numerical method for non linear wave and diffusion equations by the variational iteration method this paper applies he s variational iteration to the wave equations in an infinite one dimensional medium and some non linear diffusion equations a suitable choice of an initial solution can lead to the needed exact solution by a few iterations copyright 2007 john wiley sons ltd

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

statistical mechanics of the international trade network analyzing real data on international trade covering the time interval 1950 2000 we show that in each year over the analyzed period the network is a typical representative of the ensemble of maximally random weighted networks whose directed connections bilateral trade volumes are only characterized by the product of the trading countries gdps it means that time evolution of this network may be considered as a continuous sequence of equilibrium states i e a quasistatic process this in turn allows one to apply the linear response theory to make and also verify simple predictions about the network in particular we show that bilateral trade fulfills a fluctuation response theorem which states that the average relative change in imports exports between two countries is a sum of the relative changes in their gdps yearly changes in trade volumes prove that the theorem is valid

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

generalizations of the clustering coefficient to weighted complex networks the recent high level of interest in weighted complex networks gives rise to a need to develop new measures and to generalize existing ones to take the weights of links into account here we focus on various generalizations of the clustering coefficient which is one of the central characteristics in the complex network theory we present a comparative study of the several suggestions introduced in the literature and point out their advantages and limitations the concepts are illustrated by simple examples as well as by empirical data of the world trade and weighted coauthorship networks

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

randomizing world trade i a binary network analysis the international trade network itn has received renewed multidisciplinary interest due to recent advances in network theory however it is still unclear whether a network approach conveys additional nontrivial information with respect to traditional international economics analyses that describe world trade only in terms of local first order properties in this and in a companion paper we employ a recently proposed randomization method to assess in detail the role that local properties have in shaping higher order patterns of the itn in all its possible representations binary or weighted directed or undirected aggregated or disaggregated by commodity and across several years here we show that remarkably the properties of all binary projections of the network can be completely traced back to the degree sequence which is therefore maximally informative our results imply that explaining the observed degree sequence of the itn which has not received particular attention in economic theory should instead become one the main focuses of models of trade

weighted networks as randomly reinforced urn processes we analyze weighted networks as randomly reinforced urn processes in which the edge total weights are determined by a reinforcement mechanism we develop a new statistical test and a new procedure based on it to study the evolution of networks over time detecting the dominance of some edges with respect to the others and then assessing if a given instance of the network is taken at its steady state or not distance from the steady state can be considered as a measure of the relevance of the observed properties of the network our results are quite general in the sense that they are not based on a particular probability distribution or functional form of the random weights moreover the proposed tool can be applied also to dense networks which have received little attention by network community so far since they are often problematic we apply our procedure in the context of the international trade network determining a core of dominant edges

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

application of he s homotopy perturbation method for nth order integro differential equations abstract in this paper he s homotopy perturbation method is proposed to solve n th order integro differential equations the results reveal that the method is very effective and simple

world trade web topological properties dynamics and evolution this paper studies the statistical properties of the web of import export relationships among world countries using a weighted network approach we analyze how the distributions of the most important network statistics measuring connectivity assortativity clustering and centrality have coevolved over time we show that all node statistic distributions and their correlation structure have remained surprisingly stable in the last 20 years and are likely to do so in the future conversely the distribution of positive link weights is slowly moving from a log normal density towards a power law we also characterize the autoregressive properties of network statistics dynamics we find that network statistics growth rates are well proxied by fat tailed densities like the laplace or the asymmetric exponential power finally we find that all our results are reasonably robust to a few alternative economically meaningful weighting schemes

microcanonical optimization algorithm for the euclidean steiner problem in rn with application to phylogenetic inference the euclidean steiner tree problem in r n estp is that of finding the shortest interconnecting network spanning p given nodes in the euclidean r n with the possible use of extra nodes combinatorial explosion precludesthe use of exact methods for large high dimensional estp instances but very few heuristic approaches have so far been proposed for them here we introduce a microcanonical optimization algorithm that works over a topology describing data structure associated to the estp solutions and which is proven able to find close to minimum steiner trees in reasonable computational time even for configurations of up to p 50 points in n 50 dimensions moreover its performance is shown to increase with n which makes it especially suited for high dimensional clustering problems such as those of phylogenetic inference an instance of which is considered here

