurin series is quite limited in comparison to the adomian series obtained in the decomposition method by adding procedures from the decomposition method and the expansion of nonlinearities using the adomian polynomials as well as a recent result of adomian and rach on transformation of series using the above polynomials the maclaurin series can be made much more useful in its applicability however the convergence is still slower than for adomian s results using decomposition

on a solvable system of difference equations of kth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j

on the system of high order rational difference equations xn ayn p yn byn pxn qyn q in this paper we study the behavior of positive solutions of the system of high order rational difference equationsx n ay n p y n by n px n qy n q n 1 2 where p and q are positive integers with p

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

nonlinear transformation of series part ii abstract this paper generalizes the theorem on transformation of series presented in part i to function of several variables the result is valuable in deriving solutions of coupled ordinary differential equations with general nonlinear coupling terms such as u u w or u 1 u n using the modified decomposition method of the authors

on the fuzzy difference equation x n 1 a x n x n m in this paper we study the existence the boundedness and the asymptotic behavior of the positive solutions of the fuzzy equation xn 1 a xn xn m n 0 1 where xn is a sequence of fuzzy numbers a is a fuzzy number and m 1 2

on a system of two nonlinear difference equations abstract in this paper we study the oscillatory behavior the boundedness of the solutions and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equationsxn 1 a yn xn p yn 1 a xn yn q n 0 1 p qare positive integers

an in depth study of graph partitioning measures for perceptual organization in recent years one of the effective engines for perceptual organization of low level image features is based on the partitioning of a graph representation that captures gestalt inspired local structures such as similarity proximity continuity parallelism and perpendicularity over the low level image features mainly motivated by computational efficiency considerations this graph partitioning process is usually implemented as a recursive bipartitioning process where at each step the graph is broken into two parts based on a partitioning measure we focus on three such measures namely the minimum average and normalized cuts the minimum cut partition seeks to minimize the total link weights cut the average cut measure is proportional to the total link weight cut normalized by the sizes of the partitions the normalized cut measure is normalized by the product of the total connectivity valencies of the nodes in each partition we provide theoretical and empirical insight into the nature of the three partitioning measures in terms of the underlying image statistics in particular we consider for what kinds of image statistics would optimizing a measure irrespective of the particular algorithm used result in correct partitioning are the quality of the groups significantly different for each cut measure are there classes of images for which grouping by partitioning does not work well also can the recursive bipartitioning strategy separate out groups corresponding to k objects from each other in the analysis we draw from probability theory and the rich body of work on stochastic ordering of random variables our major conclusion is that optimization of none of the three measures is guaranteed to result in the correct partitioning of k objects in the strict stochastic order sense for all image statistics

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

a further consideration of partial solutions in the decomposition method abstract reference 1 showed that in partial differential equation problems involving linear operator terms with respect to x y z t considerable work is saved by solving only one of the linear operator equations to obtain a so called partial solution the heat equation was shown as a simple example which appeared to be a counter example because one partial solution was identically zero the l x u operator equation led to a zero solution while the l t u operator equation led to the correct solution e t sin x this is a case of asymptotic equality as t rather than equality it arises when boundary conditions are special as discussed in 1 in cases where one operator annihilates the series in a finite number of terms one partial solution may not satisfy the given conditions this paper provides further insight into cases where such an improper solution can be obtained and the resolution of such cases by appropriate expansion of the initial term without making a priori assumptions about the solution

a convenient technique for solving linear and nonlinear abel integral equations by the adomian decomposition method abstract in this paper linear and nonlinear abel integral equations are transformed in such a manner that the adomian decomposition method can be applied some examples with closed form solutions are studied in detail to further illustrate the proposed technique and the results obtained indicate this approach is indeed practical and efficient

on the system of rational difference equations xn a yn 1 xn pyn q yn a xn 1 xn ryn s abstract in this paper we study the behavior of positive solutions of the system of difference equations x n a y n 1 x n p y n q y n a x n 1 x n r y n s n 1 2 where p 2 q 2 r 2 s 2 a is a positive constant and x 1 max p r x 2 max p r x 0 y 1 max q s y 2 max q s y 0 are positive real numbers

