on a class of nonlinear dynamical systems the structure of a differential operator in the application of the decomposition method abstract a suitable expresion of a differential operator is deduced allowing the explicit calculation of the general term f h in the power series expansion of a function f x t to be utilized for solving nonlinear differential equations

polynomial nonlinearities in differential equations abstract adomian and his collaborators have found solutions for nonlinear stochastic or deterministic in the limiting case differential equations involving polynomial non linearities using the first author s a n polynomials and recently developed methods of calculating these polynomials it becomes very easy to write solutions for non linear terms such as y m for positive integers m the cases of negative integer m or even decimal powers will appear elsewhere

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

fast methods for solving nonlinearly coupled equations describing fiber amplifiers a series of fast and accurate methods for ordinary differential equations are proposed the results of solving nonlinearly coupled equations describing fiber raman amplifiers demonstrate that these methods can improve the accuracy by more than one or even two orders of magnitude in comparison with the classical methods and can increase the computing speed by more than an order of magnitude on the same accuracy the proposed methods can availably solve nonlinearly coupled equations of erbium doped fiber amplifiers e kutta method under the same accuracy model equations of erbium doped fiber amplifiers edfas can be solved by the use of the proposed methods greatly available

on the tricomi problem the tricomi equation is solved by the decomposition method 6 8 on the half space x y 0 and also for suitable boundary conditions given on the boundary of the hyperbolic elliptic domain

on the control of stochastic systems abstract a general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the iterative method of the authors or its more recent adaptations for stochastic operator equations it is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations the method therefore is applicable to the control problem application is made to the state space equation x ax bu c where a b c are stochastic matrices corresponding to stochastic operators i e involving randomly time varying elements e g a ij t a t t f a p s it is emphasized that the processes are arbitrary stochatic processes with known statistics no assumption is made of wiener or markov behavior or of smallness of fluctuations and no closure approximations are necessary the method differs in interesting aspects from volterra series expansions used by wiener and others and has advantages over the other methods because of recent progress in the solution of the nonlinear case it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages

on the convergence region for decomposition solutions abstract the author s decomposition method using his a n polynomials for the nonlinearities has been shown to apply to wide classes of nonlinear or nonlinear stochastic operator equations providing a computable accurate solution which converges rapidly in computation the above is sufficient for a rapid test of convergence region

evolution of solitary waves for a perturbed nonlinear schrodinger equation soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear schrodinger equation perturbation terms which model wide classes of physically relevant perturbations are considered an analytical solution is found for the first order correction of the evolving solitary wave this solution for the solitary wave tail is in integral form and an explicit expression is found for large time singularity theory usually used for combustion problems is applied to the large time expression for the solitary wave tail analytical results are obtained such as the parameter regions in which qualitatively different types of solitary wave tails occur the location of zeros and the location and amplitude of peaks in the solitary wave tail two examples the near continuum limit of a discrete nls equation and an explicit numerical scheme for the nls equation are considered in detail for the discrete nls equation it is found that three qualitatively different types of solitary wave tail can occur while for the explicit finite difference scheme only one type of solitary wave tail occurs an excellent comparison between the perturbation solution and numerical simulations for the solitary wave tail is found for both examples

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

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

nonlinear plasma response abstract the decomposition method g adomian stochastic systems academic press new york london 1983 g adomian stochastic systems ii in press and earlier work of adomian referenced therein allows solution without linearization of problems normally requiring the self consistent field technique the latter technique can provide a sufficiently accurate approximation in weakly coupled plasma but the decomposition method of adomian provides a superior and physically preferable methodology for the general case

direct perturbation theory for solitons of the derivative nonlinear schrodinger equation and the modified nonlinear schrodinger equation a direct perturbation theory for solitons of the derivative nonlinear schrodinger dnls equation is developed based on a closure of eigenfunctions of the linearized dnls equation around a one soliton solution the slow evolution of soliton parameters and the perturbation induced radiation are obtained under the known simple gaugelike transformation these results are transformed into those for the perturbed modified nonlinear schrodinger mnls equation describing propagation of femtosecond pulses in optical fibers a calculation of the perturbation induced radiation fields for the perturbed dnls and mnls equations is also made our results for the perturbed mnls equation can be reduced perfectly to those for the perturbed nonlinear schrodinger equation in the small nonlinear dispersion limit

nonlinear differential equations with negative power nonlinearities abstract differential equations involving the term y m where m is a positive integer are solved by the decomposition method 1 3 441 452