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

a computational method for solving nonlinear stochastic volterra integral equations in this paper an efficient method for solving nonlinear stochastic ito volterra integral equations nsivies is proposed by using new adjustment of hat basis functions and their stochastic operational matrix of integration the nsivie is reduced to a nonlinear system of algebraic equations also the error analysis of the proposed method is investigated and proved that the order of convergence is o h 4 finally numerical examples affirm the efficiency of the proposed method

scaling behavior in economics the problem of quantifying company growth inspired by work of both widom and mandelbrot we analyze the computstat database comprising all publicly traded united states manufacturing companies in the years 1974 1993 we find that the distribution of company size remains stable for the 20 years we study i e the mean value and standard deviation remain approximately constant we study the distribution of sizes of the new companies in each year and find it to be well approximated by a log normal we find i the distribution of the logarithm of the growth rates for a fixed growth period of t years and for companies with approximately the same size s displays an exponential tent shaped form rather than the bell shaped gaussian one would expect for a log normal distribution and ii the fluctuations in the growth rates measured by the width of this distribution t decrease with company size and increase with time t we find that for annual growth rates t 1 t s and that the exponent takes the same value within the error bars for several measures of the size of a company in particular we obtain 0 20 0 03 for sales 0 18 0 03 for number of employees 0 18 0 03 for assets 0 18 0 03 for cost of goods sold and 0 20 0 03 for propert plant and equipment we propose models that may lead to some insight into these phenomena first we study a model in which the growth rate of a company is affected by a tendency to retain an optimal size that model leads to an exponential distribution of the logarithm of growth rate in agreement with the empirical results then we study a hierarchical tree like model of a company that enables us to relate to parameters of a company structure we find that 1n 1nz where z defines the mean branching ratio of the hierarchical tree and is the probability that the lower levels follow the policy of higher levels in the hierarchy we also study the output distribution of growth rates of this hierarchical model we find that the distribution is consistent with the exponential form found empirically we also discuss the time dependence of the shape of the distribution of the growth rates

numerical solution of second kind fredholm integral equations system by using a taylor series expansion method in this paper we present a taylor series expansion method for a second kind fredholm integral equations system with smooth or weakly singular kernels this method reduce the system of integral equations to a linear system of ordinary differential equation after constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods finally we show the efficiency of the method using some numerical examples

a numerical solution for solving system of fredholm integral equations by using homotopy perturbation method abstract homotopy perturbation method is applied to the numerical solution for solving system of fredholm integral equations comparison of the result obtained by the present method with that obtained by taylor series expansion method k maleknejad n aghazade m rabbani numerical solution of second kind fredholm integral equations system by using a taylor series expansion method appl math comput 175 2006 1229 1234 reveals that the present method is very effective and convenient

a dynamic adaptive relaxation scheme applied to the euclidean steiner minimal tree problem the steiner problem is an np hard optimization problem which consists of finding the minimal length tree connecting a set of n points in the euclidean plane exact methods of resolution currently available are exponential in n making exact minimal trees accessible for only small size problems up to n approx 100 an acceptable suboptimal solution is provided by the minimum spanning tree mst which has been shown computable in an o n log n step we propose here an o n process that is able to relax a given initial steiner tree into a local minimum of its length this process based on a physical analogy simulates the dynamics of a fluid film which relaxes under surface tension forces and stabilizes in an equilibrium configuration minimizing its total length through purely local interactions to improve the solution to the steiner problem this o n relaxation scheme is applied to reduce the length of the mst this results in a heuristic of a very low o n log n complexity for the steiner problem whose performance is shown to compare quite favorably with that of the best available heuristics large problem sizes up to n 10000 were successfully tackled a characterization of the asymptotic behavior of the solution of the steiner problem shows a stabilization to a nonvanishing positive value of the average length reduction achieved over the mst and predicts an average length for the minimal steiner tree of about 3 below 0 65 n1 2 for large n

numerical solution of a biological population model using he s variational iteration method this paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations the variational iteration method and adomian decomposition method are used for solving this equation and then numerical results are compared with each other showing that the variational iteration method leads to more accurate results furthermore variational iteration method overcomes the difficulty arising in calculating the adomian s polynomials which is an important advantage over the adomian decomposition method

new iterative methods for nonlinear equations by modified hpm a numerical method based on modified homotopy perturbation method hpm is proposed for solving nonlinear algebric equations it is shown that the proposed method has third order convergence to assess its validity and accuracy the method is applied to solve several test problems

a third order newton type method for nonlinear equations based on modified homotopy perturbation method in this paper we present a new iterative method for solving nonlinear algebric equations by using modified homotopy perturbation method we also discuss the convergence criteria of the present method to assess its validity and accuracy the method is applied to solve several test problems

variational iteration method for solving nonlinear boundary value problems abstract in this paper he s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems the variational iteration method which produces the solutions in terms of convergent series requiring no linearization or small perturbation numerical examples are given and comparisons are made with the adomian decomposition method the fact that this method solves nonlinear equations without using adomian polynomials can be considered as an advantage of this method over adomian decomposition method

iterated he s homotopy perturbation method for quadratic riccati differential equation abstract in this paper iterated he s homotopy perturbation method is proposed to solving quadratic riccati differential equation comparisons are made between adomian s decomposition method adm and the exact solution and the proposed method the results reveal that the method is very effective and simple