curve evolution implementation of the mumford shah functional for image segmentation denoising interpolation and magnification we first address the problem of simultaneous image segmentation and smoothing by approaching the mumford shah 1989 paradigm from a curve evolution perspective in particular we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours each gradient step involves solving an optimal estimation problem for the data within each region connecting curve evolution and the mumford shah functional with the theory of boundary value stochastic processes the resulting active contour model offers a tractable implementation of the original mumford shah model i e without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner various implementations of this algorithm are introduced to increase its speed of convergence we also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions next by generalizing the data fidelity term of the original mumford shah functional to incorporate a spatially varying penalty we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing this more general model leads us to a novel pde based approach for simultaneous image magnification segmentation and smoothing thereby extending the traditional applications of the mumford shah functional which only considers simultaneous segmentation and smoothing

on the positive solutions of the system of rational difference equations xn 1 1 yn k yn 1 yn xn myn m k abstract our aim in this paper is to investigate the periodic nature of solutions of the system of rational difference equations x n 1 1 y n k y n 1 y n x n m y n m k n 0 1 where k is a nonnegative integer m is a positive integer and the initial values x m x m 1 x 0 y m k y m k 1 y 0 are positive real numbers

a new approach to the heat equation an application of the decomposition method abstract the author s decomposition method adomian stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press bellman and adomian partial differential equations methods for their treatment and solution reidel dordrecht 1984 adomian j comput appl math 2 1984 225 230 shown elsewhere to provide computable accurate approximate convergent solutions to nonlinear stochastic operator equations is applied to the heat equation

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

on some systems of difference equations abstract in this note we show that the following systems of difference equations u n 1 v n 1 v n v n 1 u n 1 u n u n 1 v n 1 u n v n 1 u n 1 v n u n 1 u n 1 v n v n 1 v n 1 u n with n n 0 and complex initial values u 0 and v 0 are solvable explicitly by means of riccati equations and based on the formulae for the general solutions we present the behaviour of their solutions as n while the result is natural for the first system it is a bit surprising for the other two systems

edge grouping combining boundary and region information this paper introduces a new edge grouping method to detect perceptually salient structures in noisy images specifically we define a new grouping cost function in a ratio form where the numerator measures the boundary proximity of the resulting structure and the denominator measures the area of the resulting structure this area term introduces a preference towards detecting larger size structures and therefore makes the resulting edge grouping more robust to image noise to find the optimal edge grouping with the minimum grouping cost we develop a special graph model with two different kinds of edges and then reduce the grouping problem to finding a special kind of cycle in this graph with a minimum cost in ratio form this optimal cycle finding problem can be solved in polynomial time by a previously developed graph algorithm we implement this edge grouping method test it on both synthetic data and real images and compare its performance against several available edge grouping and edge linking methods furthermore we discuss several extensions of the proposed method including the incorporation of the well known grouping cues of continuity and intensity homogeneity introducing a factor to balance the contributions from the boundary and region information and the prevention of detecting self intersecting boundaries

analytic solution of nonlinear boundary value problems in several dimensions by decomposition abstract nonlinear partial differential equations are systematically solved by the decomposition method of g adomian for general boundary conditions described by boundary operator equations linear partial differential equations and linear or nonlinear ordinary differential equations become special cases all dealt with by a single procedure

global attractivity in a nonlinear difference equation abstract we consider the nonlinear difference equation x n a k 1 m b k x n k n 0 1 2 where a and bk k 1 2 m are nonnegative numbers with b mk 1bk 0 and we are interested in whether all positive solutions are attracted by the positive equilibrium l a 2 a 2 2 b

on the new solutions to the fuzzy difference equation xn 1 a bxn abstract in this paper using a generalization of division for fuzzy numbers we study the existence and global behavior of the fuzzy difference equation x n 1 a b x n n 0 1 where a and b are positive fuzzy numbers some examples are presented to illustrate the applicability of our results

on first order fuzzy ricatti difference equation this paper deals with the existence the boundedness and the asymptotic behavior of the positive solutions to a first order fuzzy ricatti difference equation xn 1 a xnb xn n 0 1 where xn is a sequence of positive fuzzy numbers a b and the initial value x0 are positive fuzzy numbers moreover an example is given to demonstrate the effectiveness of the results obtained

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

on the positive solutions of the difference equation system xn 1 1yn yn 1 ynxn 1yn 1 we study the positive solutions of the difference equation systemx n 1 1y n y n 1 y nx n 1y n 1 n 0 1 2 where x 1 x 0 and y 0 y 1 are positive real numbers

