the solution of nonlinear schrodinger equations using orthogonal spline collocation abstract an orthogonal spline collocation semidiscretization previously applied in the numerical solution of the cubic schrodinger equation is extended to the solution of schrodinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives the accuracy of the method is examined as well as its ability to conserve approximations to certain invariants associated with the theoretical solution the results of numerical experiments are presented in which the integration in time is performed using a routine from a software library particular attention is paid to the value and limitations of conservation as an indicator of accuracy

design of fuzzy pid controllers using modified triangular membership functions a design method for fuzzy proportional integral derivative pid controllers is investigated in this study based on conventional triangular membership functions used in fuzzy inference systems the modified triangular membership functions are proposed to improve a system s performance according to knowledge based reasonings the parameters of the considered controllers are tuned by means of genetic algorithms gas using a fitness function associated with the system s performance indices the merits of the proposed controllers are illustrated by considering a model of the induction motor control system and a higher order numerical model

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

the conditioning of linear boundary value problems investigation is made into the sensitivity of solutions of linear boundary value problems to perturbations of the boundary condition we derive a useful quantity to decide for well or ill conditioning from this it is deduced which kind of requirements the boundary conditions should meet in order to have a well conditioned problem it is shown that this quantity can fruitfully be used to explain why e g the multiple shooting technique is stable in contrast to the single shooting one for certain well posed problems having solutions with different growth behavior the results are sustained by a number of examples

barrier methods for optimal control problems with convex nonlinear gradient state constraints in this paper we are concerned with the application of interior point methods in function space to gradient constrained optimal control problems governed by partial differential equations we will derive existence of solutions together with first order optimality conditions afterwards we show continuity of the central path together with convergence rates depending on the interior point parameter

diffusion annihilation processes in complex networks we present a detailed analytical study of the a a 0i diffusion annihilation process in complex networks by means of microscopic arguments we derive a set of rate equations for the density of a particles in vertices of a given degree valid for any generic degree distribution and which we solve for uncorrelated networks for homogeneous networks swith bounded fluctuations d we recover the standard mean field solution i e a particle density decreasing as the inverse of time for heterogeneous sscale free networksd in the infinite network size limit we obtain instead a density decreasing as a power law with an exponent depending on the degree distribution we also analyze the role of finite size effects showing that any finite scale free network leads to the mean field behavior with a prefactor depending on the network size we check our analytical predictions with extensive numerical simulations on homogeneous networks with poisson degree distribution and scalefree networks with different degree exponents

orthogonal cubic spline collocation method for the extended fisher kolmogorov equation a second order splitting combined with orthogonal cubic spline collocation method is formulated and analysed for the extended fisher kolmogorov equation with the help of lyapunov functional a bound in maximum norm is derived for the semidiscrete solution optimal error estimates are established for the semidiscrete case finally using the monomial basis functions we present the numerical results in which the integration in time is performed using radau 5 software library

fuzzy pid controller design performance evaluation and stability analysis abstract this paper presents a design for a new fuzzy logic proportional integral derivative pid controller the main motivation for this design was to control some known nonlinear systems such as robotic manipulators which violate the conventional assumption of the linear pid controller this controller is developed by first describing the discrete time linear pid control law and then progressively deriving the steps necessary to incorporate a fuzzy logic control mechanism into the modifications of the pid structure the final version of this new fuzzy pid controller is a computationally efficient analytic scheme suitable for implementation in a real time closed loop digital control numerous computer simulations are included to demonstrate the effectiveness of the controller for both linear and nonlinear systems finally a brief analysis is presented to prove that the controller has bounded input bounded output bibo stability

a nonlinear fuzzy controller with linear control rules is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral controller abstract the author analytically proves that a nonlinear fuzzy controller with linear control rules and n members for input fuzzy sets is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral pi controller which adjusts the control action generated by the global multilevel relay as n increases the resolution of the global multilevel relay is enhanced but the role of the local nonlinear pi controller in total control action is decreased as n approaches the global multilevel relay approaches a regular linear pi controller while the control action from the local nonlinear pi controller approaches zero the role of the global multilevel relay and the local nonlinear pi controller in total control action is quantitatively described as is the degree of nonlinearity of the fuzzy controllers with different n

boundary value problems and dichotomic stability since the conditioning of a boundary value problem bvp is closely related to the existence of a dichotomic fundamental solution i e where one set of modes is increasing and a complementary set is decreasing it is important to have discretization methods that conserve this dichotomy property the conditions this imposes on such a method are investigated in this paper they are worked out in more detail for scalar second order equations the central difference scheme and for linear first order systems as well for the latter type both one step methods including collocation and multistep methods those that may be used in multiple shooting are examined

on optimal shooting intervals we develop an adaptive multiple shooting strategy which is nearly optimal with respect to cpu time since the costs of integration are the most important components in this we investigate in some detail how the gridpoints are chosen by an adaptive integration routine we use this information to find out where the shooting points have to be selected we also show that our final strategy is stable in the sense that rounding errors can be kept below a given tolerance finally we pay attention to the question how the need for memory can be minimized