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

clustering in complex directed networks many empirical networks display an inherent tendency to cluster i e to form circles of connected nodes this feature is typically measured by the clustering coefficient cc the cc originally introduced for binary undirected graphs has been recently generalized to weighted undirected networks here we extend the cc to the case of binary and weighted directed networks and we compute its expected value for random graphs we distinguish between ccs that count all directed triangles in the graph independently of the direction of their edges and ccs that only consider particular types of directed triangles e g cycles the main concepts are illustrated by employing empirical data on world trade flows

newton like iterative methods for solving system of non linear equations abstract homotopy perturbation method hpm is applied to construct a new family of newton like iterative methods for solving system of non linear equations comparison of the result obtained by the present method with newton raphson method reveals that the accuracy and fast convergence of the new method

asymptotology by homotopy perturbation method in the paper an heuristical example is given to illustrate the basic idea of the homotopy perturbation method so that homotopy perturbation method has made all that is necessary simple and all that is complex unnecessary

a new family of iterative methods for solving system of nonlinear algebric equations abstract homotopy perturbation method hpm is applied to construct a new iterative method for solving system of nonlinear algebric equations comparison of the result obtained by the present method with that obtained by revised adomian decomposition method hossein jafari varsha daftardar gejji appl math comput 175 2006 1 7 reveals that the accuracy and fast convergence of the new method

a numerical simulation and explicit solutions of the generalized burgers fisher equation we consider solitary wave solutions of the generalized burgers fisher bf equation u t pu ru x u x x qu 1 u r 0 in this paper by considering the decomposition scheme we first obtain the exact solutions of the generalized bf equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some values r 2 of the generalized bf equation

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

structure and evolution of the world trade network the world trade web wtw the network defined by the international import export trade relationships has been recently shown to display some important topological properties which are tightly related to the gross domestic product of world countries while our previous analysis focused on the static undirected version of the wtw here we address its full evolving directed description this is accomplished by exploiting the peculiar reciprocity structure of the wtw to recover the directed nature of international trade channels and by studying the temporal dependence of the parameters describing the wtw topology

meshfree approach for solving multi dimensional systems of fredholm integral equations via barycentric lagrange interpolation abstract a highly accurate meshfree approach based on the barycentric lagrange basis functions is reported for solving the linear and nonlinear multi dimensional systems of fredholm integral equations fies of the second kind the method is an improved lagrange interpolation technique which possesses high precision and inexpensive procedure the systems of fies are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations the stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric lagrange formula which is dominated by barycentric weights the convergence analysis and error estimation are included some computational results indicate that the method is simple and efficient

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

numerical complexiton solutions for the complex kdv equation by the homotopy perturbation method in this paper the homotopy perturbation method is extended to investigate the numerical complexiton solutions of the complex kdv equation by constructing special forms of initial conditions three new types of realistic numerical solutions are obtained numerical positon solution expressed by the trigonometric functions numerical negaton solution expressed by the hyperbolic functions and particularly the numerical analytical complexiton solutions expressed by combinations of the two kinds of functions all these numerical solutions obtained can rapidly converge to the exact solutions obtained by lou et al illustrative numerical figures are exhibited the efficiency of the proposed method

local search for the steiner tree problem in the euclidean plane abstract most heuristics for the steiner tree problem in the euclidean plane perform a series of iterative improvements using the minimum spanning tree as an initial solution we may therefore characterize them as local search heuristics in this paper we first give a survey of existing heuristic approaches from a local search perspective by setting up solution spaces and neighbourhood structures secondly we present a new general local search approach which is based on a list of full steiner trees constructed in a preprocessing phase this list defines a solution space on which three neighbourhood structures are proposed and evaluated computational results show that this new approach is very competitive from a cost benefit point of view furthermore it has the advantage of being easy to apply to the steiner tree problem in other metric spaces and to obstacle avoiding variants

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

reconstructing the world trade multiplex the role of intensive and extensive biases in economic and financial networks the strength of each node has always an important economic meaning such as the size of supply and demand import and export or financial exposure constructing null models of networks matching the observed strengths of all nodes is crucial in order to either detect interesting deviations of an empirical network from economically meaningful benchmarks or reconstruct the most likely structure of an economic network when the latter is unknown however several studies have proved that real economic networks and multiplexes topologically differ from configurations inferred only from node strengths here we provide a detailed analysis of the world trade multiplex by comparing it to an enhanced null model that simultaneously reproduces the strength and the degree of each node we study several temporal snapshots and almost 100 layers commodity classes of the multiplex and find that the observed properties are systematically well reproduced by our model our formalism allows us to introduce the static concept of extensive and intensive bias defined as a measurable tendency of the network to prefer either the formation of extra links or the reinforcement of link weights with respect to a reference case where only strengths are enforced our findings complement the existing economic literature on dynamic intensive and extensive trade margins more generally they show that real world multiplexes can be strongly shaped by layer specific local constraints