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

a computational algebraic investigation of the decomposition method for time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat or wave equations is investigated numerically using computational algebraic techniques

an approximate method for solving a class of weakly singular volterra integro differential equations abstract in this paper we present a new approach to resolve linear and nonlinear weakly singular volterra integro differential equations of first or second order by first removing the singularity using taylor s approximation and then transforming the given first or second order integro differential equations into an ordinary differential equation such as the well known legendre degenerate hypergeometric euler or abel equations in such a manner that adomian s asymptotic decomposition method can be applied which permits convenient resolution of these equations some examples with closed form solutions are studied in detail to further illustrate the proposed technique and the results obtained demonstrate this approach is indeed practical and efficient

invariants for difference equations and systems of difference equations of rational form abstract we present some invariants for difference equations and systems of difference equations of rational form we examine two cases in the first case the coefficients are constant and in the second are positive periodic sequences of certain periods

a variational framework for multiregion pairwise similarity based image segmentation variational cost functions that are based on pairwise similarity between pixels can be minimized within level set framework resulting in a binary image segmentation in this paper we extend such cost functions and address multi region image segmentation problem by employing a multi phase level set framework for multi modal images cost functions become more complicated and relatively difficult to minimize we extend our previous work proposed for background foreground separation to the segmentation of images in more than two regions we also demonstrate an efficient implementation of the curve evolution which reduces the computational time significantly finally we validate the proposed method on the berkeley segmentation data set by comparing its performance with other segmentation techniques

on the system of rational difference equations abstract in this paper we study the behavior of the positive solutions of the system of two difference equations x n a 1 y n p y n a y n 1 x n r y n s n 1 2 where p 1 r 1 s 1 a 0 and x1 r x2 r x0 y1 max p s y2 max p s y0 are positive real numbers

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

an optimal graph theoretic approach to data clustering theory and its application to image segmentation a novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated the data to be clustered are represented by an undirected adjacency graph g with arc capacities assigned to reflect the similarity between the linked vertices clustering is achieved by removing arcs of g to form mutually exclusive subgraphs such that the largest inter subgraph maximum flow is minimized for graphs of moderate size approximately 2000 vertices the optimal solution is obtained through partitioning a flow and cut equivalent tree of g which can be efficiently constructed using the gomory hu algorithm 1961 however for larger graphs this approach is impractical new theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size this algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices the new clustering algorithm is applied to the image segmentation problem the segmentation is achieved by effectively searching for closed contours of edge elements equivalent to minimum cuts in g which consist mostly of strong edges while rejecting contours containing isolated strong edges this method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours

a coupled system of rational difference equations abstract we investigate the global stability properties and asymptotic behavior of solutions of the recursive sequence where the parameters a b c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers

image segmentation with ratio cut this paper proposes a new cost function cut ratio for segmenting images using graph based methods the cut ratio is defined as the ratio of the corresponding sums of two different weights of edges along the cut boundary and models the mean affinity between the segments separated by the boundary per unit boundary length this new cost function allows the image perimeter to be segmented guarantees that the segments produced by bipartitioning are connected and does not introduce a size shape smoothness or boundary length bias the latter allows it to produce segmentations where boundaries are aligned with image edges furthermore the cut ratio cost function allows efficient iterated region based segmentation as well as pixel based segmentation these properties may be useful for some image segmentation applications while the problem of finding a minimum ratio cut in an arbitrary graph is np hard one can find a minimum ratio cut in the connected planar graphs that arise during image segmentation in polynomial time while the cut ratio alone is not sufficient as a baseline method for image segmentation it forms a good basis for an extended method of image segmentation when combined with a small number of standard techniques we present an implemented algorithm for finding a minimum ratio cut prove its correctness discuss its application to image segmentation and present the results of segmenting a number of medical and natural images using our techniques

approximations to the solution of cauchy problem for a linear evolution equation via the space shift operator second order equation example abstract we present a general method of solving the cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two one and zero the method is based on the chernoff approximation procedure applied to a specially constructed shift operator it is proven that approximations converge uniformly to the exact solution

