oscillation theorems for differential equations with a nonlinear damping term in this paper we are concerned with a class of nonlinear second order differential equations with a nonlinear damping term passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity and as a consequence our results apply to wider classes of nonlinear differential equations two illustrative examples are considered

on oscillation of nonlinear second order differential equation with damping term we present new oscillation criteria for the second order nonlinear differential equation with damping our theorems are stated in general form they complement and extend related results known in the literature the relevance of our results is illustrated with a number of examples to facilitate computations in these examples we use mathematica r a symbolic computer language

new kamenev type oscillation criteria for second order neutral nonlinear differential equations abstract in this paper some new oscillation criteria for second order neutral nonlinear differential equation r t y t p t y t i 1 n q i t f i y i t 0 t t 0 are established new oscillation criteria are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

oscillation criteria of yan type for linear hamiltonian systems abstract new oscillation criteria of yan type are established for the linear hamiltonian system x x a t x b t y t t0 y c t x a t y where a t b t and c t are continuous n n matrix valued functions with b t b t 0 c t c t the criteria obtained generalize and improve some previous results for the second order linear matrix differential systems in literature

thermostochastics heat conduction and temperature fluctuations abstract a stochastic approach to the energy balance equation of nonequilibrium thermodynamics is suggested the stochastic formulation interprets the temperature field as a multivariate stochastic process which is governed by a master equation by means of a systematic expansion in powers of the inverse number of degrees of freedom per volume element it is shown that the expectation value of the stochastic process obeys the macroscopic fourier equation the expansion reveals that in the linear noise approximation the fluctuations superimposed on the macroscopic dynamics are governed by the equations of fluctuating hydrodynamics the master equation formulation gives rise to a new stochastic simulation method which is illustrated by applying it to heat conduction and temperature fluctuations in a fluid between two infinite parallel planes which are kept at different temperatures

oscillation of second order functional differential equations with damping abstract using a class of new test functions t s r defined in our recent work y g sun new kamenev type oscillation criteria for second order nonlinear differential equations with damping j math anal appl 291 2004 341 351 we establish some new oscillation criteria for the second order functional differential equation with damping x t p t x t a b q t f x g 1 t x g m t d 0 our results are different from most known ones and can be applied to many cases which are not covered by existing results

fluctuations in the lattice gas for burgers equation the langevin equation for the biased random hopping of particles onto unoccupied sites of a lattice is obtained from the master equation for finite systems by invoking a limit theorem due to kurtz for a weakly asymmetric process with diffusive scaling the continuum limit of this equation yields a deterministic equation that is equivalent to burgers equation the stochastic equation for the leading lattice corrections is derived and the approach to the continuum limit is illustrated with simulations

oscillation results for a second order damped differential equation with nonmonotonous nonlinearity abstract we present new oscillation criteria for a nonlinear second order differential equation with a damping term an essential feature of our results is that we do not require the nonlinearity to be nondecreasing furthermore as opposed to the recent results by s r grace 1989 math nachr 141 117 127 1992 j math anal appl 171 220 241 and s r grace and b s lalli 1990 j math anal appl 149 277 311 we do not impose any additional condition on the damping term p t consequently our theorems apply to new classes of equations which are not covered by the known criteria the relevance of the results obtained is illustrated with a number of carefully selected examples

stationary turbulent closure via the hopf functional equation exact closed form solutions are exhibited for the hopf equation for stationary incompressible 3d navier stokes flow for the cases of homogeneous forced flow including a solution with depleted nonlinearity and inhomogeneous flow with arbitrary boundary conditions this provides an exact method for computing two and higher point moments given the mean flow

oscillation criteria for second order nonlinear damped differential equations abstract this paper discusses a class of second order nonlinear damped differential equations by using the generalized riccati technique and the integral averaging technique new oscillation criteria are obtained

oscillation criteria for even order half linear distributed delay differential equations with damping by using the generalized riccati technique and an inequality due to hardy littlewood and polya several new oscillation criteria are established for a class of even order half linear distributed for a differential equation with damping the results obtained extend and improve earlier results in existing literature

interval criteria for oscillation of second order half linear differential equations by employing a generalized riccati technique and an integral averaging method interval oscillation criteria are established for the second order half linear differential equation r t x t 1x t q t x t 1x t 0 these criteria are different from most known ones in the sense that they are based on information only on a sequence of subintervals of t0 rather than on the whole half line they also extend improve and complement a number of existing results and can be applied to extreme cases such as t0 q s ds in particular several interesting examples that illustrate the importance of our results are included

