the variational iteration method for solving two forms of blasius equation on a half infinite domain the variational iteration method is applied for a reliable treatment of two forms of the third order nonlinear blasius equation which comes from boundary layer equations the study shows that the series solution is obtained without restrictions on the nonlinearity behavior the obtained series solution is combined with the diagonal pade approximants to handle the boundary condition at infinity for only one of these forms

adomian decomposition method for a reliable treatment of the emden fowler equation in this paper we present a reliable algorithm to determine new exact and new approximate solutions of the generalized emden fowler equation the algorithm rests mainly on adomian decomposition method with an alternative framework designed to overcome the difficulty of the singular point at x 0 the proposed framework reveals quite a number of remarkable features of the scheme used here

an averaging theorem for hamiltonian dynamical systems in the thermodynamic limit it is shown how to perform some steps of perturbation theory if one assumes a measure theoretic point of view i e if one renounces to control the evolution of the single trajectories and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase space for a system of coupled rotators estimates uniform in n for finite specific energy can be obtained in quite a direct way this is achieved by making reference not to the sup norm but rather following koopman and von neumann to the much weaker l2 norm

the variational iteration method for a reliable treatment of the linear and the nonlinear goursat problem abstract the linear and the nonlinear goursat problems are investigated the he s variational iteration method vim is used to construct exact solutions to these problems the linear and the nonlinear structures are handled in a like manner without any need to restrictive assumptions the accuracy level of the obtained results reveal the power of this method over existing numerical methods

time scales to equipartition in the fermi pasta ulam problem finite size effects and thermodynamic limit we investigate numerically the common and the pure fpu models as well as some higher order generalizations we consider initial conditions in which only low frequency normal modes are excited and perform a very accurate systematic study of the equilibrium time as a function of the number n of particles the specific energy e and the parameters and while at any fixed n the equilibrium time is found to be a stretched exponential in 1 e in the thermodynamic limit i e for n at fixed e we observe a crossover to a power law concerning the usually disregarded dependence of teq on and we find it is nontrivial and propose and test a general law a central role is played by the comparison of the fpu models with the toda model

the variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics variational iteration method has been used to handle linear and nonlinear differential equations the main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently in this work a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics the fractional derivatives are described in the caputo sense numerical illustrations that include the fractional wave equation fractional burgers equation fractional kdv equation fractional klein gordon equation and fractional boussinesq like equation are investigated to show the pertinent features of the technique comparison of the results obtained by the variational iteration method with those obtained by adomian decomposition method reveals that the first method is very effective and convenient the basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus

the variational iteration method a powerful scheme for handling linear and nonlinear diffusion equations in this work we introduce a framework for obtaining exact solutions to linear and nonlinear diffusion equations exact solutions are developed for some diffusion processes of power law diffusitivies he s variational iteration method vim is used for analytic treatment of these equations the powerful vim method is capable of handling both linear and nonlinear equations in a direct manner

proof of nishida s conjecture on anharmonic lattices we prove nishida s 1971 conjecture stating that almost all low energetic motions of the anharmonic fermi pasta ulam lattice with fixed endpoints are quasi periodic the proof is based on the formal computations of nishida the kam theorem discrete symmetry considerations and an algebraic trick that considerably simplifies earlier results

the variational iteration method a reliable analytic tool for solving linear and nonlinear wave equations in this work he s variational iteration method vim is used for analytic treatment of the linear and the nonlinear wave equations in bounded and unbounded domains wave like equations are also investigated the method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or adomian polynomials the study highlights the power of the vim method

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

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

extensive adiabatic invariants for nonlinear chains we look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit considering the quadratic part of the klein gordon hamiltonian by a linear change of variables we transform it into a sum of two parts in involution at variance with the usual method of introducing normal modes our constructive procedure allows us to exploit the complete resonance while keeping the extensive nature of the system next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear schrodinger model the fluctuations of this quantity are controlled via gibbs measure estimates independent of the system size for a large set of initial data at low specific energy finally by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory with fluctuation much smaller than expected according to standard statistical estimates

low dimensional q tori in fpu lattices dynamics and localization properties abstract recent studies on the fermi pasta ulam fpu paradox like the theory of q breathers and the metastability scenario dealing mostly with the energy localization properties in the fpu space of normal modes q space motivated our first work on q tori in the fpu problem christodoulidi et al 2010 19 the q tori are low dimensional invariant tori hosting trajectories that present features relevant to the interpretation of fpu recurrences as well as the energy localization in q space the present paper is a continuation of our work in christodoulidi et al 2010 19 our new results are we extend a method of analytical computation of q tori using poincare lindstedt series from the to the fpu and we reach significantly higher expansion orders using an improved computer algebraic program we probe numerically the convergence properties as well as the level of precision of our computed series we develop an additional algorithm in order to systematically locate values of the incommensurable frequencies used as an input in the pl series construction of q tori corresponding to progressively higher values of the energy we generalize a proposition proved in christodoulidi et al 2010 19 regarding the so called sequence of propagation of an initial excitation in the pl series we show by concrete examples how the latter interprets the localization patterns found in numerical simulations we focus in particular on various types of extensive initial excitations that lead to q tori solutions with exponentially localized profiles finally we discuss the relation between q tori q breathers viewed as one dimensional q tori and the so called fpu trajectories invoked in the original study of the fpu problem

