an accurate solution to the multispecies lotka volterra equations the decomposition method is applied to the solution of the lotka volterra equations modelling the dynamic behaviour of an arbitrary number of species the analytical solution derived is an infinite power series for each species where the n term is given by a recurrence relation as particular examples the cases of one two and three species are considered for these cases comparisons between the present semi analytical solution and a fully numerical solution or an exact one for one species show exc ellent agreement

existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations in this paper the banach fixed point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference email protected n dx n b n x email protected email protected n x h 1 n x h 2 n x h k n f n x f 1 n x f 2 n x f k n c n n n 0 where t email protected n n email protected n 0 a b c n n 0 r with a n 0 for email protected n n 0 h f n n 0xr k r and h l f l n n 0 z withlimn h l n limn f l n email protected 1 2 k a few mann type iterative approximation schemes with errors are suggested and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed seven nontrivial examples are given to illustrate the advantages of our results

oscillation for difference equations with continuous variable consider the delay difference equation y t y t p t h y t t t 0 where 0 0 and p c r r f c r r h c r r some oscillation criteria for this equation are obtained

the persistence of nonoscillatory solutions of difference equations under impulsive perturbations we obtain a sufficient condition for the persistence of nonoscillatory solutions of the difference equation with continuous variable x t t x t i 1mp i t x t c i t 0 under the impulsive perturbations x t k t x t k i k x t k k n 1

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

new approaches for the study of non linear oscillators abstract this paper presents a power series approach to accurately obtain the damped separatrices for a class of unforced non linear oscillators this in turn delineates the basins of attraction of two or more stable attractors in the phase space the method is illustrated by its applications to the unforced duffing holmes and blacklash oscillators next a novel semi analytical integration scheme called the phase space linearization method psl is developed to obtain stable and unstable periodic solutions of forced as well as unforced non linear oscillators and also the damped separatrices the performance of the proposed method has been tested against periodic solutions of three oscillators namely ueda s duffing holmes and van der pol s oscillators obtained using a fourth order runge kutta method with a sufficiently small time step moreover the separatrices obtained using the psl method are compared with those obtained via the power series method as developed earlier the issue of accumulation of error in the psl method as against the fourth order runge kutta scheme is also described numerically through an example of a first order non linear equation having closed form solution

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

a generalized local linearization principle for non linear dynamical systems tangent spaces in non linear dynamical systems are state dependent hence it is generally not possible to exactly represent a non linear dynamical system by a linear one over finite segments of the evolving trajectories in the phase space it is known from the well known theorem of hartman and grobman that a non linear oscillator admits a linearization within a neighbourhood of any hyperbolic fixed point in the sense that the linearized flow bears a topological conjugacy to the non linear flow within the same neighbourhood this linearization is based on the construction of the tangent space to the non linear vector field at the hyperbolic fixed point even though useful in bifurcation analysis such a linearization procedure cannot be successfully applied to identically simulate the non linear flow over the phase space because the functional form of the topological conjugacy is not known with this in mind an implicit approach to local linearization is evolved in this paper such that the tangent space of the linearized equation transversally intersects the tangent space to the non linear dynamical system at that point in the state space where the solution vector is desired several numerical schemes for the implementation of this implicit and local linearization procedure are explored and illustrated with several numerical results it is shown that the locally transversal linearization ltl procedure finally reduces the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations with the desired solution vector as the unknown the variables in the state space appear as unknown quantities in this approach finally one arrives at non linear poincare maps for the non linear oscillator in this scheme the characteristic time interval for the map is arbitrarily fixed the methodology is found to be quite versatile for handling non linear dynamical systems in particular it is verified that these schemes are capable of accurately predicting a wide spectrum of typically non linear response characteristics such as limit cycles multi periodicity almost periodicity and chaos for a limited class of response patterns the principle proposed has another advantage in that the time step for integrating the non linear odes need not be small an improved and a more general higher order version of the ltl method is also considered a distinct advantage of this higher version is in its improved ability for a closer simulation of phase dependent time histories where a difference in the initial conditions leads to a phase difference in the realized solution history