oscillation criteria for certain nonlinear differential equations we present new oscillation criteria for certain classes of second order nonlinear differential equations and delay differential equations obtained by using an integral averaging technique our theorems complement a number of existing results and handle the cases which are not covered by known criteria

oscillation criteria for second order nonlinear delay dynamic equations on time scales by employing the generalized riccati transformation we establish some new oscillation criteria for second order delay dynamic equations on time scales the obtained results essentially improve the well know oscillation results for half linear dynamic equations such as kamenev type and philos type oscillation criteria some interesting examples are given to illustrate the versatility of our results

asymptotic behavior of solutions of second order nonlinear delay differential equations with impulses this paper is concerned with second order nonlinear delay differential equations with impulses of the form r t x t p t x t f t x email protected 0 t t k x t k i k x t k x t k j k x t k t t k k 1 2 t t 0 by impulsive differential inequality and riccati transformation sufficient conditions of asymptotic behavior of all solutions of second order nonlinear delay differential equations with impulses are obtained some examples are also inserted to illustrate the impulsive effect

strong property fluctuation theory for homogenization of bianisotropic composites formulation the strong property fluctuation theory is developed for the homogenization of the linear dielectric magnetic and magnetoelectric properties of a two constituent bianisotropic composite the notion of a bianisotropic comparison medium bcm is introduced to serve as a springboard for the dyson equation satisfied by the ensemble averaged electromagnetic field with the constitutive properties of the bcm serving as the zeroth order solution of the dyson equation the first order correction known as the bilocal approximation is obtained wave propagation in the composite can be described in this manner by a nonlocal effective medium containing information about the spatial correlations of the constitutive properties for scales larger than the correlation length the nonlocality vanishes and a local effective medium emerges analytical results for the local effective constitutive properties are presented after assuming a spherical particulate topology for the constituent mediums illustrative numerical results are provided

numerical methods for high dimensional probability density function equations in this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables these types of equations arise naturally in many different areas of mathematical physics e g in particle systems liouville and boltzmann equations stochastic dynamical systems fokker planck and dostupov pugachev equations random wave theory malakhov saichev equations and coarse grained stochastic systems mori zwanzig equations we propose three different classes of new algorithms addressing high dimensionality the first one is based on separated series expansions resulting in a sequence of low dimensional problems that can be solved recursively and in parallel by using alternating direction methods the second class of algorithms relies on truncation of interaction in low orders that resembles the bogoliubov born green kirkwood yvon bbgky framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations the third class of algorithms is based on high dimensional model representations e g the anova method and probabilistic collocation methods a common feature of all these approaches is that they are reducible to the problem of computing the solution to high dimensional equations via a sequence of low dimensional problems the effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations with up to 120 variables

the distance between zeros of an oscillatory solution to a half linear differential equation consider the oscillatory equation u t 1u t q t u t 1u t 0 where q t a r is locally integrable for some a 0 we prove some results on the distance between consecutive zeros of a solution of we apply also the results to the following equations r t u t 1u t q t u t 1u t 0 and i 1ndi du n 2diu c x u n 2u 0 a where i r c 0 0 and ar t 1 ii di xi d d1 dn a x rn x a is an exterior domain and c c a 0 iii 0 n 1 and n 2

oscillation criteria for second order quasi linear neutral delay differential equations abstract in this paper some new oscillation criteria for second order neutral nonlinear differential equation r t x t p t x t m 1 x t p t x t q t f x t 0 are established new oscillation criteria are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel

properties of the dyadic green s function for an unbounded anisotropic medium radiation in an unbounded anisotropic medium is treated analytically by studying the dyadic green s function of the problem initially expressed as a triple fourier integral which is next reduced to a double one under certain conditions the existence of incoming waves is verified it is also found that exponentially decaying waves are possible in such media finally the existence of branch points in the remaining integrand function is investigated and appropriate branch cuts are proposed

oscillation of second order nonlinear ode with damping we are concerned with a class of general type second order differential equations with a nonlinear damping term new oscillation criteria are established which are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel

sturmian comparison theorem for half linear second order differential equations let be defined by s s p 2 s with p 1 a fixed number we extend sturm comparison theorem of the linear differential equation to the nonlinear differential equation by using the wirtinger inequality a lyapunov inequality and some oscillation criteria of e are also given

oscillation of certain second order nonlinear differential equations abstract in this paper we consider the second order nonlinear differential equation formula where 0 is any quotient of odd integers a c 1 r 0 q c r r f c r r xf x 0 f x 0 for x 0 some new sufficient conditions for the oscillation of all solutions of are obtained several examples that dwell upon the importance of our results are also included

oscillation theorems for certain nonlinear differential equations of second order abstract we present new oscillation criteria for the second order nonlinear differential equations these criteria involve the use of averaging functions our theorems are stated in general form they complement and extend related results known in the literature also the relevance of our results is to show that some of manojlovic s results contain the superfluous condition

asymptotic behavior of solutions of second order nonlinear impulsive differential equations in this work the asymptotic behavior of all solutions of second order nonlinear ordinary differential equations with impulses is investigated by impulsive differential inequality and riccati transformation sufficient conditions of asymptotic behavior of all solutions of second order nonlinear ordinary differential equations with impulses are obtained an example is also inserted to illustrate the impulsive effect

interval criteria for oscillation of second order functional differential equations by using averaging functions new interval oscillation criteria are established for the second order functional differential equation r t x t a 1x t f t x t x t t x t x t t 0 t t 0that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of t 0 rather than on the whole half line our results can be applied to three cases ordinary delay and advance differential equations in the case of half linear functional differential equations our criteria implies that the t t t delay and gt t t advance cases do not affect the oscillation in particular several examples are given to illustrate the importance of our results

oscillation criteria for second order nonlinear neutral variable delay dynamic equations abstract this paper is concerned with oscillation of second order nonlinear neutral dynamic equations on time scales with a variable delay by using the generalized riccati technique and integral averaging technique new oscillation criteria are obtained for all solutions of the equation some results extend known results for difference equations when the time scale is the set z of positive integers and for differential equations when the time scale is r several examples are given to illustrate the results of the paper

on oscillation of nonlinear second order differential equations with damping term the paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term first it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question and vice versa advantages and drawbacks of such translation are carefully examined then an oscillation criterion for the new class of equations is established the principal result of the paper is compared to those reported in the literature and an illustrative example to which known oscillation criteria fail to apply is provided

new kamenev type oscillation criteria for second order nonlinear differential equations with damping abstract some new oscillation criteria are established for the nonlinear damped differential equation r t y p t y q t f y 0 that are different from most known ones in the sense that they are based on a class of new functions t s r defined in the sequel our results are sharper than some previous results which can be seen by the examples at the end of this paper

interval oscillation criteria for second order neutral nonlinear differential equations some oscillation criteria for the second order neutral nonlinear differential equation y t p t y s t email protected i 1 nq i t f i y t i t 0 t t 0are established new oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem

oscillation criteria for second order neutral equations with distributed deviating argument abstract we present new oscillation criteria for the second order neutral delay differential equations with distributed deviating argument r t x t z t a b p t f x g t d 0 t t 0 where z t x t q t x t new oscillation criteria are established which are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

some oscillation criteria for a class of second order nonlinear damped differential equations a class of second order nonlinear damped differential equations is studied and several new oscillation conditions are obtained our results extend improve and unify a number of existing results and handle the cases which are not covered by known criteria

new oscillation criteria for second order nonlinear neutral delay differential equations in this paper several new oscillation criteria for the second order nonlinear neutral delay differential equation r t x t p t x s t m 1 x t p t x s t q t f x t t 0 t t 0 are established these oscillation criteria extend and improve some known results an interesting example illustrating the importance of our results is also provided

response to comment on application of the extended lie group analysis to the hopf functional formulation of the burgers equation j math phys 57 034102 2016 we address the criticism of frewer et al concerning the paper application of the extended lie group analysis to the hopf functional formulation of the burgers equation j math phys 54 072901 2013 most importantly we stress that we never claimed that any new statistical symmetries were found in this paper the aim of this paper was to apply the lie group analysis to an equation with functional derivatives and derive invariant solutions for this equation these results still stand as they are most important mathematically correct we address also other critical statements of frewer et al and show that there is a connection between the translational invariance of statistics and transformations of the functional to sum up key ideas and fundamental result in the work of waclawczyk and oberlack are still unaffected

oscillation theorems for nonlinear differential equations of second order abstract in this paper we present some new criteria for the oscillation of the differential equation a t x t x t p t x t q t f x t 0 the obtained results extend improve and correlate a number of existing criteria