on the system of rational difference equations xn a yn 3 yn byn 3 xn qyn q abstract in this paper we investigate the behaviour of the positive solutions of the system of rational difference equation x n a y n 3 y n by n 3 x n q y n q n 1 2 where q 3 is a positive integer with 3 q a and b are positive constants and the initial values x q 1 x q 2 x 0 y q 1 y q 2 y 0 are positive real numbers

geometric flow approach for region based image segmentation geometric flows have been successfully used for surface modeling and designing largely because they are inherently good at controlling geometric shape evolution variational image segmentation approaches on the other hand detect objects of interest by deforming certain given shapes this motivates us to revisit the minimal partition problem for segmentation of images and propose a new geometric flow based formulation and solution to it our model intends to derive a mapping that will evolve given contours or piecewise constant regions toward objects in the image the mapping is approximated by b spline basis functions and the positions of the control points are to be determined starting with the energy functional based on intensity averaging we derive a euler lagrange equation and then a geometric evolution equation the linearized system of equations is efficiently solved via a special matrix vector multiplication technique furthermore we extend the piecewise constant model to a piecewise smooth model which effectively handles images with intensity inhomogeneity

globally optimal regions and boundaries as minimum ratio weight cycles we describe a new form of energy functional for the modeling and identification of regions in images the energy is defined on the space of boundaries in the image domain and can incorporate very general combinations of modeling information both from the boundary intensity gradients etc and from the interior of the region texture homogeneity etc we describe two polynomial time digraph algorithms for finding the global minima of this energy one of the algorithms is completely general minimizing the functional for any choice of modeling information it runs in a few seconds on a 256 spl times 256 image the other algorithm applies to a subclass of functionals but has the advantage of being extremely parallelizable neither algorithm requires initialization

a dynamic compensator for in domain control of burgers equation this paper addresses the problem of output regulation control of a burgers equation under pointwise in domain actuation a dynamic compensator that generates the in domain control is developed for

multi region active contours with a single level set function segmenting an image into an arbitrary number of coherent regions is at the core of image understanding many formulations of the segmentation problem have been suggested over the past years these formulations include among others axiomatic functionals which are hard to implement and analyze and graph based alternatives which impose a non geometric metric on the problem we propose a novel method for segmenting an image into an arbitrary number of regions using an axiomatic variational approach the proposed method allows to incorporate various generic region appearance models while avoiding metrication errors in the suggested framework the segmentation is performed by level set evolution yet contrarily to most existing methods here multiple regions are represented by a single non negative level set function the level set function evolution is efficiently executed through the voronoi implicit interface method for multi phase interface evolution the proposed approach is shown to obtain accurate segmentation results for various natural 2d and 3d images comparable to state of the art image segmentation algorithms

global asymptotic behavior of a two dimensional difference equation modelling competition we investigate the global asymptotic behavior of solutions of the system of difference equations xn 1 xn a cyn yn 1 yn b dxn n 0 1 where the parameters a and b are in 0 1 c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers we show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior in the case where a b we find an explicit equation for the stable manifold

convenient analytic recurrence algorithms for the adomian polynomials abstract in this article we present four analytic recurrence algorithms for the multivariable adomian polynomials as special cases we deduce the four simplified results for the one variable adomian polynomials these algorithms are comprised of simple orderly and analytic recurrence formulas which do not require time intensive operations such as expanding regrouping parametrization and so on they are straightforward to implement in any symbolic software and are shown to be very efficient by our verification using mathematica 7 0 we emphasize that from the summation expressions a n k 1 n u n k for the multivariable adomian polynomials and an k 1 n f k u 0 c n k for the one variable adomian polynomials we obtain the recurrence formulas for the u n k and the c n k these provide a theoretical basis for developing new algorithmic approaches such as for parallel computing in particular the recurrence process of one particular algorithm for the one variable adomian polynomials does not involve the differentiation operation but significantly only the arithmetic operations of multiplication and addition are involved precisely c n 1 u n n 1 and c n k 1 n j 0 n k j 1 u j 1 c n 1 j k 1 2 k n we also discuss several other algorithms previously reported in the literature including the adomian rach recurrence algorithm 1 and this author s index recurrence algorithm 23 36