a new numeric analytical principle for nonlinear deterministic and stochastic dynamical systems a new implicit local linearization method termed as the locally transversal linearization ltl is developed in the present paper for a numeric analytical integration of nonlinear dynamical systems under deterministic and or stochastic excitations the ltl procedure is developed in such a way that at a chosen time instant the ltl solution manifold transversally intersects the nonlinear solution manifold at that particular point or cross section in the state space where the solution vector needs to be obtained in general construction of the ltl equation is non unique and thus it is generally needed to substitute the conditionally linear ltl based solution into the given nonlinear ordinary differential equation ode at a specified time instant to ensure a transversal intersection in the present study however it is demonstrated that a transversal intersection at a given time instant may as well be ensured by just choosing a few specific forms of the ltl equation thereby eliminating the need for substitution this new variant of ltl is particularly suitable for discretization of stochastic differential equations sdes in that one simultaneously avoids dealing with formal derivatives of wiener processes while satisfying the nonlinear sde at discrete time points the stochastic ltl procedure is adaptable for obtaining strong pathwise solutions of a general nonlinear dynamical system with continuous but not necessarily differentiable vector fields under additive and or multiplicative excitations a specific advantage of the ltl method is that for a class of response regime herein referred to as the phase independent regime it can obtain an accurate description of response irrespective of the choice of the time step and without a propagation of local errors presently the method has been numerically illustrated for strong solutions of the hardening duffing oscillator under deterministic sinusoidal or additive stochastic white noise excitations or their combination limited numerical results on periodic solutions chaos with or without noise and stochastic resonance are included in this study the results presented are indicative of the sample path accuracy consistency stochastic numerical stability and a partial insensitivity to time step sizes of the proposed method

oscillation of second order nonlinear difference equation with continuous variable in this paper we are mainly concerned with the second order nonlinear difference equation with continuous variable here by using the iterated integral transformations generalized riccati transformations and integrating factors we give some oscillatory criteria for this equation

a note on oscillation of second order nonlinear difference equation with continuous variable abstract in this paper by improving the proofs of some theorems in j math anal appl 255 2001 349 357 we obtain some new oscillation criteria for the second order nonlinear difference equation with continuous variable

comparison of the solutions obtained by b spline fem and adm of kdv equation a numerical solution to a generalized korteweg de vries kdv equation is obtained using the galerkin method with quadratic b spline finite element method fem over which the nonlinear term is locally linearized and using the adomian s decomposition method adm test problems concerning the motion and interaction of soliton solutions are used to compare the fem with the adm the present methods extremely well in terms of accuracy efficiency simplicity stability and reliability

oscillations of delay difference equations with oscillating coefficients abstract in this paper we establish sufficient conditions for the oscillation of all solutions of the delay difference equation y n 1 y n p n y n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer an open problem in ladas in international conference on differential equations stability and control dekker new york 1990 is solved

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

a new symbolic computational approach to singular initial value problems in the second order ordinary differential equations this paper presents a new symbolic algorithm to compute the singular initial value problem of second order ordinary differential equations using adomian decomposition method the algorithm has been implemented in the computer algebra system maple to facilitate the application of this method a number of examples are presented to demonstrate the computation efficiency of the program

solving delay differential systems with history functions by the adomian decomposition method abstract a number of nonlinear phenomena in many branches of the applied sciences and engineering are described in terms of delay differential equations which arise when the evolution of a system depends both on its present and past time in this work we apply the adomian decomposition method adm to obtain solutions of several delay differential equations subject to history functions and then investigate several numerical examples via our subroutines in maple that demonstrate the efficiency of our new approach in our approach history functions are continuous across the initial value and its derivatives must be equal to the initial conditions see section 3 so that our results are more efficient and accurate than previous works