oscillation of even order half linear functional differential equations with damping in this paper a class of even order half linear functional differential equations with damping are studied by using the generalized riccati transformation and the integral averaging technique the new oscillation criteria which generalize and improve some known results are obtained for all solutions of the equations

integral averages and oscillation criteria for half linear partial differential equation the technique of weighted integral averages known in the oscillation theory of ordinary differential equations is extended to the half linear partial differential equation e div email protected email protected p email protected u c x u p 2u 0 p 1 this technique is used to obtain new oscillation criteria for e on the unbounded domains

oscillation criteria for second order sublinear differential equation abstract the basic purpose of this paper is to present a new oscillation criteria for second order sublinear differential equation a t x t x t q t t 0 t t 0 0 where a a c 1 t 0 is positive function q c t 0 has no restriction on its sign c 1 r are such that x 0 x x 0 for x 0

interval oscillation criteria for second order nonlinear differential equations with damping new oscillation criteria are established for the nonlinear damped differential equation ry py qf y 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t0 rather than on the whole half line our results are sharper than some previous results and extend and improve the recent interval criteria of kong and huang and can be applied to extreme cases such as t0 q s ds in particular several examples that dwell upon the sharp conditions of our results are also included

oscillation criteria for second order nonlinear perturbed differential equations abstract we present new oscillation criteria for the second order nonlinear perturbed differential equations these criteria are of a high degree of generality and they extend and unify a number of existing results

oscillation criteria for half linear dynamic equations on time scales this paper is concerned with oscillation of the second order half linear delay dynamic equation r t x p t x t 0 on a time scale t where 1 is the quotient of odd positive integers p t and t t are positive rd continuous functions on t r t is positive and delta differentiable t t and limt t we establish some new sufficient conditions which ensure that every solution oscillates or converges to zero our results in the special cases when t r and t n involve and improve some oscillation results for second order differential and difference equations and when t hn t q n 0 and t n 2 our oscillation results are essentially new some examples illustrating the importance of our results are also included

oscillation and asymptotics for second order half linear differential equations this paper discusses a class of second order half linear differential equations by using the generalized riccati technique and the averaging technique new oscillation criteria are obtained for all solutions of the equations to be oscillatory asymptotic behavior for forced equations is also discussed

interval oscillation criteria for second order nonlinear perturbed differential equations abstract we present new interval oscillation criteria for certain classes of second order nonlinear perturbed differential equations which are based on the information only on a sequence of subintervals of t o rather than on the whole half line our results are sharper than those known in the literature and deal with the cases which are not covered by the known criteria in particular our results can be applied to extreme cases such as q s ds finally two examples which dwell upon the sharpness of the conditions of our results are also illustrated

kamenev type oscillation criteria for second order nonlinear dynamic equations on time scales the purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation r t x t f t x g t 0 on an arbitrary time scale t where is a quotient of odd positive integers and r is a positive rd continuous function on t the function g t t satisfies g t t and limt g t and f c t r r we establish some new sufficient conditions such that the above equation is oscillatory by using generalized riccati transformation our results in the special cases when t r and t n involve and improve some oscillation results for second order differential and difference equations and when t hn t qn0 and t n2 our oscillation results are essentially new some examples illustrating the importance of our results are included

oscillation criteria for second order nonlinear differential equations with damping abstract for a second order nonlinear differential equation with a damping term we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations and both sets of oscillation criteria established in the paper are of independent interest several examples are provided to illustrate the relevance of the new theorems

ordinary differential equations in the oscillation theory of partial half linear differential equation abstract in the paper we study the damped half linear partial differential equation div a x u p 2 u b x u p 2 u c x u p 2 u 0 using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations using careful radialization we improve several known oscillation criteria

oscillation and asymptotics for nonlinear second order differential equations this paper discusses a class of second order nonlinear differential equations by using the generalized riccati technique and the averaging technique new oscillation criteria are obtained for all solutions of the equation to be oscillatory asymptotic behavior for forced equations is also discussed

oscillation criteria for second order nonlinear perturbed differential equations abstract some criteria are obtained which guarantee that all solutions of the second order nonlinear perturbed differential equations r t y t p t y t q t y t h t y t y t are oscillatory

integral averages and oscillation criteria of second order nonlinear differential equations the purpose of this paper is to present some new sufficient criteria for the oscillation of all solutions of the second order nonlinear differential equation a t j x x q t f x 0 t t 0 0 where a c 1 t 0 is a positive function qc t 0 has no restriction on its sign j c r c 1 r are such that 0 and j 0 email protected 0 for 0 these criteria are obtained by using a general class of the parameter functions in the averaging techniques and represent extension as well as improvement of known oscillation criteria of several authors for the generalized emden fowler equation

third order implementation and convergence of the strong property fluctuation theory in electromagnetic homogenization the strong property fluctuation theory spft has been widely applied under the second order approximation also known as the bilocal approximation to estimate the constitutive properties of effectively homogeneous composite mediums a third order mass operator approximation is developed here the convergence of the long wavelength bilocally approximated spft is demonstrated for isotropic chiral composite mediums as well as for chiroferrite composite mediums

lie symmetry analysis of the lundgren monin novikov equations for multi point probability density functions of turbulent flow the problem of turbulence statistics described by the lundgren monin novikov lmn hierarchy of integro differential equations is studied in terms of its group properties for this we perform a lie group analysis of a truncated lmn chain which presents the first two equations in an infinite set of integro differential equations for the multi point probability density functions pdf s of velocity a complete set of point transformations is derived for the one point pdf s and the independent variables sample space of velocity space and time for this purpose we use a direct method based on the canonical lie backlund operator due to the one way coupling of correlation equations the present results are complete in the sense that no additional symmetries exist for the first leading equation even if the full infinite hierarchy is considered

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

a master equation representation of two dimensional turbulence a stochastic representation of the turbulent dynamics of the two dimensional navier stokes equation is developed this stochastic formulation interprets the fluctuating velocity field as a discrete multivariate stochastic process which is governed by a master equation by deriving the hopf functional equation of statistical fluid mechanics it is shown that this approach yields a complete description of the stochastic properties of the turbulent velocity field on the basis of the multivariate master equation a stochastic simulation method for the two dimensional navier stokes equation is constructed this method is applied to the stochastic simulation of high reynolds number turbulent flows

interval oscillation of a general class of second order nonlinear differential equations with nonlinear damping abstract the paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping terms several new interval oscillation criteria are established for such a class of differential equations under quite general assumptions examples are also given to illustrate the results in particular it is shown that under some very mild conditions on k 1 k 2 and f the equation k 1 x x cos tk 2 x x x sin tf x 0 is oscillatory which extends the result obtained by kwong and wong j math anal appl 288 2003 15

statistical symmetries of the lundgren monin novikov hierarchy it was shown by oberlack and rosteck discr cont dyn sys s 3 451 2010 that the infinite set of multipoint correlation mpc equations of turbulence admits a considerable extended set of lie point symmetries compared to the galilean group which is implied by the original set of equations of fluid mechanics specifically a new scaling group and an infinite set of translational groups of all multipoint correlation tensors have been discovered these new statistical groups have important consequences for our understanding of turbulent scaling laws as they are essential ingredients of e g the logarithmic law of the wall and other scaling laws which in turn are exact solutions of the mpc equations in this paper we first show that the infinite set of translational groups of all multipoint correlation tensors corresponds to an infinite dimensional set of translations under which the lundgren monin novikov lmn hierarchy of equations for the probability density functions pdf are left invariant second we derive a symmetry for the lmn hierarchy which is analogous to the scaling group of the mpc equations most importantly we show that this symmetry is a measure of the intermittency of the velocity signal and the transformed functions represent pdfs of an intermittent i e turbulent or nonturbulent flow interesting enough the positivity of the pdf puts a constraint on the group parameters of both shape and intermittency symmetry leading to two conclusions first the latter symmetries may no longer be lie group as under certain conditions group properties are violated but still they are symmetries of the lmn equations second as the latter two symmetries in its mpc versions are ingredients of many scaling laws such as the log law the above constraints implicitly put weak conditions on the scaling parameter such as von karman constant ensuremath kappa as they are functions of the group parameters finally let us note that these kind of statistical symmetries are of much more general type i e not limited to mpc or pdf equations emerging from navier stokes but instead they are admitted by other nonlinear partial differential equations like for example the burgers equation when in conservative form and if the nonlinearity is quadratic