analytical solutions for steady state responses of an infinite euler bernoulli beam on a nonlinear viscoelastic foundation subjected to a harmonic moving load abstract the paper deals with the steady state responses of an infinite euler bernoulli beam resting on a nonlinear foundation under a harmonic moving load greatly different from previous works the nonlinear partial differential governing equation of the beam motion is converted to two nonlinear volterra integral equations by using the fourier transform residues theorem and the convolution theorem the volterra integral equations have four different expressions depending on the damping coefficient of the foundation the linear part of the foundation stiffness and the frequency of the moving load the modified adomian decomposition method in conjunction with a simple iterative formula derived from the integral equation theorems are applied to obtain analytical solutions for the steady state responses of the beam the closed form solutions presented in this paper do not contain complicated infinite integrals which are better to show the influence of the nonlinear part of foundation stiffness the parametric study shows that the nonlinear part of foundation stiffness affects not only qualitative but also quantitative analysis results of the infinite euler bernoulli beam under a harmonic moving load

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

a scaled gauss newton primal dual search direction for semidefinite optimization interior point methods for semidefinite optimization sdo have recently been studied intensively due to their polynomial complexity and practical efficiency most of these methods are extensions of linear optimization lo algorithms as opposed to the lo case there are several different ways of constructing primal dual search directions in sdo the usual scheme is to apply linearization in conjunction with symmetrization to the perturbed optimality conditions of the sdo problem symmetrization is necessary since the linearized system is overdetermined a way of avoiding symmetrization is to find a least squares solution of the overdetermined system such a gauss newton direction was investigated by kruk et al the gauss newton direction in semidefinite programming research report corr 98 16 university of waterloo waterloo canada 1998 without giving any complexity analysis in this paper we present a similar direction where a local norm is used in the least squares formulation and we give a polynomial complexity analysis and computational evaluation of the resulting primal dual algorithm

the analytical evolution of nls solitons due to the numerical discretization error soliton perturbation theory is used to obtain analytical solutions describing solitary wave tails or shelves due to numerical discretization error for soliton solutions of the nonlinear schrodinger equation two important implicit numerical schemes for the nonlinear schrodinger equation with second order temporal and spatial discretization errors are considered these are the crank nicolson scheme and a scheme due to taha 1 based on the inverse scattering transform the first order correction for the solitary wave tail or shelf is in integral form and an explicit expression is found for large time the shelf decays slowly at a rate of which is characteristic of the nonlinear schrodinger equation singularity theory usually used for combustion problems is applied to the explicit large time expression for the solitary wave tail analytical results are then obtained such as the parameter regions in which qualitatively different types of solitary wave tails occur the location of zeros and the location and amplitude of peaks it is found that three different types of tail occur for the crank nicolson and taha schemes and that the taha scheme exhibits some unusual symmetry properties as the tails for left and right moving solitary waves are different optimal choices of the discretization parameters for the numerical schemes are also found which minimize the amplitude of the solitary wave tail the analytical solutions are compared with numerical simulations and an excellent comparison is found

application of the decomposition method to inversion of matrices abstract the decomposition method g adomian stochastic systems academic press new york london 1983 is applied to inversion of matrices by approximation

on the nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

inverse scattering transform for the derivative nonlinear schrodinger equation with nonvanishing boundary conditions an inverse scattering transform for the derivative nonlinear schr odinger equation with nonvanishing boundary conditions is derived by introducing an affine parameter to avoid constructing riemann sheets a one soliton solution simpler than that in the literature is obtained which is a breather and degenerates to a bright or dark soliton as the discrete eigenvalue becomes purely imaginary the solution is mapped to that of the modified nonlinear schr odinger equation by a gaugelike transformation predicting some sub picosecond solitons in optical fibers

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

transient behaviour of semilinear stochastic systems with random parameters abstract the transient solution of a class of nonautonomous stochastic differential equations with given random initial conditions is studied the considered evolution equation is characterized by the presence of a deterministic linear term and of a random nonlinear term whose parameters can be modeled by random processes of rice noise type analytical approximated expressions for the first and second order moments and for the probability density of the solution process are derived and are applied to determine the statistical properties of a class of stochastic nonlinear oscillators

on composite nonlinearities and the decomposition method abstract accurate convergent computable solutions using the decomposition method have been demonstrated in and papers for wide classes of nonlinear and or stochastic differential partial differential or algebraic equations it is shown specifically in this paper that composite nonlinearities of the form nx n0 n1 n2 x appearing in such equations where the ni are nonlinear operators can also be handled with the adomian an polynomials