oscillation for linear non autonomous systems of difference equations with continuous arguments abstract in this paper we establish sufficient conditions for the oscillation of the linear non autonomous systems of difference equations with continuous arguments n x t p x t j 1 n q j t x t j 0 where x t x t x t n 1 p r m m q j c r r m m j 1 2 n and j j 1 2 n are non negative constants furthermore assume q j t q j l k t where q j l k t l k 1 2 m j 1 2 n are continuous bounded and do not change signs on r

global solvability for a second order nonlinear neutral delay difference equation this paper studies the global existence of solutions of the second order nonlinear neutral delay difference equation d a email protected x n bx n t f n x n d 1 n x n d 2 n x n d k n c n n n 0 with respect to all email protected r a few results on global existence of uncountably many bounded nonoscillatory solutions are established for the above difference equation several nontrivial examples which dwell upon the importance of the results obtained in this paper are also included

revisit on partial solutions in the adomian decomposition method solving heat and wave equations the adomian decomposition method is considered in application to heat and wave equations the so called partial solution technique is used it is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions in cases involving inhomogeneous boundary conditions improper results may be obtained by former method this paper presents a further insight into partial solutions in the decomposition method and the resolution of such cases

oscillation and comparison results for delay difference equations abstract we establish a necessary and sufficient condition for the existence of positive solutions and obtain sturm type comparison theorem for the difference equation x n 1 x n i 1 m pi n x n ki o n 0 we also obtain the following comparison result equation oscillates if and only if x n 1 x n i 1 m qi n x n ki o n 0 oscillates provided n lim pi n qi i 1 2 m

an aftertreatment technique for improving the accuracy of adomian s decomposition method abstract adomian s decomposition method adm is a nonnumerical method which can be adapted for solving nonlinear ordinary differential equations in this paper the principle of the decomposition method is described and its advantages as well as drawbacks are discussed then an aftertreatment technique at is proposed which yields the analytic approximate solution with fast convergence rate and high accuracy through the application of pade approximation to the series solution derived from adm some concrete examples are also studied to show with numerical results how the at works efficiently

bounded oscillation criteria for certain third order nonlinear difference equations with several delays and advances this paper deals with the third order nonlinear difference equations with several delays and advances d a email protected 2x n f n x n x n d 1 n x n d k n 0 n n 0 and d a email protected 2x n b ng x n x n d 1 n x n d k n 0 n n 0 sufficient conditions of the existence of uncountably many bounded positive negative and nonoscillatory solutions respectively and necessary and sufficient conditions of the bounded oscillation of and with certain asymptotic behavior are given in addition a sufficient condition of the oscillatory behavior of all bounded solutions of with a special of asymptotic property is also presented nine nontrivial examples which dwell upon the importance of the results obtained in this paper are also included

an adaptation of adomian decomposition for numeric analytic integration of strongly nonlinear and chaotic oscillators a novel form of an explicit numeric analytic technique is developed for solving strongly nonlinear oscillators of engineering interest the analytic part of this technique makes use of adomian decomposition method adm but unlike other analytical solutions it does not rely on the functional form of the solution over the whole domain of the independent variable instead it discretizes the domain and solves multiple ivps recursively adm uses a rearranged taylor series expansion about a function and finds a series of functions which add up to generate the required solution the present method discretizes the axis of the independent variable and only collects lower powers of the chosen step size in series solution each function constituting the series solution is found analytically it is next shown that the modified adm can be used to obtain the analytical solution in a piecewise form for nonlinear oscillators such a piecewise solution is valid only within a chosen time step an attempt has been made to address few issues like the order of local error and convergence of the method emphasis has been on the application of the present method to a number of well known oscillators the method has the advantage of giving a functional form of the solution within each time interval thus one has access to finer details of the solution over the interval this is not possible in purely numerical techniques like the runge kutta method which provides solution only at the two ends of a given time interval provided that the interval is chosen small enough for convergence it is shown that the present technique successfully overcomes many limitations of the conventional form of adm the present method has the versatility and advantages of numerical methods for being applied directly to highly nonlinear problems and also have the elegance and other benefits of analytical techniques

exact and numerical soliton solutions of some nonlinear physical models we consider two types of variant water wave equations and the solitary wave solutions of them are constructed by using the adomian decomposition we also made a numerical comparison of the equations

a new method for solving singular initial value problems in the second order ordinary differential equations singular initial value problems linear and nonlinear homogeneous and nonhomogeneous are investigated by using adomian decomposition method the solutions are constructed in the form of a convergent series a new general formula is established the approach is illustrated with few examples

locally convex quasi algebras with sufficiently many representations the main aim of this paper is the investigation of conditions under which a locally convex quasi algebra a a0 attains sufficiently many tw continuous representations in l d h to separate its points having achieved this a usual notion of bounded elements on a rises on the other hand a natural order exists on a a0 related to the topology that also leads to a kind of bounded elements which we call order bounded what is important is that under certain conditions the latter notion of boundedness coincides with the usual one several nice properties of order bounded elements are extracted that enrich the structure of locally convex quasi algebras

experimentation with two step adomian decomposition method to solve evolution models the two step adomian decomposition method tsadm is derived by separate the standard adomian decomposition method into two steps although the tsadm needs only a slight variation from the standard adomian decomposition method experimentation shows that the tsadm is also promising for systems of inhomogeneous differential and integral equations hyperbolic partial differential equations and singular initial value problems

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

tensor products of generalized b algebras abstract generalized b algebras shortly gb algebras are generalizations of c algebras and belong to the class of abstract unbounded operator algebras these algebras have interesting applications in mathematical physics and this gives a strong impetus for studying them to our knowledge up to now there is nothing in the literature about tensor products of gb algebras for this reason this paper is devoted to the investigation of this subject matter

linear oscillation of second order nonlinear difference equations with damped term abstract in this paper we shall establish relations between the oscillation of second order nonlinear difference equations with damped term and the oscillation of their linear limiting equations

an explicit and numerical solutions of some fifth order kdv equation by decomposition method in this paper by considering the adomian decomposition method explicit and numerical solutions are calculated for a various fifth order korteweg de vries equations with initial condition the method does not need linearization or weak nonlinearity assumptions perturbation theory the decomposition series explicit solution of the equation is quickly obtained by observing the existence of the self canceling noise terms where sum of components vanishes in the limit

spline methods for the solution of fourth order parabolic partial differential equations in this paper a fourth order non homogeneous parabolic partial differential equation that governs the behaviour of a vibrating beam is solved by using a new three level method based on parametric quintic spline in space and finite difference discretization in time stability analysis of the method has been carried out it has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non homogeneous cases can be derived from our method we also obtain two new high accuracy schemes of o k 4 h 6 and o k 4 h 8 and two new schemes which are analogues of jain s formula for the non homogeneous case comparison of our results with those of some known methods show the superiority of the present approach

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

the noisy convergence phenomena in decomposition method solutions abstract a damped oscillating convergence to a correct solution sometimes observed in decomposition method solutions of differential and partial differential equation is investigated

bounded continuous nonoscillatory solutions of second order nonlinear difference equations with continuous variable in this paper by using the fixed point theory under quite general condition on the nonlinear term we obtain an existence result concerning bounded continuous nonoscillatory solutions of a second order nonlinear difference equation with continuous variable

chaos in numerical analysis of ordinary differential equations abstract the discretisation of the ordinary nonlinear differential equation d y d t y 1 y by the entral difference scheme is studied for fixed mesh size in the usual numerical computation this method produces some ghost solution for the long range calculation regarding this discretisation as a dynamical system in r2 these pathological behaviors are shown to be a kind of chaos in the dynamical system for any mesh size moreover some combination of the central difference scheme and the euler s scheme is studied for the above equation it gives some motivation for henon s model the usual discretisation of a second order differential equation are studied also it gives some chaotic behaviors numerically which is similar to the behavior of the orbits of the system of differential equations proposed by henon heiles