integral averaging techniques for the oscillation of second order nonlinear differential equations abstract in this paper we enlist some known and establish some new oscillation criteria for the second order nonlinear differential equations of the form if a t x t t su p t t q t f x t 0 where the coefficients p t and q t are not assumed to be nonnegative for all large values of t these criteria are obtained by using integral averaging techniques and can be applied to some special cases where other classical oscillation results are not applicable a systematic study is attempted to extend improve and correlate a number of existing results

application of the extended lie group analysis to the hopf functional formulation of the burgers equation a study concerning the lie group analysis of the functional differential equations has been performed this is a continuation of the previous common work of oberlack and waclawczyk on the extension of lie group analysis to functional differential equations arch mech 58 597 2006 where lie group theory has been extended to functional differential equations and a special solution of the functional formulation of the burgers equation was derived based on the calculated set of infinitesimals here we derive an infinite set of symmetry transformations of this equation and find new and more general invariant solutions with this we get a step closer to solutions of functional differential equations which e g give a complete statistical description of turbulence in case of the famous hopf novikov stokes functional differential equations

an exact representation of the space time characteristic functional of turbulent navier stokes flows with prescribed random initial states and driving forces by adopting a formal operator viewpoint the space time characteristic functional associated with navier stokes turbulence is expressed in terms of a linear operator acting on the space of functionals obtained by a simple similarity transformation of the local translation operator generated by the nonlinear terms in the navier stokes equation this operator is unitary with respect to the formal scalar product of functionals the equivalence of this operator representation to the functional integral representation of rosen is shown and for gaussian initial velocity and external force fields some consequences of this representation are presented

integral averages and oscillation of second order nonlinear differential equations new oscillation criteria are given for the second order sublinear differential equation left a left t right psi left x left t right right x left t right right q left t right f left x left t right right 0 t geqslant t 0 0 where a c 1 t 0 is a nonnegative function f c with x 0 xf x x 0 for x 0 f have continuous derivative on 0 with f x x03c8 x 0 for x 0 and q c t 0 has no restriction on its sign this oscillation criteria involve integral averages of the coefficients q and a and extend known oscillation criteria for the equation x t q t x t 0

oscillatory behavior of solutions of certain second order nonlinear differential equations abstract in this paper some simple inequalities are used to offer sufficient conditions for the oscillation of all solutions of the differential equation formula where 0 p q with p q odd integers or p even and q odd integers several examples which dwell upon the importance of our results are also included

averaging technique and nonlinear second order elliptic oscillation abstract by using the refinement of the standard integral averaging technique we obtain new oscillation criteria for certain nonlinear second order damped elliptic differential equations we do not impose any additional conditions on the damping terms except for the continuity in particular when n 1 our results extend those of rogovchenko and tuncay yu v rogovchenko f tuncay oscillation criteria for second order nonlinear differential equations with damping nonlinear analysis 69 2008 208 221 for second order nonlinear differential equations with damping to damped elliptic differential equations when n 2 our results improve the oscillation criteria of xu jia ma z xu b jia d ma oscillation theorems for elliptic equations with damping appl math comput 156 2004 93 106 for second order elliptic equations with damping four examples are provided to illustrate the relevance of the main results

interval criteria for oscillation of certain matrix differential systems by employing a generalized riccati technique and an averaging technique interval oscillation criteria are established for the second order matrix differential system of the form p t y q t y 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t0 rather than on the whole half line our results are presented in the form of a high degree of generality handle the cases which are not covered by known criteria and can be applied to extreme cases such as limt 1 t0tq s ds in particular several interesting examples that dwell upon the importance of our results are included

enhanced group velocity in composite media of particles with non spherical shape or shape distribution effective medium approximation is derived to estimate the effective refractive index ne and the group velocity vg in two phase composites in which one of the components is non spherical and even distributed in shape enhanced group velocity may be achieved through the suitable adjustment of the particles shape and shape distribution the range of the volume fractions in which the group velocity is enhanced will expand once we take into account the non spherical shape or the shape distribution therefore the non spherical shape and shape distribution play crucial roles in determining the group velocity in two phase composites

burger s model of turbulence as a stochastic process a recently proposed mesoscopic description of fluid dynamics leads to a new approach to turbulence in contrast to the classical statistical theory of turbulence the new approach introduces a probabilistic time evolution of the random velocity governed by a master equation the mesoscopic approach is explained by means of the 1 1 dimensional burger s model of turbulence by a continuous time stochastic simulation realizations of turbulent velocity fields are generated correlation functions and energy spectra are evaluated from appropriate ensemble averages