application of the adomian decomposition method for two dimensional parabolic equation subject to nonstandard boundary specifications in this work we are concerned with the solution of a nonlocal boundary value problem an approach is presented for solving the two dimensional parabolic partial differential equation subject to integral boundary specifications the main objective is to propose an alternative method of solution one not based on finite difference methods or finite element schemes or spectral techniques the aim of the present paper is to investigate the application of the adomian decomposition method for solving the two dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions the adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques the introduction of this idea as will be discussed not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume the solutions will be handle more easily quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions in this approach the solution is found in the form of a convergent power series with easily computed components the adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization to give a clear overview of the methodology we have selected illustrative example

winding number constrained contour detection salient contour detection can benefit from the integration of both contour cues and region cues however this task is difficult due to different nature of region representations and contour representations to solve this problem this paper proposes an energy minimization framework based on winding number constraints in this framework both region cues such as color texture homogeneity and contour cues such as local contrast and continuity are represented in a joint objective function which has both region and contour labels the key problem is how to design constraints that ensure the topological consistency of the two kinds of labels our technique is based on the topological concept of winding number using a fast method for winding number computation a small number of linear constraints are derived to ensure label consistency our method is instantiated by ratio based energy functions by successfully integrating both region and contour cues our method shows advantages over competitive methods our method is extended to incorporate user interaction which leads to further improvements

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

comparing numerical methods for the solutions of two dimensional diffusion with an integral condition in this paper we present a numerical comparison between the adomian decomposition method adm and a fourth order explicit finite difference method for solving two dimensional diffusion problems with an integral condition the numerical results show that the adm is quite accurate and easily implemented

stepwise stability for the heat equation with a nonlocal constraint the authors prove the stepwise stability for a finite difference scheme for the heat equation with an integral constraint the resulting matrix is nonsymmetric and does not have the usual band structure the proof is based on the method of matrix analysis the eigenvalues of several matrices are found explicitly or their location described precisely this method relies upon the relationship of the characteristic polynomials of these matrices with orthogonal polynomials

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

a new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations abstract in this paper we propose a new modified recursion scheme for the resolution of multi order and multi point boundary value problems for nonlinear ordinary and partial differential equations by the adomian decomposition method adm our new approach including duan s convergence parameter provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series we utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components whereas several prior recursion schemes have done so this modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots which is required to complete calculation of the solution by several prior modified recursion schemes using the adm

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

inversion of nonlinear stochastic operators abstract the operator theoretic method adomian and malakian j math anal appl 76 1 1980 183 201 recently extended adomian s solutions of nonlinear stochastic differential equations g adomian stochastic systems analysis in applied stochastic processes nonlinear stochastic differential equations j math anal appl 55 1 1976 441 452 on the modeling and analysis of nonlinear stochastic systems in proceeding international conf on mathematical modeling vol 1 pp 29 40 to provide an efficient computational procedure for differential equations containing polynomial exponential and trigonometric nonlinear terms n y the procedure depends on the calculation of certain quantities an and bn this paper generalizes the calculation of the an and bn to much wider classes of nonlinearities of the form n y y essentially the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance this procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities an and bn for any order n in a convenient form this paper also demonstrates the significance of the iterative series decomposition proposed by adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators since both the nonlinear and stochastic behavior is quite general the results are extremely significant for applications processes need not for example be limited to either gaussian processes white noise or small fluctuations

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied

adomian method for a class of hyperbolic equations with an integral condition abstract in this paper adomian decomposition method adm is applied to the solvability of a boundary problem with a nonlocal condition for a class of hyperbolic equations and illustrated with a few simple examples

graph partitioning active contours gpac for image segmentation in this paper we introduce new types of variational segmentation cost functions and associated active contour methods that are based on pairwise similarities or dissimilarities of the pixels as a solution to a minimization problem we introduce a new curve evolution framework the graph partitioning active contours gpac using global features our curve evolution is able to produce results close to the ideal minimization of such cost functions new and efficient implementation techniques are also introduced in this paper our experiments show that gpac solution is effective on natural images and computationally efficient experiments on gray scale color and texture images show promising segmentation results

solving nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the adomian decomposition method in this paper we present a new approach to solve nonlocal initial boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type we first transform the given nonlocal initial boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local dirichlet initial boundary value problems and then use a relatively new modified adomian decomposition method adm furthermore we investigate the fourier adomian method which also does not require any a priori assumptions on the solution for the solution of nonlocal initial boundary value problems combined with our new approach several examples are presented to demonstrate the efficiency of the adm