necessary and sufficient conditions for the oscillation of systems of difference equations with continuous arguments abstract in this paper the oscillation of all solutions of system of the form y t i 1 n p i y t i 0 where p i r m m i r i 1 2 n is studied necessary and sufficient conditions in terms of the characteristic equation and explicit conditions in terms of the eigenvalues or the logarithmic norms of the coefficient matrices are established

investigation on non linear vibration in arched beam for bridges construction via agm method analyzing and modeling the vibrational behavior of arched bridges during earthquake in order to decrease the exerted damages to the structure is a very hard task to do in the present paper this item has been performed analytically for the first time due to the importance of arched bridges building as a great structure in the human being civilization and its specifications such as transferring vertical loads to its arcs and the lack of bending moments and shearing forces this case study is devoted to this special issue therefore the nonlinear vibration of arched bridges has been modeled and simulated by an arched beam with harmonic vertical loads and its behavior has been investigated by analyzing a nonlinear partial differential equation governing on the system it is notable that the procedure has been done analytically by agm akbari ganji s method it has found that angular frequency is increased significantly by escalating the amount of the center outlet furthermore comparisons have been made between the obtained results by numerical method rkf 45 and agm in order to assess the scientific validity obtained results indicate the precision of the achieved answers by agm method

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

a multistep transversal linearization mtl method in non linear dynamics through a magnus characterization a new form of a multi step transversal linearization mtl method is developed and numerically explored in this study for a numeric analytical integration of non linear dynamical systems under deterministic excitations as with other transversal linearization methods the present version also requires that the linearized solution manifold transversally intersects the non linear solution manifold at a chosen set of points or cross section in the state space however a major point of departure of the present method is that it has the flexibility of treating non linear damping and stiffness terms of the original system as damping and stiffness terms in the transversally linearized system even though these linearized terms become explicit functions of time from this perspective the present development is closely related to the popular practice of tangent space linearization adopted in finite element fe based solutions of non linear problems in structural dynamics the only difference is that the mtl method would require construction of transversal system matrices in lieu of the tangent system matrices needed within an fe framework the resulting time varying linearized system matrix is then treated as a lie element using magnus characterization w magnus on the exponential solution of differential equations for a linear operator commun pure appl math vii 1954 649 673 and the associated fundamental solution matrix fsm is obtained through repeated lie bracket operations or nested commutators an advantage of this approach is that the underlying exponential transformation could preserve certain intrinsic structural properties of the solution of the non linear problem yet another advantage of the transversal linearization lies in the non unique representation of the linearized vector field an aspect that has been specifically exploited in this study to enhance the spectral stability of the proposed family of methods and thus contain the temporal propagation of local errors a simple analysis of the formal orders of accuracy is provided within a finite dimensional framework only a limited numerical exploration of the method is presently provided for a couple of popularly known non linear oscillators viz a hardening duffing oscillator which has a non linear stiffness term and the van der pol oscillator which is self excited and has a non linear damping term

numerical solution of two dimensional nonlinear differential equation by homotopy perturbation method abstract in this paper an application of homotopy perturbation method is applied to solve the nonlinear two dimensional wave equation the analytic solution of the nonlinear wave equation is calculated in the form of a series with easily computable components the non homogenous equation is effectively solved by employing the phenomena of the self canceling noise terms where sum of components vanishes in the limit comparing the methodology with some known techniques shows that the present approach is powerful and reliable its remarkable accuracy properties are finally demonstrated by an example

application of modified decomposition method for the analytical solution of space fractional diffusion equation abstract spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow finance and others areas of application this paper presents the analytical solutions of the space fractional diffusion equations by modified decomposition method mdm by using initial conditions the explicit solutions of the equations have been presented in the closed form the decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self cancelling noise phenomenon two examples the first one is one dimensional and the second one is two dimensional fractional diffusion equation are presented to show the application of the present technique the present method performs extremely well in terms of efficiency and simplicity

equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations abstract we consider the solution of partial differential equations for initial boundary conditions using the decomposition method the partial solutions obtained from the seperate equations for the highest ordered linear operator terms are shown to be identical when the boundary conditions are general and asymptotically equal when the boundary conditions in one independent variable are independet of other variables

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

states and derivations on quasi algebras the gel fand naimark segal representation is extended to quasi algebras provided that a suitable choice of states is made derivations of quasi algebras and in particular of left hilbert quasi algebras are studied

the modified homotopy perturbation method for solving strongly nonlinear oscillators in this paper we propose a reliable algorithm for the solution of nonlinear oscillators our algorithm is based upon the homotopy perturbation method hpm laplace transforms and pade approximants this modified homotopy perturbation method mhpm utilizes an alternative framework to capture the periodic behavior of the solution which is characteristic of oscillator equations and to give a good approximation to the true solution in a very large region the current results are compared with those derived from the established runge kutta method in order to verify the accuracy of the mhpm it is shown that there is excellent agreement between the two sets of results results also show that the numerical scheme is very effective and convenient for solving strongly nonlinear oscillators

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

the oscillation of partial difference equations with continuous arguments abstract for the partial difference equations and we shall obtain sufficient conditions for the oscillation of all solutions of these equations

a semi analytical locally transversal linearization method for non linear dynamical systems an numeric analytical implicit and local linearization methodology called the locally transversal linearization ltl is developed in the present paper for analyses and simulations of non linear oscillators the ltl principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non linear dynamical system at that particular point in the state space where the solution vector is sought for purposes of numerical implementation two different numerical schemes namely ltl 1 and ltl 2 schemes based on the ltl methodology are presented both ltl 1 and ltl 2 procedures finally reduce the given set of non linear ordinary differential equations odes to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space while in the ltl 1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter in the ltl 2 scheme a set of unknown residues enters the linearized system as parameters a limited set of examples involving a few well known single degree of freedom sdof non linear oscillators indicate that the ltl methodology is capable of accurately predicting many complicated non linear response patterns including limit cycles quasi periodic orbits and even strange attractors copyright 2001 john wiley sons ltd

oscillation of solutions for second order nonlinear difference equations with nonlinear neutral term abstract some riccati type difference inequalities are given for the second order nonlinear difference equations with nonlinear neutral term and using these inequalities we obtain some oscillation criteria for the above equation

oscillation criteria for difference equations with unbounded delay consider the delay difference equation n 1 n pn n 0 n 0 1 2 where n z is nondecreasing n n for n n and limn n pn is a nonnegative sequence some oscillation criteria for this equation are obtained

lp spaces as quasi algebras abstract the banach space l p x for x a compact hausdorff measure space is considered as a special kind of quasi algebra called cq algebra over the c algebra c x of continuous functions on x it is shown that for p 2 l p x c x is semisimple in a generalized sense some consequences of this fact are derived

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed

phase space linearization for non linear oscillators deterministic and stochastic systems abstract a new and efficient semi analytical phase space linearization psl scheme for a class of non linear oscillators is developed in this paper the method is based on replacement of the non linear vector field by a set of linear ones each valid over a short segment of the evolving trajectory or over sufficiently small interval of time based on this concept a few explicit and implicit integration schemes are first proposed and applied to a class of low dimensional non linear dynamical systems to accurately determine their response trajectories this approach of local linearization is further extended to non linear oscillators excited by formal derivatives of one or a combination of gauss markov processes since the present methodology reduces the non linear operator by a set of linear operators it is also demonstrated that the principles of linear random vibration may be suitable exploited to arrive at a faster and semi analytical monte carlo scheme for computing the response statistics both in stationary and non stationary regimes limited examples are presented and compared with exact solutions whenever available to illustrate the efficiency and versatility of the proposed schemes

a two step adomian decomposition method the adomian decomposition method adm is a nonnumerical method for solving a wide variety of functional equations and usually gets the solution in a series form in this paper we propose an efficient modification of adm namely two step adomian decomposition method tsadm that will facilitate the calculations we then conduct a comparative study between the tsadm and previous methods with the help of several illustrative examples the results obtained indicate that the tsadm is effective and promising

exact solutions for variable coefficients fourth order parabolic partial differential equations in higher dimensional spaces in this paper analytic solution of variable coefficient fourth order parabolic partial differential equation in two and three space variables are developed the calculations are accelerated by using the noise terms phenomenon for nonhomogeneous problems numerical examples are investigated to illustrate the pertinent features of the proposed algorithm

solutions to the non homogeneous parabolic problems by the extended hadm in this paper we extend the homogeneous adomain decomposition method hadm introduced by jang bongsoo jang exact solutions to one dimensional non homogeneous parabolic problems by the homogeneous decomposition method appl math comput 186 2 2007 969 979 to solve non homogeneous parabolic partial differential equations with variable coefficients the new modified technique provides the less computational work as well as easy calculation in obtaining the accurate approximations compared with the standard adm illustrative examples show the excellent performance for the new modified decomposition method

modified adomian decomposition method for specific second order ordinary differential equations in this paper an efficient modification of adomian decomposition method is introduced for solving second order ordinary differential equations the proposed method can be applied to singular and nonsingular problems the scheme is tested for some examples and the obtained results demonstrate efficiency of the proposed method

a new approach to the nonlinear advection problem an application of the decomposition method abstract in this paper the nonlinear advection problem is solved easily and elegantly by using the adomian s decomposition method the exact solution is quickly obtained by observing the appearance of the self canceling noise terms whose sum vanishes in the limit comparison with other existing techniques shows that the present approach is highly accurate

oscillation criteria for a class of neutral difference equations with continuous variable in this paper we are mainly concerned with oscillatory behaviour of solutions for a class of second order nonlinear neutral difference equations with continuous variable using an integral transformation the riccati transformation and iteration some oscillation criteria are obtained

tensor products of normed and banach quasi algebras abstract quasi algebras form an essential class of partial algebras which are algebras of unbounded operators in this work we aim to construct tensor products of normed respectively banach quasi algebras and study their capacity to preserve some important properties of their tensor factors like for instance semisimplicity and full representability

a modification of adomian s solution for nonlinear oscillatory systems abstract in this paper we consider nonlinear oscillatory equations of the form d 2 u dt 2 2 u ef u du dt whose solutions can be obtained by the decomposition method but these solutions do not exhibit periodicity which is characteristic of oscillatory systems we use an alternative technique by which the solution obtained by the decomposition method is made periodic the method is described and illustrated with examples there is no smallness assumption on the parameter e occuring in the equation

extensions of the phase space linearization psl technique for non linear oscillators abstract this paper is devoted to further exploration of the psl technique developed earlier in its companion paper to start with it is shown that the method is not only applicable for obtaining one periodic orbits but can be made use of for obtaining every conceivable orbit including subharmonic or periodic doubled orbits quasi periodic orbits and even chaotic orbits this is numerically illustrated by constructing various orbits of ueda s duffing holmes and van der pol s oscillators next the separatrix that separates the basins of attraction of the stable limit cycles of duffing holmes oscillator is constructed using the psl procedure a possibility of predicting the chaotic diffusion of trajectories based on a heuristic argument of a near tangency of stable and unstable limit cycles of duffing holmes oscillator is also discussed finally the psl scheme is made use of to compute various characteristic quantities such as fourier spectra liapunov characteristic exponents and probability density functions many new results are presented to establish the versatility of the psl method