the fermi pasta ulam problem and its underlying integrable dynamics an approach through lyapunov exponents fpu models in dimension one are perturbations either of the linear model or of the toda model perturbations of the linear model include the usual beta model perturbations of toda include the usual alpha beta model in this paper we explore and compare two families or hierarchies of fpu models closer and closer to either the linear or the toda model by computing numerically for each model the maximal lyapunov exponent chi more precisely we consider statistically typical trajectories and study the asymptotics of chi for large n the number of particles and small varepsilon the specific energy e n and find for all models asymptotic power laws chi simeq c varepsilon a c and a depending on the model the asymptotics turns out to be in general rather slow and producing accurate results requires a great computational effort we also revisit and extend the analytic computation of chi introduced by casetti livi and pettini originally formulated for the beta model with great evidence the theory extends successfully to all models of the linear hierarchy but not to models close to toda

a new algorithm for solving differential equations of lane emden type in this paper a reliable algorithm is employed to investigate the differential equations of lane emden type the algorithm rests mainly on the adomian decomposition method with an alternate framework designed to overcome the difficulty of the singular point the proposed framework is applied to a generalization of lane emden equations so that it can be used in differential equations of the same type

birkhoff coordinates for the toda lattice in the limit of infinitely many particles with an application to fpu abstract in this paper we study the birkhoff coordinates cartesian action angle coordinates of the toda lattice with periodic boundary condition in the limit where the number n of the particles tends to infinity we prove that the transformation introducing such coordinates maps analytically a complex ball of radius r n in discrete sobolev analytic norms into a ball of radius r n with r r 0 independent of n if and only if 2 then we consider the problem of equipartition of energy in the spirit of fermi pasta ulam we deduce that corresponding to initial data of size r n 2 0 r 1 and with only the first fourier mode excited the energy remains forever in a packet of fourier modes exponentially decreasing with the wave number finally we consider the original fpu model and prove that energy remains localized in a similar packet of fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet the proof of the theorem on birkhoff coordinates is based on a new quantitative version of a vey type theorem by kuksin and perelman which could be interesting in itself

chaos and the approach to equilibrium in a discrete sine gordon equation abstract in this paper we study the long time dynamics of a hamiltonian system with many degrees of freedom we numerically investigate the system s approach to equipartation of energy when the intial energy is confined to one or a small set of fourier modes we find that there is a transition from a low energy regime in which energy does not spread appreciably among the modes to a high energy regime in which the system rapidly approaches equipartition and that this transition coincides with the onset of chaos in the system as evidenced by a sharp rise in the largest lyapunov exponent for low frequency initial conditions the critical parameter for this transition is the scaled energy e1 l n 2e using a generalization of the traditional chirikov resonance overlap calculation on a three mode subset of the full system we predict the onset of widespread chaos and the transition to equipartition on relatively short timescales

on the numerical integration of fpu like systems abstract this paper concerns the numerical integration of systems of harmonic oscillators coupled by nonlinear terms like the common fpu models we show that the most used integration algorithm namely leap frog behaves very gently with such models preserving in a beautiful way some peculiar features which are known to be very important in the dynamics in particular the selection rules which regulate the interaction among normal modes this explains why leap frog in spite of being a low order algorithm behaves so well as numerical experimentalists always observed at the same time we show how the algorithm can be improved by introducing at a low cost a counterterm which eliminates the dominant numerical error

multi threshold dimension vector for texture analysis and its application to liver tissue classification abstract a new feature set derived from the fractal geometry called the multi threshold dimension vector mtdv is proposed for texture analysis the major properties of the approach are i it is easy to compute ii it is geometrically invariant i e it is invariant under the transformations of translation rotation reflection and scaling and iii it is invariant under the linear gray level transformation these properties have enhanced the practical applications of the features the mtdv is applied to the classification of ultrasonic liver images and produces about 88 of the correct classification rate this suggests that the mtdv is an excellent tool for medical image processing

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

the variational iteration method for analytic treatment for linear and nonlinear odes in this work the variational iteration method vim is used for analytic treatment of the linear and nonlinear ordinary differential equations homogeneous or inhomogeneous the method is capable of reducing the size of calculations and handles both linear and nonlinear equations homogeneous or inhomogeneous in a direct manner however for concrete problems a huge number of iterations are needed for a reasonable level of accuracy

the fermi pasta ulam problem and its underlying integrable dynamics this paper is devoted to a numerical study of the familiar fpu model precisely we here discuss revisit and combine together two main ideas on the subject i in the system at small specific energy e e n two well separated time scales are present in the former one a kind of metastable state is produced while in the second much larger one such an intermediate state evolves and reaches statistical equilibrium ii fpu should be interpreted as a perturbed toda model rather than as is typical as a linear model perturbed by nonlinear terms in the view we here present and support the former time scale is the one in which fpu is essentially integrable its dynamics being almost indistinguishable from the toda dynamics the toda actions stay constant for fpu too while the usual linear normal modes do not the angles fill their almost invariant torus and nothing else happens the second time scale is instead the one in which the toda actions significantly evolve and statistical equilibrium is possible we study both fpu like initial states in which only a few degrees of freedom are excited and generic initial states extracted randomly from an approximated microcanonical distribution the study is based on a close comparison between the behavior of fpu and toda in various situations the main technical novelty is the study of the correlation functions of the toda constants of motion in the fpu dynamics such a study allows us to provide a good definition of the equilibrium time i e of the second time scale for generic initial data our investigation shows that is stable in the thermodynamic limit i e the limit of large n at fixed e and that by reducing e ideally the temperature approximately grows following a power law e a with a 5 2