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

on green s function in higher order stochastic differential equations abstract in adomian s solution of linear or nonlinear deterministic or stochastic differential equations the decomposition of the linear part into l r can be made so as to simplify green s function and consequently the resulting computations

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

an application of the decomposition method to the matrix riccati equation in a neutron transport process abstract the differential matrix riccati equation with matrix variable coefficients arising in transport theory is considered an analytical approximation of the solution of the non linear equation is investigated using the adomian decomposition method and an application to neutron steady state process is proposed to test the accuracy of the method

on a class of nonlinear stochastic dynamical systems analysis of the transient behaviour abstract a class of nonlinear dynamical systems with random parameters and initial conditions is considered the transient time evolution of the mean value of the dependent variable is analyzed both by means of the stochastic averaging method and by adomian s decomposition method a proof of the equivalence of the decomposition method to the time series approximation of the dynamical response is provided an application is made to the van der pol equation and finally a comparison between the two previously discussed methods is given

stochastic model for colored noise abstract the input output point of view for the ito equation to avoid stochastic integration is related to the stochastic operator approach for a physically reasonable robust mathematical model

collisions of optical ultra short vector pulses asymptotic two pulse solutions of a modified via attaching a nonlinear derivative term vector nonlinear schrodinger equation mvnse are analyzed using the hirota bilinearization method this equation is known to describe the propagation of ultra short optical pulses in nonlinear fibers in particular in the optical crystal fibers some pulse parameter regimes of applicability of the solutions to study the pulse collisions are estimated it is found for these regimes that the collision of ultra short vector pulses of almost equal velocities results in a transformation of their polarizations similar as that of the manakov solitons whenever the pulse interaction effects can be neglected

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

green s function method for perturbed dnls equation with the squared spectrum parameter abstract a general direct perturbation theory for the perturbed dnls equation is developed the completeness relation is given by the green s function method similar to that used for the linear equations the effect of damping on a single soliton is calculated as an example and the result is good with the squared spectrum parameter

squared eigenfunctions for the sasa satsuma equation squared eigenfunctions are quadratic combinations of jost functions and adjoint jost functions which satisfy the linearized equation of an integrable equation they are needed for various studies related to integrable equations such as the development of its soliton perturbation theory in this article squared eigenfunctions are derived for the sasa satsuma equation whose spectral operator is a 3 3 system while its linearized operator is a 2 2 system it is shown that these squared eigenfunctions are sums of two terms where each term is a product of a jost function and an adjoint jost function the procedure of this derivation consists of two steps first is to calculate the variations of the potentials via variations of the scattering data by the riemann hilbert method the second one is to calculate the variations of the scattering data via the variations of the potentials through elementary calculations while this procedure has been used before on other integrable equations it is shown here for

numerical solution of a viscous incompressible flow problem through an orifice by adomian decomposition method a steady flow problem of a viscous incompressible fluid through an orifice is widely applicable to many physical phenomena and has been studied previously by many researchers a problem of such type has been solved by applying lad method given by roache comp fluids 3 1975 179 the resulting system of linear equations is solved by adomian decomposition method solving frontier problems of physics the decomposition method kluwer academic publishers boston ma 1994 we study an analytic solution of a steady flow problem of a viscous incompressible fluid through an orifice by using the adomian s decomposition method the numerical results show the effectiveness of the method for this type of equations they show that the present approach is relatively easy and highly accurate

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

nonlinear stochastic differential delay equations abstract the methods used by adomian and his co workers to solve linear and nonlinear stochastic differential equations will be demonstrated to be applicable to differential equations deterministic or stochastic involving delays constant time dependent or random

semilinear stochastic systems analysis with the method of the stochastic green s function and application in mechanics abstract this work deals with the analysis of a class of semilinear vector stochastic operator equations the method of the stochastic green s function due to adomian is extended to the above mentioned class of equations some perturbation techniques are also developed and a pertinent application in nonlinear mechanics is considered

coupled differential equations and coupled boundary conditions abstract it is demonstrated that the high accuracy for approximations requiring only a few terms which is typical of the decomposition method for nonlinear stochastic operator equations or special cases linear or deterministic holds for coupled equations and coupled boundary conditions as well

coupled nonlinear stochastic differential equations abstract systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co workers examples include two coupled riccati equations coupled linear equations stochastic coupled equations with product terms and n coupled stochastic differential equations

solving nonlinear differential equations with decimal power nonlinearities abstract the decomposition method stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press is applied to the solution of nonlinear differential equations of the form ly ny x t where l is a linear differential operator and ny is a nonlinear term of the form y with a decimal number