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

rate equation approach for correlations in growing network models we propose a rate equation approach to compute two vertex correlations in scale free growing network models based on the preferential attachment mechanism the formalism based on previous work of szabo et al fphys rev e 67 056102 s2002dg for the clustering spectrum measuring three vertex correlations is based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree k with an appropriate boundary condition we study the properties of both two and three vertex correlations for linear preferential attachment models and also for a model yielding a large clustering coefficient the expressions obtained are checked by means of extensive numerical simulations the rate equation proposed can be generalized to more sophisticated growing network models and also extended to deal with related correlation measures as an example we consider the case of a recently proposed model of weighted networks for which we are able to compute a weighted two vertex correlation function taking into account the strength of the interactions between connected vertices

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated

some aspects of using iv software to solve bvp via a riccati transformation we consider a numerical method for the solution of linear boundary value problems bvps with separated boundary conditions bcs for ordinary differential equations odes by using a particular linear transformation the problem is reformulated as three initial value problems ivps one of them being a riccati differential equation the nonlinearity of the riccati de makes it necessary to deal with the eventuality of modifying the structure of integration so called multiple embedding equations must thus be derived and a marching algorithm is incorporated a stability analysis of the three ivps indicates clearly for what class of problems the method is suited and why stiff iv methods should then frequently be used this leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the riccati de we interpret this occurrence in terms of the original bvp and the resulting analysis suggests some final considerations including the possibility of corrective measures

numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

split step orthogonal spline collocation methods for nonlinear schrodinger equations in one two and three dimensions abstract split step orthogonal spline collocation osc methods are proposed for one two and three dimensional nonlinear schrodinger nls equations with time dependent potentials firstly the nls equation is split into two nonlinear equations and one or more one dimensional linear equations commonly the nonlinear subproblems could be integrated directly and accurately but it fails when the time dependent potential cannot be integrated exactly in this case we propose three approximations by using quadrature formulae but the split order is not reduced discrete time osc schemes are applied for the linear subproblems in numerical experiments many tests are carried out to prove the reliability and efficiency of the split step osc ssosc methods solitons in one two and three dimensions are well simulated and conservative properties and convergence rates are demonstrated we also apply the ways of solving the nonlinear subproblems to the split step finite difference ssfd methods and the time splitting spectral tssp methods and the approximate ways still work well finally we apply the ssosc methods to solve some problems of bose einstein condensates

localization transition lifschitz tails and rare region effects in network models effects of heterogeneity in the suspected infected susceptible model on networks are investigated using quenched mean field theory the emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare region effects appearing in simulations and in the lifschitz tails the latter in the linear approximation is related to the spectral density of the laplacian matrix and to the time dependent order parameter i show that these approximations indicate correctly griffiths phases both on regular one dimensional lattices and on small world networks exhibiting purely topological disorder i discuss the localization transition that occurs on scale free networks at 3 degree exponent

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

the simplest fuzzy controllers using different inference methods are different nonlinear proportional integral controllers with variable gains abstract the author analytically proves that the simplest fuzzy controllers using different inference methods are different nonlinear proportional integral pi controllers with proportional gains and integral gains changing with inputs of the controllers the inference methods involved are mamdani s minimum inference method larsen s product inference method the drastic product inference method and the bounded product inference method configuration of the fuzzy controllers is minimal which includes two input fuzzy sets three output fuzzy sets four control rules zadeh fuzzy logic and lukasiewicz fuzzy logic or and a center of gravity defuzzification algorithm after analytically investigating properties of the nonlinear pi controllers the author reveals that the bounded product inference method is inappropriate for the control purpose while the other three inference methods are appropriate dynamic and static control behaviors of the fuzzy controllers with the appropriate inference methods are analytically compared with each other and are also compared with those of the linear pi controller finally it is analytically proven that the fuzzy control systems have the same local stability at the equilibrium point as the corresponding linear pi control system does

conditioning and dichotomy in differential algebraic equations the relationship between the dichotomy of the solution space for a differential algebraic system and the well conditioning of associated boundary value problems is investigated this work extends results that are known for regular explicit ordinary differential equations and it follows the presentation by de hoog and mattheij siam j numer anal 24 1987 pp 89 105

analytic formulation of linguistic rules for fuzzy controller in this paper an analytic formulation of a fuzzy controller using rules with two conditions is given this formulation will be exploited in order to prove the nonlinearity of the controller and to establish equivalence between fuzzy controllers and pi ones

strong stability of compact discrete boundary value problems via exact discretizations this paper is concerned with strong stability inequalities with respect to discrete and continuous sobolev norms for general linear mth order two point boundary value problems and their discretizations by compact finite difference schemes the well known stability of compact consistent approximations on irregular grids is demonstrated here by a direct connection with the strong stability inequality of the continuous problem the connection is made by means of exact discretizations which are compact schemes that are exactly satisfied by the true solution of the original problem the analysis involves certain generalized splines related to the green s function for the problem

a note on conditioning stability and collocation matrices the concept of a well conditioned bvp as considered in recent literature essentially corresponds to the concept of a uniformly stable ivp but this terminology suggests a shift from absolute to relative quantities which makes it more difficult to understand the conditioning of numerical discretizations especially for higher order differential operators in this note we wish to make it clear that the notion of well conditioning of bvps is not always a direct analogue of the u the condition number of a discretization of a well conditioned bvp generally grows unboundedly when higher accuracy is desired because it contains a discretization of a differential operator this condition number gets large particularly rapidly if a direct discretization of a higher order differential operator is attempted for collocation an implementation which uses a local runge kutta solution representation produces a much better conditioning because the repeated numerical differentiation is avoided instead the highest derivative is approximated locally by a discontinuous piece wise polynomial and local integrations yield lower solution derivatives upon imposing the collocation equations the highest derivative approximation can be locally eliminated resulting in a sophisticated discretization of the corresponding first order system

quasistationary analysis of the contact process on annealed scale free networks we present an analysis of the quasistationary qs state of the contact process cp on annealed scale free networks using a mapping of the cp dynamics in a one step process and analyzing numerically and analytically the corresponding master equation the relevant qs quantities determined via the master equation exhibit an excellent agreement with direct qs stochastic simulations of the cp the high accuracy of the resulting data allows a probe of the strong corrections to scaling present in both the critical and supercritical regions corrections that mask the correct finite size scaling obtained analytically by applying an exact heterogeneous mean field approach our results represent a promising starting point for a deeper understanding of the contact process and absorbing phase transitions on real quenched complex networks

rare regions of the susceptible infected susceptible model on barabasi albert networks i extend a previous work to susceptible infected susceptible sis models on weighted barab asi albert scale free networks numerical evidence is provided that phases with slow power law dynamics emerge as the consequence of quenched disorder and tree topologies studied previously with the contact process i compare simulation results with spectral analysis of the networks and show that the quenched mean field qmf approximation provides a reliable relatively fast method to explore activity clustering this suggests that qmf can be used for describing rare region effects due to network inhomogeneities finite size study of the qmf shows the expected disappearance of the epidemic threshold ensuremath lambda c in the thermodynamic limit and an inverse participation ratio ensuremath sim 0 25 meaning localization in case of disassortative weight scheme contrarily for the multiplicative weights and the unweighted trees this value vanishes in the thermodynamic limit suggesting only weak rare region effects in agreement with the dynamical simulations strong corrections to the mean field behavior in case of disassortative weights explains the concave shape of the order parameter ensuremath rho ensuremath lambda at the transition point application of this method to other models may reveal interesting rare region effects griffiths phases as the consequence of quenched topological heterogeneities

an alternating direction implicit method for orthogonal spline collocation linear systems an alternating direction implicit method is analyzed for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to some separable second order linear elliptic partial differential equations in rectangles on annxn partition with jordan s selection of the acceleration parameters the method requireso n 2 ln 2 n arithmetic operations to produce an approximation whose accuracy in theh 1 norm is that of the collocation solution

three dimensional min max gravity based fuzzy pid inference analysis and tuning this paper proposes a novel algorithm to produce an analytical solution for three input fuzzy pid systems the analysis is based on the most common inference method zadeh mamdani s min max gravity fuzzy reasoning two types of fuzzy systems are investigated both types use unevenly distributed triangular shaped fuzzy input memberships in the first type the output fuzzy sets are considered to be uniformly distributed symmetrical triangular type memberships and in the second type the output fuzzy sets are considered unevenly distributed singleton functions a new input transformation technique is developed to reduce the number of input conditions required in defining the fuzzy output the multi phase complexity that exists in conventional fuzzy reasoning analysis is thus avoided an efficient solution algorithm is outlined to produce the general fuzzy output solution using a minimum number of nonlinear expressions with the proposed method the fuzzy output solution for a three input fuzzy system having uniform output membership functions requires only two nonlinear expressions when the output fuzzy sets are unevenly distributed the general three input based solution requires only four nonlinear expressions the fuzzy controller output is finally decomposed into linear and nonlinear parts the output decomposition is useful in defining tuning gains for fuzzy pid systems

a stable elemental decomposition for dynamic process optimization in cervantes and biegler a i ch e j 44 1998 1038 we presented a simultaneous nonlinear programming problem nlp formulation for the solution of dae optimization problems here by applying collocation on finite elements the dae system is transformed into a nonlinear system the resulting optimization problem in which the element placement is fixed is solved using a reduced space successive quadratic programming rsqp algorithm the space is partitioned into range and null spaces this partitioning is performed by choosing a pivot sequence for an lu factorization with partial pivoting which allows us to detect unstable modes in the dae system the system is stabilized without imposing new boundary conditions the decomposition of the range space can be performed in a single step by exploiting the overall sparsity of the collocation matrix but not its almost block diagonal structure in order to solve larger problems a new decomposition approach and a new method for constructing the quadratic programming qp subproblem are presented in this work the decomposition of the collocation matrix is now performed element by element thus reducing the storage requirements and the computational effort under this scheme the unstable modes are considered in each element and a range space move is constructed sequentially based on decomposition in each element this new decomposition improves the efficiency of our previous approach and at the same time preserves its stability the performance of the algorithm is tested on several examples finally some future directions for research are discussed

multiple shooting with dichotomically stable formulae for linear boundary value problems abstract this paper describes an improvement of england and mattheij s code mutssym for solving linear boundary value problems for ordinary differential equations with may or not give rise to sharp boundary layers the method is based on multiple shooting with a decoupling strategy allowing the calculation of stable solutions according to the increasing or decreasing fundamental modes the integration of the associated initial value problems is performed using a 4 th order symmetric implicit runge kutta method with the dichotomic stability property if the problem is well conditioned the method calculates discrete decaying growing modes controlled by initial terminal conditions corresponding to similar continuous modes a special step size control strategy permits efficient calculation of the numerical solution throughout the interval 1999 elsevier science ltd all rights reserved

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

slow dynamics and rare region effects in the contact process on weighted tree networks we show that generic slow dynamics can occur in the contact process on complex networks with a tree like structure and a superimposed weight pattern in the absence of additional nontopological sources of quenched disorder the slow dynamics is induced by rare region effects occurring on correlated subspaces of vertices connected by large weight edges and manifests in the form of a smeared phase transition we conjecture that more sophisticated network motifs could be able to induce griffiths phases as a consequence of purely topological disorder

general framework stability and error analysis for numerical stiff boundary value methods this paper provides a general framework called theoretical multiple shooting within which various numerical methods for stiff boundary value ordinary differential problems can be analyzed a global stability and error analysis is given allowing as much as possible the specificities of an actual numerical method to come in only locally we demonstrate the use of our results for both one sided and symmetric difference schemes the class of problems treated includes some with internal e g turning point layers

on the convergence of stability constants the stability constants of finite difference schemes approximating linear mth order integro differential equations are shown to converge to their continuous counterparts

on stability constants and condition numbers of discretisation methods discretizations lead to systems of algebraic equations of steadily growing size if the accuracy of the approximation is increasing inverse stability constants and condition numbers describe the sensitivity of the solution of these equations with respect to perturbations of the data starting from an abstract result in this paper the inverse stability constants of finite difference and collocation methods for bvodes as well as of ritz galerkin methods for elliptic variational problems are shown to converge to those of their continuous counterparts as the discretization becomes finer and finer for two choices of basis systems the exact asymptotic behavior of the collocation method condition number is given

on optimally scaled systems for second order scalar singularity perturbed problems the conditioning of the singularly perturbed scalar dirichlet problem is considered it is shown how this is related to the conditioning of an appropriate associated first order system through this the dichotomy of the solution space a concept that only makes sense in a vectorial setting can be investigated two typical equations are studied in more detail one with possible boundary layers on both sides of the interval and one with an internal layer i e the turning point case the results are applied to obtain estimates of global discretization errors for difference methods several examples illustrate the analysis

conditions and conditioning stability and stabilization an overview is given of several concepts that are useful when dealing with numerical bvp in particular a generalisation of the well conditioning concept for nonlinear problems is considered this is done to be able to investigate a class of second order scalar nonlinear problems a detailed study of the solution structure is made for this class the results are applied to two specific problems korteweg de vries and burgers and a way is indicated to stabilize these ill posed bvp

a general technique for deriving analytical structure of fuzzy controllers using arbitrary trapezoidal input fuzzy sets and zadeh and operator deriving the analytical structure of fuzzy controllers is very important as it creates a solid foundation for better understanding insightful analysis and more effective design of fuzzy control systems we previously developed a technique for deriving the analytical structure of the fuzzy controllers that use zadeh fuzzy and operator and the symmetric identical trapezoidal or triangular input fuzzy sets many fuzzy controllers use arbitrary trapezoidal triangular input fuzzy sets that are asymmetric at present there exists no technique capable of deriving the analytical structure of these fuzzy controllers extending our original technique we now present a novel method that can accomplish rigorously the structure derivation for any fuzzy controller mamdani type or ts type that employs the arbitrary trapezoidal input fuzzy sets and zadeh fuzzy and operator the new technique contains our original technique as a special case given the importance of pid control we focus on mamdani fuzzy pi and pd controllers in this paper and show in detail how to use the new technique for different configurations of the fuzzy pi pd controllers the controllers use two arbitrary trapezoidal fuzzy sets for each input variable four arbitrary singleton output fuzzy sets four fuzzy rules zadeh fuzzy and operator and the centroid defuzzifier this configuration is more general and complicated than the mamdani fuzzy pi pd controllers in the current literature it actually contains them as special cases we call this configuration the generalized fuzzy pi pd controller

decomposition property of fuzzy systems and its applications this paper presents the decomposition property of fuzzy systems using a simple constructive decomposition procedure that is by properly dividing the input space into sub input spaces a general fuzzy system is decomposed into several sub fuzzy systems which are the simplest fuzzy systems in the sub input spaces based on the decomposition property of fuzzy systems the analysis of fuzzy systems can be divided into two steps first analyze the properties of the simplest fuzzy systems and then use the decomposition property to extend the results to general fuzzy systems using this idea two applications of the decomposition property are given the first is the application to the representation capability analysis of fuzzy systems the second is the application to the analysis of a class of nonlinear control systems then based on the piecewise affine fuzzy system model the existence condition and the design of a stable control for a class of single input single output siso nonlinear systems are presented

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library

the conditioning of boundary value problems in transferable differential algebraic equations the well conditioning of boundary value problems for transferable differential algebraic equations is shown to be related as in the case of explicit ordinary differential equations to the boundary conditions and the green function in this case other quantities related to the well transferability and the movement speed of the null space of the leading coefficient also appear in the bound the condition number of the related multiple shooting matrix is also bounded in terms of the quantities mentioned above

langevin approach for the dynamics of the contact process on annealed scale free networks we study the dynamics of the contact process one of the simplest nonequilibrium stochastic processes taking place on a scale free network we consider the network topology as annealed i e all links are rewired at each microscopic time step so that no dynamical correlation can build up this is a practical implementation of the absence of correlations assumed by mean field approaches we present a detailed analysis of the contact process in terms of a langevin equation including explicitly the effects of stochastic fluctuations in the number of particles in finite networks this allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs the fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has a hard upper bound when the upper bound is soft the presence of outliers with huge connectivity perturbs the picture even more inducing an apparent shift of the critical point in light of these findings recent theoretical and numerical results in the literature are critically reviewed

rare region effects in the contact process on networks networks and dynamical processes occurring on them have become a paradigmatic representation of complex systems studying the role of quenched disorder both intrinsic to nodes and topological is a key challenge with this in mind here we analyze the contact process i e the simplest model for propagation phenomena with node dependent infection rates i e intrinsic quenched disorder on complex networks we find griffiths phases and other rare region effects leading rather generically to anomalously slow algebraic logarithmic etc relaxation on erd ifmmode mbox h o else h o fi s r enyi networks we predict similar effects to exist for other topologies as long as a nonvanishing percolation threshold exists more strikingly we find that griffiths phases can also emerge char22 even with constant epidemic rates char22 as a consequence of mere topological heterogeneity in particular we find griffiths phases in finite dimensional networks as for instance a family of generalized small world networks these results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks and are relevant for the analysis of both models and empirical data

conditioning of collocation matrices and discrete green s functions the conditioning of spline collocation matrices is considered in the case of a boundary value problem for an mth order equation for bases with continuity built in which corresponds to most current implementations of splines for solving odes the collocation matrix a is related to the corresponding multiple shooting matrix for a first order system this gives the explicit form for a 1 as a discrete green function we briefly discuss the implications for mesh selection and how our results relate to previous ones for finite differences

a comparison of some global methods for solving two point boundary value problems a numerical comparison is made between three codes which implement global methods for solving two point boundary value problems the codes compared are a collocation method colsys a deferred correction method pasva 3 and a deferred correction method based on mono implicit runge kutta formulae hagron although the difficulty of conducting a fair comparison of complex codes is widely recognised it is nevertheless of great practical importance to try to gauge the relative performance of different codes and the three codes considered in this paper are particularly appropriate for numerical comparison by choosing a test set which includes both smooth and increasingly stiff problems some interesting characteristics of the codes are noted and some recommendations regarding their use are made

stability of finite difference schemes for two point boundary value problems in this follow up to a paper on the stability of one step schemes for first order problems the stability of more general finite difference schemes is related explicitly to the stability of the continuous mth order system being solved at times this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem the argument uses a novel smooth approximant f to given data on an arbitrary mesh for which d m f infty m is bounded by the absolutely largest mth divided difference of the data

epidemic spreading in correlated complex networks we study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node s connectivities for the case of markovian complex networks showing only correlations between pairs of nodes we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links which from a node with connectivity k go to nodes with connectivity k ensuremath numerical simulations on a correlated growing network model provide support for our conclusions

on spline basis selection for solving differential equations the suitability of b splines as a basis for piecewise polynomial solution representation for solving differential equations is challenged two alternative local solution representations are considered in the context of collocating ordinary differential equations hermite type and lmonomial both are much easier and shorter to implement and somewhat more efficient than b splines a new condition number estimate for the b splines and hermite type representations is presented one choice of the hermite type representation is experimentally determined to produce roundoff errors at most as large as those for b splines the monomial representation is shown to have a much smaller condition number than the other ones and correspondingly produces smaller roundoff errors especially for extremely nonuniform meshes the operation counts for the two local representations considered are about the same the hermite type representation being slightly cheaper it is concluded that both representations are preferable

voter models on weighted networks we study the dynamics of the voter and moran processes running on top of complex network substrates where each edge has a weight depending on the degree of the nodes it connects for each elementary dynamical step the first node is chosen at random and the second is selected with probability proportional to the weight of the connecting edge we present a heterogeneous mean field approach allowing to identify conservation laws and to calculate exit probabilities along with consensus times in the specific case when the weight is given by the product of nodes degree raised to a power we derive a rich phase diagram with the consensus time exhibiting various scaling laws depending on and on the exponent of the degree distribution numerical simulations give very good agreement for small values of an additional analytical treatment heterogeneous pair approximation improves the agreement with numerics but the theoretical understanding of the behavior in the limit of large remains an open challenge

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

sampling methods for the quasistationary regime of epidemic processes on regular and complex networks a major hurdle in the simulation of the steady state of epidemic processes is that the system will unavoidably visit an absorbing disease free state at sufficiently long times due to the finite size of the networks where epidemics evolves in the present work we compare different quasistationary qs simulation methods where the absorbing states are suitably handled and the thermodynamical limit of the original dynamics can be achieved we analyze the standard qs sqs method where the sampling is constrained to active configurations the reflecting boundary condition rbc where the dynamics returns to the pre absorbing configuration and hub reactivation hr where the most connected vertex of the network is reactivated after a visit to an absorbing state we apply the methods to the contact process cp and susceptible infected susceptible sis models on regular and scale free networks the investigated methods yield the same epidemic threshold for both models for cp that undergoes a standard collective phase transition the methods are equivalent for sis whose phase transition is ruled by the hub mutual reactivation the sqs and hr methods are able to capture localized epidemic phases while rbc is not we also apply the autocorrelation time as a tool to characterize the phase transition and observe that this analysis provides the same finite size scaling exponents for the critical relaxation time for the investigated methods finally we verify the equivalence between rbc method and a weak external field for epidemics on networks

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

the stability of one step schemes for first order two point boundary value problems the stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved at times this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem

fuzzy controllers synthesis and equivalences it has been proved that fuzzy controllers are capable of approximating any real continuous control function on a compact set to arbitrary accuracy in particular any given linear control can be achieved with a fuzzy controller for a given accuracy the aim of this paper is to show how to automatically build this fuzzy controller the proposed design methodology is detailed for the synthesis of a sugeno or mamdani type fuzzy controller precisely equivalent to a given pi controller the main idea is to equate the output of the fuzzy controller with the output of the pi controller at some particular input values called modal values the rule base and the distribution of the membership functions can thus be deduced the analytic expression of the output of the generated fuzzy controller is then established for sugeno type fuzzy controllers precise equivalence is directly obtained for mamdani type fuzzy controllers the defuzzification strategy and the inference operators have to be correctly chosen to provide linear interpolation between modal values the usual inference operators satisfying the linearity requirement when using the center of gravity defuzzification method are proposed

fuzzy control theory a nonlinear case abstract sources of nonlinearity in a fuzzy controller include the fuzzification algorithm used for the controller inputs fuzzy control rules type of fuzzy logic used for evaluating the fuzzy control rules and the defuzzification algorithm used for the controller output we analyze the performance of a simple fuzzy controller with linear and nonlinear defuzzification algorithms we prove theoretically that such a fuzzy controller the smallest possible with two inputs error and rate change of error and a nonlinear defuzzification algorithm is equivalent to a nonfuzzy nonlinear proportional integral pi controller with proportional gain and integral gain changing with error and rate change of error about a setpoint furthermore this fuzzy controller is precisely equivalent to a conventional linear pi controller if a linear defuzzification algorithm is employed computer simulation showed that the performance of the fuzzy controller was almost the same as that of the pi controller when first order and second order linear processes were used furthermore the fuzzy controller was significantly better when a first order with a time delay model was used more importantly the simulated result illustrated that the fuzzy controller was stable when a nonlinear process model was controlled but the pi controller was unstable

on dichotomy and well conditioning in two point boundary value problems we investigate the close relationships between the stability constants and the growth behavior of the fundamental matrix to the general boundary value problem we also show that conditioning number is the right criterion to indicate possible error amplification of the perturbed boundary conditions

accurate estimates for the fundamental solutions of discrete boundary value problems it is investigated how the solutions of a discretized ode can be related to solutions of the ode by using a perturbation analysis based on boundary value formulations in particular it is indicated how one can find estimates for discrete modes that correspond to certain continuous modes these estimates are fairly sharp in a relative sense as is shown by some examples as a consequence one can deduce that certain properties of the one solution space like a dichotomy carry over to the other

mean field diffusive dynamics on weighted networks diffusion is a key element of a large set of phenomena occurring on natural and social systems modeled in terms of complex weighted networks here we introduce a general formalism that allows to easily write down mean field equations for any diffusive dynamics on weighted networks we also propose the concept of annealed weighted networks in which such equations become exact we show the validity of our approach addressing the problem of the random walk process pointing out a strong departure of the behavior observed in quenched real scale free networks from the mean field predictions additionally we show how to employ our formalism for more complex dynamics our work sheds light on mean field theory on weighted networks and on its range of validity and warns about the reliability of mean field results for complex dynamics

an algebraic approximation for the matrix exponential in singularly perturbed boundary value problems a symmetric difference scheme for linear stiff or singularly perturbed boundary value problems of first order with constant coefficients is constructed being based on a stability function containing a matrix square root its essential feature is the unconditional stability in the absence of purely imaginary eigenvalues of the coefficient matrix local damping of errors uniform stability and uniform second order convergence are proved the computation of the specific matrix square root by a well known stable variant of newton s method is discussed a numerical example confirming the results is given

fuzzy control rules and their natural laws abstract we consider the convergence of fuzzy controllers as the number of fuzzy rules tends to infinity we assume that the fuzzy rules are regular and show the existence of limit laws for five different fuzzy control rules

spectral analysis and slow spreading dynamics on complex networks the susceptible infected susceptible sis model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous annihilation reactions the effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean field qmf approximations in scale free networks qmf can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix therefore it can provide predictions on possible rare region effects thus on the occurrence of slow dynamics i compare qmf results of sis with simulations on various large dimensional graphs in particular i show that for erd s renyi graphs this method predicts correctly the occurrence of rare region effects it also provides a good estimate for the epidemic threshold in case of percolating graphs griffiths phases emerge if the graph is fragmented or if we apply a strong exponentially suppressing weighting scheme on the edges the latter model describes the connection time distributions in the face to face experiments in case of a generalized barabasi albert type of network with aging connections strong rare region effects and numerical evidence for griffiths phase dynamics are shown the dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

a survey on free boundary identification of the truncated boundary in numerical bvps on infinite intervals a free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in fazio siam j numer anal 33 1996 1473 we consider here a survey on recent developments related to the free boundary identification of the truncated boundary the goals of this survey are to recall the reasoning for a free boundary identification of the truncated boundary to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line

comparison of voter and glauber ordering dynamics on networks we study numerically the ordering process of two very simple dynamical models for a two state variable on several topologies with increasing levels of heterogeneity in the degree distribution we find that the zero temperature glauber dynamics for the ising model may get trapped in sets of partially ordered metastable states even for finite system size and this becomes more probable as the size increases voter dynamics instead always converges to full order on finite networks even if this does not occur via coherent growth of domains the time needed for order to be reached diverges with the system size in both cases the ordering process is rather insensitive to the variation of the degreee distribution from sharply peaked to scale free

a computational study of finite element methods for second order linear two point boundary value problems a computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear separated boundary conditions is described in each method the approximate solution is a piecewise polynomial expressed in terms of a b spline basis and is determined by solving a system of linear algebraic equations with an almost block diagonal structure the aim of the investigation is twofold to determine if the theoretical orders of convergence of the methods are realized in practice and to compare the methods on the basis of cost for a given accuracy in this study three parametrized families of test problems containing problems of varying degrees of difficulty are used the conclusions drawn are rather straightforward collocation is the cheapest method for a given accuracy and the easiest to implement also for solving the linear algebraic equations the use of a special purpose solver which takes advantage of the structure of the equations is advisable 1 introduction finite element methods for two point boundary value problems for a single ordinary differential equation may take many forms the simplest type of method is probably collocation 3 where an approximate solution is sought in some finite space subject to the constraint that it satisfy the differential equation at certain specified points this type of method has been applied very successfully in the case of mixed order systems of boundary value problems see for example 1 other methods can be described based on the standard l2 galerkin approximation 9 or on a combination of this and the collocation approach 7 13 25 in addition different weak formulations of the boundary value problem lead to two other techniques the hx and galerkin methods 12 14 15 17 how one would apply these methods to a system of differential equations with the exception of the collocation case is not immediately clear in addition it is not apparent what the relative advantages are of these various methods even when applied to a single equation in this paper we restrict our attention to the case of a single linear differential equation our aim is to investigate numerically five finite element techniques for a second order linear two point boundary value problem with separated boundary conditions of a general nature in particular we consider i the organization of the methods ii the treatment of the boundary conditions

numerical methods in seismology a survey is given of the numerical methods used for computing synthetic seismograms in particular the properties of finite differences finite elements integral transforms and ray tracing are described each of these methods is capable of giving good results for its own range of frequencies and wave numbers thus for complete seismograms one should be prepared to use all of the available tools

use of hamiltonian mechanics in systems driven by colored noise the evaluation of the path integral representation for stochastic processes in the weak noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics we show that even when the noise is colored these may be put into a hamiltonian form which leads to better insights and improved numerical treatments we concentrate on solving hamilton s equations over an infinite time interval in order to determine the leading order contribution to the mean escape time for a bistable potential the paths may be oscillatory and inherently unstable in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy we look at two systems in some detail the underdamped langevin equation driven by external exponentially correlated noise and the overdamped langevin equation driven by external quasi monochromatic noise we deduce that caustics focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes

second order splitting combined with orthogonal cubic spline collocation method for the kuramoto sivashinsky equation in this paper a second order splitting method is applied to the kuramoto sivashinsky equation and then an orthogonal cubic spline collocation procedure is employed to the approximate resulting system this semidiscrete method yields a system of differential algebraic equations daes of index 1 error estimates in l 2 and l c norms are obtained for the semidiscrete approximation for the time discretization the time integrator radau5 is used the results of numerical experiments are presented to validate the theoretical findings

a systematic study of fuzzy pid controllers function based evaluation approach a function based evaluation approach is proposed for a systematic study of fuzzy proportional integral derivative pid like controllers this approach is applied for deriving process independent design guidelines from addressing two issues simplicity and nonlinearity to examine the simplicity of fuzzy pid controllers we conclude that direct action controllers exhibit simpler design properties than gain scheduling controllers then we evaluate the inference structures of direct action controllers in five criteria control action composition input coupling gain dependency gain role change and rule parameter growth three types of fuzzy pid controllers using one two and three input inference structures are analyzed the results according to the criteria demonstrate some shortcomings in mamdani s two input controllers for keeping the simplicity feature like a linear pid controller a one input fuzzy pid controller with one to three mapping inference engine is recommended we discuss three evaluation approaches in a nonlinear approximation study function estimation based generalization capability based and nonlinearity variation based approximations the study focuses on the last approach a nonlinearity evaluation is then performed for several one input fuzzy pid controllers based on two measures nonlinearity variation index and linearity approximation index using these quantitative indices one can make a reasonable selection of fuzzy reasoning mechanisms and membership functions without requiring any process information from the study we observed that the zadeh mamdani s max min gravity scheme produces the highest score in terms of nonlinearity variations which is superior to other schemes such as mizumoto s product sum gravity and takagi sugeno kang schemes

collocation for singular perturbation problems i first order systems with constant coefficients the application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated collocation at gauss radau and lobatto points is considered for both initial and boundary value problems for first order systems with constant coefficients particular attention is paid to symmetric schemes for boundary value problems these problems may have boundary layers at both interval ends br our analysis shows that certain collocation schemes in particular those based on gauss or lobatto points do perform very well on such problems provided that a fine mesh with steps proportional to the layers width is used in the layers only and a coarse mesh just fine enough to resolve the solution of the reduced problem is used in between ways to construct appropriate layer meshes are proposed of all methods considered the lobatto schemes appear to be the most promising class of methods as they essentially retain their usual superconvergence power for the smooth reduced solution whereas gauss legendre schemes do not br we also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem for a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number which indicates that these systems can be solved safely by gaussian elimination with scaled partial pivoting

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

random walks on complex trees we study the properties of random walks on complex trees we observe that the absence of loops is reflected in physical observables showing large differences with respect to their looped counterparts first both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case second the mean first passage time mfpt displays a logarithmic degree dependence in contrast to the inverse degree shape exhibited in looped networks this deviation can be ascribed to the dominance of source target topological distance in trees to show this we study the distance dependence of a symmetrized mfpt and derive its logarithmic profile obtaining good agreement with simulation results these unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees

on the computation of solutions of boundary value problems on infinite intervals for solutions of linear boundary value problems defined on lbrack0 infty one has to study the stable or bounded solution manifold a characterization of these manifolds is investigated here a multiple shooting type algorithm is then developed to compute such solutions this algorithm is fully adaptive and also covers problems where the ode matrix does not tend to a limit as is usually assumed if the unstable manifold consists only of exponentially growing solutions if the latter manifold also contains polynomially growing solutions an extrapolation type approach is suggested the theory is illustrated by a number of examples

multilevel preconditioners for non self adjoint or indefinite orthogonal spline collocation problems efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation osc discretization of a dirichlet boundary value problem with a non self adjoint or an indefinite operator the osc solution is sought in the hermite space of piecewise bicubic polynomials it is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal osc equation the preconditioners are used with the preconditioned conjugate gradient method and numerical results are presented that demonstrate their efficiency

indirect multiple shooting for nonlinear parabolic optimal control problems with control constraints we discuss the indirect multiple shooting approach for the solution of pde based parabolic optimal control problems with control constraints the method is formulated within an abstract function space setting and uses a space time galerkin finite element discretization the emphasis is on the embedding of indirect multiple shooting into the optimal control framework as well as the detailed description of the discretization within the pde context numerical results for linear and nonlinear model problems with and without control constraints illustrate the efficient use of indirect multiple shooting particularly in cases where other standard methods fail

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough