violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

a simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint in this paper a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity for various end boundary conditions we transform the governing equation with varying coefficients to linear algebraic equations then a characteristic equation in critical buckling loads will be obtained several examples of estimating buckling loads under typical end supports are discussed by comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams it can be found that our method has fast convergence and the obtained numerical results have high accuracy moreover the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties the effects of gradient parameters on the critical buckling loads are elucidated finally we give an example to illustrate the enhancement of the load carrying capacity of tapered beams for admissible shape profiles with constant volume or weight the proposed method is of benefit to optimum design of beams against buckling in engineering applications

scaling in the aging dynamics of the site diluted ising model we study numerically the phase ordering kinetics of the two dimensional site diluted ising model the data can be interpreted in a framework motivated by renormalization group concepts apart from the usual fixed point of the nondiluted system there exist two disorder fixed points characterized by logarithmic and power law growth of the ordered domains this structure gives rise to a rich scaling behavior with an interesting crossover due to the competition between fixed points and violation of superuniversality

partial regularity of stable solutions to the supercritical equations and its applications abstract we prove the partial regularity of stable solutions of supercritical elliptic equations as an application we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric

stable and finite morse index solutions of toda system abstract we prove a liouville type theorem and a partial regularity result for stable and finite morse index solutions of toda system the main tools are integral estimates a monotonicity formula and blowing down analysis for such systems they may be split into two sub systems during the blowing down process some e regularity theorems are developed to deal with this problem

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

liouville theorems for stable solutions of biharmonic problem we prove some liouville type results for stable solutions to the biharmonic problem delta 2 u u q u 0 in mathbb r n where 1 q infty for example for n ge 5 we show that there are no stable classical solution in mathbb r n when frac n 4 n 4 q le left frac n 8 n right 1

liouville type theorems for stable solutions of the weighted elliptic system abstract we examine the weighted elliptic system u 1 x 2 2 v v 1 x 2 2 u p in r n and prove liouville type theorems for the classical positive and nonnegative stable solutions in higher dimension in particular there are no positive stable solutions for any n 12 5 p 1 and 0 our proof is based on a combination of the bootstrap argument souplet s inequality and intermediate stability criterion which is used to obtain sharp results

multiplicity of solutions for a fourth order equation with power type nonlinearity let b be the unit ball in mathbb r n n 3 and n be the exterior unit normal vector on the boundary we consider radial solutions to delta 2 u lambda 1 rm sign p u p quad rm in b quad u 0 quad frac partial u partial n 0 quad rm on partial b where 0 for positive p we assume 5 n 12 and p frac n 4 n 4 or n 13 and frac n 4 n 4 0 such that if s there exists a radial weakly singular solution for s there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as s

crossover in growth law and violation of superuniversality in the random field ising model we study the nonconserved phase ordering dynamics of the d 2 3 random field ising model quenched to below the critical temperature motivated by the puzzling results of previous work in two and three dimensions reporting a crossover from power law to logarithmic growth together with superuniversal behavior of the correlation function we have undertaken a careful investigation of both the domain growth law and the autocorrelation function our main results are as follows we confirm the crossover to asymptotic logarithmic behavior in the growth law but at variance with previous findings we find the exponent in the preasymptotic power law to be disorder dependent rather than being that of the pure system furthermore we find that the autocorrelation function does not display superuniversal behavior this restores consistency with previous results for the d 1 system and fits nicely into the unifying scaling scheme we have recently proposed in the study of the random bond ising model

buckling and post buckling analysis of extensible beam columns by using the differential quadrature method buckling and post buckling analysis of extensible beam columns is performed numerically in this paper it was experienced earlier that in some cases the numerical integration would not produce the convergent post buckling solutions especially under high loads therefore a new differential quadrature dq based iterative numerical integration method is proposed to solve post buckling differential equations of extensible beam columns six cases including five classical euler buckling cases are analyzed critical loads and convergent post buckling solutions under different applied loads are obtained the results are compared with the existing multiple scales solutions it is found that under high applied loads the small rotation assumption in obtaining multiple scales solutions is no longer valid the proposed iterative dq based numerical integration method can yield reliable and accurate post buckling solutions even at high applied loads for the cases investigated

a monotonicity formula and a liouville type theorem for a fourth order supercritical problem we consider liouville type and partial regularity results for the nonlinear fourth order problem

a regularity theory of biharmonic maps in this article we prove the regularity of weakly biharmonic maps of domains in euclidean four space into spheres as well as the corresponding partial regu larity result of stationary biharmonic maps of higher dimensional domains into spheres c 1999 john wiley sons inc in this article we consider for simplicity the class of bihar monic maps from eu clidean domains to spheres we realize the standard spheres s k as unit vectors in r k 1 and consider maps u s k as vector valued functions that are contained in s k the energy functional for biharmonic maps is then j uj 2 dx a locally de fined biharmonic map is a map that is critical with respect to c ompactly supported variations we note that in the case where the domain has dimension four this en ergy functional is conformally invariant and hence conformal maps of euclidean four space are biharmonic in this sense we remark that this definition of bihar monic map depends on the embedding of the target space in euclidean space we

preface the variational iteration method reliable efficient and promising in the last few decades considerable work has been invested in developing new methods for analytical and numerical solutions of differential and integral equations linear or nonlinear there has been a great deal of research work done to address the issues of nonlinearity and singularity phenomena that arise in many scientific and engineering problems moreover there has been a great need for effective algorithms to avoid the onerous work required by traditional techniques as a result of this need many analytical and numerical tools such as grid points techniques perturbation techniques spline solutions the adomian method and others have been developed however each of these methods suffers from one or more limitations the grid points techniques define the solution at grid points only the spline solution requires restrictions on boundary points the perturbation method suffers from a high computational workload especially when the degree of nonlinearity increases in addition the adomian method suffers from the complication of the algorithms needed to calculate adomian polynomials that are necessary for nonlinear problems this special issue of computers and mathematics with applications conveys a strong reliable efficient and promising development ji huan he developed the variational iteration method vim in 1999 in this method the equations are initially approximated with possible unknowns a correction functional is established by a general lagrange multiplier which can be identified optimally via the variational theory the method gives rapidly convergent successive approximations of the exact solution the vim has no restrictions or unrealistic assumptions such as linearization or small parameters that are used for nonlinear operators the vim handles linear and nonlinear problems in a like manner the convergence concept has been proved a substantial amount of research work has been directed to the study of the variational iteration method by many well known researchers several studies have been conducted to compare the vim method with existing techniques and it was shown by all that this method gives exact solutions faster than other methods the method has been applied in a wide variety of scientific and engineering applications these works confirmed the fact that this methodology is reliable efficient and promising as well as having a wider applicability included herein is a collection of original refereed research papers by well established researchers in the field of applied mathematics we hope that this issue will prove to be a timely and valuable reference for researchers in this area special thanks go to the referees for their valuable work we here thank dr ervin y rodin for providing us with the opportunity to produce this special issue on this exciting method

determination of critical buckling load for elastic columns of constant and variable cross sections using variational iteration method in this paper variational iteration method vim is applied to the problem of determination of critical buckling loads for euler columns with constant and variable cross sections vim is a powerful method for the solution of nonlinear ordinary and partial differential equations and integral equations hence it is a suitable approach for the analysis of engineering problems where an exact solution is difficult to obtain this study presents the application of vim to various buckling cases and results are produced for columns with different support conditions and with different variation of cross sections the results obtained are accurate which show that variational iteration method is a very efficient technique in the analysis of elastic stability problems

variational iteration method for solving integral equations in this paper several integral equations are solved by he s variational iteration method comparison with exact solution shows that the method is very effective and convenient for solving integral equations

a new dq based method and its application to the stability of columns a numerical picard like method which combines successive approximations with differential quadrature is discussed for application to the stability of columns unlike other methods the present method satisfies all the boundary conditions without any additional computation for each iteration and it does not require any subdivision of the problem domain despite of its recursive nature the approximate solution can be expressed by operational matrices and vectors of known quantities such as the solution obtained by advanced polynomial expansions several examples show the satisfactorily behaviour of the proposed method in comparison to the existing methods both in the linear and nonlinear range

stability analysis of bars with varying cross section abstract the exact solutions for stability analysis of bars with varying cross sections subjected to simple or complicated loads including concentrated and variably distributed axial loads are presented in this paper the distribution of flexural stiffness of the bar and that of axial loads acting on the bar are expressed as power functions or exponential functions also the extracted exact solutions are expressed in terms of bessel functions and super geometric series

some nonexistence theorems for semilinear fourth order equations in this paper we analyse the semilinear fourth order problem 2 u g u in exterior domains of n assuming the function g is nondecreasing and continuous in 0 and positive in 0 we show that positive classical supersolutions u of the problem which additionally verify u 0 exist if and only if n 5 and int 0 delta displaystyle g s over s 2 n 2 n 4 rm d s lt infty for some 0 when only radially symmetric solutions are taken into account we also show that the monotonicity of g is not needed in this result finally we consider the same problem posed in n and show that if g is additionally convex and lies above a power greater than one at infinity then all positive supersolutions u automatically verify u 0 in n and they do not exist when the previous condition fails

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

the gel fand problem for the biharmonic operator we study stable and finite morse index solutions of the equation delta 2 u e u if the equation is posed in mathbb r n we classify radial stable solutions we then construct nonradial stable solutions and we prove that unlike the corresponding second order problem no liouville type theorem holds unless additional information is available on the asymptotics of solutions at infinity thanks to this analysis we prove that stable solutions of the equation on a smoothly bounded domain supplemented with navier boundary conditions are smooth if and only if n leqq 12 we find an upper bound for the hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded morse index provided they are controlled in a suitable morrey norm

partial regularity of finite morse index solutions to the lane emden equation we prove regularity and partial regularity results for finite morse index solutions u h1 lp to the lane emden equation u u p 1u in

new applications of variational iteration method abstract the variational iteration method is used for solving three types of nonlinear partial differential equations such as coupled schrodinger kdv generalized kdv and shallow water equations the exact and numerical solutions obtained by variational iteration method are compared with that obtained using adomian decomposition method the comparison shows that the two solutions obtained are excellent agreement the method is made showing that the former is more effective than the latter in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

liouville type results for semi stable solutions of the weighted lane emden system abstract we examine the weighted lane emden system u 1 x 2 2 v p v 1 x 2 2 u q in r n where 1 p q and 0 and the weighted lane emden equation u 1 x 2 2 u p in r n where p 4 3 and 0 we prove that the system equation does not have the classical positive semi stable solution in dimension n 2 4 2 q 1 p q 1 0 n 2 2 2 p 1 p p 2 p in particular there is no positive semi stable solution of the weighted lane emden system if n 10 4 and 2 p q

off equilibrium response function in coarsening systems abstract in the late stage dynamics of systems evolving via domain coarsening the linear response function splits into the sum of a stationary term due to thermal fluctuations within domains and of an aging term due to off equilibrium interface dynamics the latter contribution is analyzed in the pure ising model under variation of space dimensionality and in the random field ising model in one dimension the variety of behaviors observed can be accounted for in terms of the effective response associated to a single interface implications for the connection between static and dynamic properties are discussed

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

monotonicity formulas for extrinsic triharmonic maps and the triharmonic lane emden equation abstract we derive monotonicity formulas and e regularity results for extrinsic triharmonic maps and triharmonic lane emden equations as an application we prove partial regularity results for both types of equations in the supercritical regime

off equilibrium response function in the one dimensional random field ising model a thorough numerical investigation of the slow dynamics in the d 1 random field ising model in the limit of an infinite ferromagnetic coupling is presented in this paper crossovers from the preasymptotic pure regime tothe asymptotic sinai regime are investigated for the average domain size the autocorrelation function and staggered magnetization by switching on an additional small random field at the time t w the linear off equilibrium response function is obtained which displays as well the crossover from the nontrivial behavior of the d 1 pure ising model to the asymptotic behavior where it vanishes identically