a microscopic study of the fitness dependent topology of the world trade network abstract previous studies have suggested that the world trade network belongs to the class of static hidden variable models in this article we investigate the microscopic structure of the world trade network that is the hidden variable correlation matrix of the network the hidden variable is defined as a rank ordering of gross domestic products this choice significantly reduces the noise in the statistical analysis found in previous studies the hidden variable correlation matrix that expresses the probability that a trade relationship between two countries of given fitness exists suggests an attachment kernel that at least partially favours trading pairs or dissimilar fitness rather than the purely multiplicative one found previously additionally we provide an in depth look at the data source and reveal that first order results such as the degree distribution exhibit significant qualitative differences depending on the data provider furthermore we shed light on the intertemporal activity of international trade and point out that fluctuations occur mostly between countries with strong dissimilarities of fitness and connectivity

invariant distributions and the limiting behavior of markovian economic models equilibria in stochastic economic models are often time series which fluctuate in complex ways but it is sometimes possible to summarize the long run average characteristics of these fluctuations for example if the law of motion determined by economic interactions is markovian and if the equilibrium time series converges in a specific probabilistic sense then the long run behavior is completely determined by an invariant probability distribution this paper develops and unifies a number of results found in the probability literature which enable one to prove under very general conditions the existence of an invariant distribution and the convergence of the corresponding markov process virtually all of economic theory focuses upon the study of economic equilibrium this concept has recently undergone several subtle elaborations no longer must a system of markets in equilibrium be thought of as one at rest in a static steady state instead there is a growing body of literature e g 4 5 12 16 20 21 which defines equilibrium as a stochastic process of market clearing prices and quantities which is consistent with the self interested behavior of economic agents needless to say equilibrium stochastic processes can be very complex time series which fluctuate in irregular ways for theoretical and econometric purposes it is useful to have a convenient way of summarizing the average behavior of such processes over time this paper draws together and unifies a number of fundamental results from the probability literature which enable one to do this for discrete time markov processes on general state spaces the starting point of the analysis is a set s of economic states e g prices and or quantities the only technical restriction placed upon s is that it be a borel subset of a complete separable metric space the second datum is a transition probability p s on s the number p s a records the probability that the economic system moves from the state s to some state in the borel subset a of s during one unit of elapsed time in economic applications the transition probability is usually derived from hypotheses about market clearing and the maximizing behavior of economic agents the transition probability together with an initial probability measure on s defines a discrete time markov process one way of summarizing the dynamic behavior implied by p is to look for an invariant probability a probability measure x on s is invariant for p if for all borel subsets a of s one has the equality f p s a x ds x a an invariant probability is a kind of probabilistic steady state for the dynamics defined by p of course there may be no invariant probability for p at all and even if one exists it may convey no information about the average behavior of the process over time except under very special initial conditions there is a second way of summarizing the behavior of markov processes defined by the transition probability p let p s a denote the n step transition

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

convergence and error estimation of homotopy perturbation method for fredholm and volterra integral equations abstract in this paper an application of the homotopy perturbation method for solving fredholm and volterra integral equations of the second kind is presented discussed method consists in constructing the functional series sum of which determines the function giving the solution of considered problem conditions under which the constructed series is convergent are formulated and proved in the paper in general the composed series is rather fast convergent thanks to which calculation of few first terms ensures a very good approximation of the sought solution estimation of errors of approximate solution obtained by taking the partial sum of the series is also elaborated in the paper

empirical study on clique degree distribution of networks the community structure and motif modular network hierarchy are of great importance for understanding the relationship between structures and functions we investigate the distribution of clique degrees which are an extension of degree and can be used to measure the density of cliques in networks empirical studies indicate the extensive existence of power law clique degree distributions in various real networks and the power law exponent decreases with an increase of clique size

iterative methods to nonlinear equations homotopy perturbation method is applied to construct a new iterative method for solving nonlinear algebraic equations the convergence of the new scheme is proposed an at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of the new method

applications of the annealing algorithm to combinatorial problems in statistics summary there are several problems in statistics which can be formulated so that the desired solution is the global minimum of some explicitly defined objective function in many cases the number of candidate solutions increases exponentially with the size of the problem making exhaustive search impossible but descent procedures devised to reduce the number of solutions examined can terminate with local minima in this paper we describe the annealing algorithm a widely applicable stochastic search procedure which can escape local optima and we use the evolutionary tree problem to illustrate the method of application

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation