the 2 1 dimensional sine gordon equation integrability and localized solutions in this paper the 2 1 dimensional sine gordon equation 2dsg introduced by konopelchenko and rogers is investigated and is shown to satisfy the painleve property a variable coefficient hirota bilinear form is constructed by judiciously using the painleve analysis with a non conventional choice of the vacuum solutions first the line kinks are constructed then exact localized coherent structures in the 2dsgi equation are generated by the collision of two non parallel ghost solitons which drive the two non trivial boundaries present in the system also the reason for the difficulty in identifying localized solutions in the 2dsgii equation is indicated we also highlight the significance of the asymptotic values of the boundaries of the system

a norm relaxed method of feasible directions for finely discretized problems from semi infinite programming abstract in this paper a class of finely discretized semi infinite programming sip problems is discussed combining the idea of the norm relaxed method of feasible directions mfd and the technique of updating discretization index set we present a new algorithm for solving the discretized semi infinite dsi problems from sip at each iteration the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem i e a quadratic program and some appropriate constraints are chosen to reduce the computational cost a high order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints under weak conditions such as mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses weak global convergence moreover the superlinear convergence is obtained under linearly independent constraint qualification licq and other assumptions in the end some elementary numerical experiments are reported

variational bounds on energy dissipation in incompressible flows ii channel flow a variational principle for lower bounds on the time averaged mass flux for newtonian fluids driven by a pressure gradient in a channel is derived from the incompressible navier stokes equations when supplied with appropriate test background flow fields the variational formulation produces explicit estimates for the friction coefficient these rigorous bounds are compared with the predictions of conventional turbulence theory

bounds on dissipation for navier strokes flow with kologorov forcing abstract in this paper bounds on average viscous dissipation are derived for kolmogorov flow in a spatially periodic domain with steady and unsteady forcing at arbitrarily large grashof number g for a force of the form f 0 sin mz i or f 0 sin mz cos t i we derive various bounds on the total dissipation in the flow d u as well as on the dissipation d m obtained from the x velocity averaged over the x y plane the mean velocity of the flow we derive upper bounds on d u and d v d u d m as well as lower bounds on d m and d m d u adopting constraints of the kind introduced by howard and busse and assuming a steady force the background flow method introduced by doering and constantin is used to obtain an improved lower bound on d m d u of o g 1 and a lower bound on d u of o g 1 2 where g f0l3 2 is the grashof number some of these results are then generalized to time periodic forcing direct numerical simulation of the flow indicates that these bounds leave substantial gaps at large grashof number g the calculated d m g and d u g being o g 1 2 and o 1 respectively as g our theoretical bounds on d m d u are shown to be attained by steady laminar type flows for neighboring forcing functions which seems to indicate that these bounds cannot be improved by adding further dynamical constraints however our elementary upper bound on d v can probably be improved by placing more constraints on the flows these results serve to emphasize the difference between boundary driven turbulence and body force driven turbulence where the appropriate dissipation bound is believed saturated at least up to logarithms

verhulst lotka volterra vlv model of ideological struggle abstract a model for ideological struggles is formulated the underlying set is a closed one like a country but in which the population size is variable in time the dynamics of the struggle is described by model equations of verhulst lotka volterra kind several ideologies compete to increase their number of adepts such followers can be either converted from one ideology to another or become followers of an ideology though being previously ideologically free a reverse process is also allowed two kinds of conversions are considered unitary conversion e g by means of mass communication tools or binary conversion e g by means of interactions between people it is found that the steady state when it exists depends on the number of ideologies moreover when the number of ideologies increases some tension arises between them this tension can change in the course of time we propose to measure the ideology tensions through an appropriately defined scale index finally it is shown that a slight change in the conditions of the environment can prevent the extinction of some ideology after almost collapsing the ideology can spread again and can affect a significant part of the country s population two kinds of such resurrection effects are described as phoenix effects

on solitary wave solutions of a class of nonlinear partial differential equations based on the function 1 coshn αx βt abstract the method of simplest equation is applied for obtaining exact solitary traveling wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives the used simplest equation is f 2 n 2 f 2 f 2 n 2 n the developed methodology is illustrated on examples of classes of nonlinear partial differential equations that contain i only monomials of odd grade with respect to participating derivatives ii only monomials of even grade with respect to participating derivatives iii monomials of odd and monomials of even grades with respect to participating derivatives the obtained solitary wave solution for the case i contains as particular cases the solitary wave solutions of korteweg devries equation and of a version of the modified korteweg devries equation one of the obtained solitary wave solutions for the case ii is a solitary wave solution of the classic version of the boussinesq type equation

micropolar meets newtonian the rayleigh benard problem abstract we consider the rayleigh benard problem for the two dimensional boussinesq system for the micropolar fluid our main goal is to compare the value of the critical rayleigh number and estimates of the nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical navier stokes system our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous navier stokes fluid in particular the critical rayleigh number for the micropolar model is larger than that for the navier stokes one and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction the estimate of the fractal dimension of the global attractor for the considered problem is better than that for the same problem for the navier stokes system besides we obtain also other results which relate the heat convection in the micropolar fluids to the newtonian ones among them relations between the nusselt number and the energy dissipation rate and upper semicontinuous convergence of global attractors to the rayleigh benard attractor for the considered model

an exact penalty function for semi infinite programming this paper introduces a global approach to the semi infinite programming problem that is based upon a generalisation of the l1 exact penalty function the advantages are that the ensuing penalty function is exact and the penalties include all violations the merit function requires integrals for the penalties which provides a consistent model for the algorithm the discretization is a result of the approximate quadrature rather than an a priori aspect of the model

upper bound on the heat transport in a layer of fluid of infinite prandtl number rigid lower boundary and stress free upper boundary we obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite prandtl number with rigid lower boundary and stress free upper boundary because of the asymmetric boundary conditions the solutions of the euler lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on the upper and lower boundary of the fluid the obtained bound on the convective heat transport and the corresponding wave number are between the values for a fluid layer with two rigid boundaries and a fluid layer with two stress free boundaries

upper bounds on the heat transport in a porous layer abstract numerical and asymptotic analytical upper bounds on the convective heat transport in a fluid saturated porous layer of infinite darcy prandtl number are obtained by the howard busse variational method of the optimum theory of turbulence the nonlinear euler lagrange equations of the variational problem possess different numerical solutions as well as different approximate analytical solutions in the region of high rayleigh numbers we use symmetric fields for the numerical investigation and obtain slightly different numerical bounds on the heat transport in comparison to the numerical bounds obtained by gupta and joseph j fluid mech 57 1973 491 514 as well as different numerical values of the wave numbers connected to the optimum fields for the analytical investigation we use new approximate solution of the euler lagrange equations for the intermediate layers of the optimum fields this solution leads to different thickness of the boundary layers of the optimum fields as well as to a lower analytical upper bounds on the heat transport in comparison to the boundary layer thicknesses and analytical upper bounds obtained by gupta and joseph

rayleigh benard convection improved bounds on the nusselt number we consider rayleigh benard convection as modelled by the boussinesq equations in the infinite prandtl number limit we are interested in the scaling of the average upward heat transport the nusselt number nu in terms of the non dimensionalized temperature forcing the rayleigh number ra experiments asymptotics and heuristics suggest that nu ra1 3 this work is mostly inspired by two earlier rigorous work on upper bounds of nu in terms of ra 1 the work of constantin and doering establishing nu ra1 3ln 2 3ra with help of a logarithmically failing maximal regularity estimate in l on the level of the stokes equation 2 the work of doering reznikoff and the first author establishing nu ra1 3ln 1 3ra with help of the background field method the paper contains two results 1 the background field method can be slightly modified to yield nu ra1 3ln 1 15ra 2 the estimates behind the background field method can be combined with the maximal regularity in l to yield nu ra1 3ln 1 3ln ra

the 3 2 mixed refined zigzag theory for generally laminated beams theoretical development and c0 finite element formulation abstract a third order refined zigzag theory for multilayered composite and sandwich beams is developed based on the reissner mixed variational theorem the assumed kinematics enriches the timoshenko s beam theory a second and a third order smeared polynomial terms along with a piece wise linear contributions are added to the axial displacement whereas the transverse displacement is approximated with a power series expansion up to the second order term transverse shear and normal stress continuous and able to satisfy the boundary stress conditions on the outer beam surfaces are assumed by the model the transverse shear stress is derived with the aid of the elasticity equations whereas a third order power series expansion is adopted for the transverse normal stress based on the proposed model an efficient c 0 continuous beam element is formulated by adopting the anisoparametric interpolation strategy to avoid the shear locking phenomenon the accuracy of the proposed model and the finite element implementation is assessed by solving problems concerning the elasto static behavior of generally laminated beams both simply supported and clamped at the ends comparison with reference solution elasticity or high fidelity fe model demonstrates the remarkable accuracy of the proposed model both in terms of displacements and stresses distributions as well as the finite element implementation efficiency

an unconstrained convex programming approach to linear semi infinite programming in this paper an unconstrained convex programming dual approach for solving a class of linear semi infinite programming problems is proposed both primal and dual convergence results are established under some basic assumptions numerical examples are also included to illustrate this approach

on the backward behavior of some dissipative evolution equations abstract we prove that every solution of a kdv burgers sivashinsky type equation blows up in the energy space backward in time provided the solution does not belong to the global attractor this is a phenomenon contrast to the backward behavior of the periodic 2d navier stokes equations studied by constantin et al 1997 but analogous to the backward behavior of the kuramoto sivashinsky equation discovered by kukavica and malcok 2005 also we study the backward behavior of solutions to the damped driven nonlinear schrodinger equation the complex ginzburg landau equation and the hyperviscous navier stokes equations in addition we provide some physical interpretation of various backward behaviors of several perturbations of the kdv equation by studying explicit cnoidal wave solutions furthermore we discuss the connection between the backward behavior and the energy spectra of the solutions the study of backward behavior of dissipative evolution equations is motivated by the investigation of the bardos tartar conjecture on the navier stokes equations stated in bardos and tartar 1973

kolmogorov s theory of turbulence and inviscid limit of the navier stokes equations in mathbb r 3 we are concerned with the inviscid limit of the navier stokes equations to the euler equations in mathbb r 3 we first observe that a pathwise kolmogorov hypothesis implies the uniform boundedness of the th order fractional derivatives of the velocity for some 0 in the space variables in l2 which is independent of the viscosity 0 then it is shown that this key observation yields the l2 equicontinuity in the time variable and the uniform bound in lq for some q 2 of the velocity independent of 0 these results lead to the strong convergence of solutions of the navier stokes equations to a solution of the euler equations in mathbb r 3 we also consider passive scalars coupled to the incompressible navier stokes equations and in this case find the weak star convergence for the passive scalars with a limit in the form of a young measure pdf depending on space and time not only do we offer a framework for mathematical existence theories but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place with the bounds that are independent of 0 that is in the high reynolds number limit

application of the extended simplest equation method to the coupled schrodinger boussinesq equation in this paper we successfully construct the new exact traveling wave solutions of the coupled schrodinger boussinesq equation by using the extended simplest equation method the exact traveling wave solutions with double arbitrary parameters are expressed by the hyperbolic functions the trigonometric functions and the rational functions respectively when the parameters are taken as special values some solitary wave solutions are derived from the hyperbolic function solutions

on locking free shear deformable beam finite elements abstract this is an elementary exposition of beam finite element models an overview of existing and new displacement finite element models of the euler bernoulli and shear deformation beam theories is presented a locking free finite element model using the form of the exact solution of the timoshenko beam theory is developed the element contains the euler bernoulli beam element as a special case and yields exact nodal values for the generalized displacements for constant material and geometric properties of beams the procedure is extended to develop a locking free finite element model of the simplified third order theory of beams the element contains the euler bernoulli and the locking free timoshenko beam finite elements as special cases the element stiffness matrix of the unified beam bending element is 4 4 and yet has the superconvergence character the advantage of this element is that it can be used to obtain exact nodal displacements and forces from the equilibrium equations of the element in a frame structural analysis using one finite element per a structural member this cannot be achieved with the traditional equal interpolation reduced integration element because the element exhibits shear locking for thin members unless two or more elements per a structural member are used

symmetry analysis and exact solutions of the 2 1 dimensional sine gordon system according to the space variable exchange symmetry we change the 2 1 dimensional sine gordon system obtained by konopelchenko and rogers kr to a variant form some types of similarity reductions are obtained by using some lie symmetry analysis from these similarity reductions we find that the soliton structure of the system possesses quite a rich structure the line solitons parallel to the lines x y 0 and x y 0 may have an arbitrary shape in addition to the well known dromion solution which is constructed by two line solitons one may find many other kinds of soliton solutions localized in all directions for instance some kinds of ring type basin like plateau like and bowl like and instanton type soliton solutions can be found directly by selecting the arbitrary functions included in the single soliton solution the dromion solutions may also be constructed by straight line and curved line solitons

breather and soliton wave families for the sine gordon equation the approach of steuerwaldlamb for obtaining exact solutions of the sinegordon equation is further developed through transformations by means of which complicated solutions of the corresponding odes are obtained without any explicit integration polynomial and nonpolynomial transformations are discussed and several of them are used to obtain breather and soliton solutions of the sinegordon equation the obtained solutions are discussed from the point of view of the electrodynamics of the long josephson junction

optimization with affine homogeneous quadratic integral inequality constraints we introduce a new technique to optimize a linear cost function subject to an affine homogeneous quadratic integral inequality i e the requirement that a homogeneous quadratic integral functional affine in the optimization variables is nonnegative over a space of functions defined by homogeneous boundary conditions such problems arise in control and stability or input to state output analysis of systems governed by partial differential equations pdes particularly fluid dynamical systems we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities lmis and show that a convergent sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs sdps we also obtain inner approximations in terms of lmis and sum of squares constraints so upper bounds for the optimal cost and strictly feasible points for the integral inequality can be computed with sdps we present quinopt an open source add on to yalmip to aid the formulation and solution of our sdps and demonstrate our techniques on problems arising from the stability analysis of pdes

analytical properties and exact solutions of the lotka volterra competition system the lotka volterra competition system with diffusion is considered the painleve property of this system is investigated exact traveling wave solutions of the lotka volterra competition system are found periodic solutions expressed in terms of the weierstrass elliptic function are also given

the energy operator and new scaling relations for the incompressible navier stokes equations the incompressible navier stokes equations have been widely studied due to their vast ability to model hydrodynamic phenomena the nonlinear character of these equations produces a hierarchical system of dynamical scales that offer challenging barriers to computation simulation and scale elimination modeling here the incompressible navier stokes equations are decomposed and studied under the context of global eigensystems that have a direct relation to the dynamical character of the scales of motion in particular the self adjoint linear reynolds orr operator is shown to correspond to the energy operator of this nonlinear system of equations and has the distinguishing property of having an eigensystem that orders the time derivative of the integral kinetic energy for the nonlinear navier stokes operator the eigensystem of the reynolds orr operator is shown to yield a dynamical range of scales that can be characterized by an energy containing range an inertial range and a dissipation range it i

a barrier function method for minimax problems this paper presents an algorithm based on barrier functions for solving semi infinite minimax problems which arise in an engineering design setting the algorithm bears a resemblance to some of the current interior penalty function methods used to solve constrained minimization problems global convergence is proven and numerical results are reported which show that the algorithm is exceptionally robust and that its performance is comparable while its structure is simpler than that of current first order minimax algorithms

improved rigorous upper bounds for transport due to passive advection described by simple models of bounded systems the work of j a krommes and r a smith on rigorous upper bounds for the turbulent transport of a passively advected scalar is extended in two directions 1 for their reference model improved upper bounds are obtained by utilizing more sophisticated two time constraints which include the effects of cross correlations up to fourth order numerical solutions of the model stochastic differential equation are also obtained they show that the new bounds compare quite favorably with the exact results even at large reynolds and kubo numbers 2 the theory is extended to take account of afinite spatial autocorrelation lengthlc as a reasonably generic example the problem of particle transport due to statistically specified stochastic magnetic fields in a collisionless turbulent plasma is revisited a bound is obtained which reduces for smalllc to the quasilinear limit and for largelc to the strong turbulence limit and which provides a reasonable and rigorous interpolation for intermediate values oflc

variational bounds on shear driven turbulence and turbulent boussinesq convection abstract based on earlier studies by hopf 1941 doering and constantin 1992 1994 1995 have recently formulated a new background technique for obtaining upper bounds on turbulent fluid flow quantities this method produces upper bounds on the limit supremum of long time averages making no statistical assumptions about the flow in contrast to the well known howard busse approach the full optimisation problems posed by this method for the momentum transport in turbulent couette flow and the heat transport with zero background flow in turbulent boussinesq convection are solved here for the first time at asymptotically large reynolds number and rayleigh number within busse s multiple boundary layer approximation to extract the best lowest possible upper bounds available intriguingly the original bounds isolated by busse 1969 1970 within the confines of statistical stationarity are recovered exactly using this new formalism the optimal background velocity profile for turbulent couette flow is found to be shearless in the interior thus differing from busse s 1 4 mean shear result in the convective case an interesting degeneracy in the formulation of the background variational problem leads to an indeterminacy in the optimal background temperature profile only for one special choice is the isothermal core feature of busse s mean profile recovered

a globally convergent sqp method for semi infinite nonlinear optimization abstract a new approach for semi infinite programming problems is presented which belongs to the class of successive quadratic programming sqp methods with trust region technique the proposed algorithm employs the exact l penalty function as a criterion function and incorporates an appropriate scheme for estimating active constraints it is proved that the algorithm is globally convergent under some assumptions numerical experiments show that the algorithm is very promising in practice

adaptation and its impact on the dynamics of a system of three competing populations abstract we investigate how the adaptation of the competition coefficients of the competing populations for the same limited resource influences the system dynamics in the regions of the parameter space where chaotic motion of shilnikov kind exists we present results for two characteristic values of the competition coefficient adaptation factor the first value 0 05 belongs to the small interval of possible negative values of for this a transition to chaos by period doubling bifurcations occurs and a window of periodic motion exists between the two regions of chaotic motion with increasing the system becomes more dissipative and the number of the windows of periodic motion increases when 1 0 a region of transient chaos is observed after the last window of periodic motion we verify the picture of the system dynamics by power spectra histograms and autocorrelations and calculate the lyapunov exponents and kaplan yorke dimension finally we discuss the eligibility of the investigated system for a topological analysis

convective heat transport in a rotating fluid layer of infinite prandtl number optimum fields and upper bounds on nusselt number by means of the howard busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress free vertical boundaries rotating about a vertical axis we discuss the case of infinite prandtl number 1 ensuremath ensuremath alpha solution of the obtained variational problem and optimum fields possessing internal intermediate and boundary layers we investigate regions of rayleigh and taylor numbers r and ta where no analytical bounds can be derived and compare the analytical and numerical bounds for these regions of r and ta where such comparison is possible the increasing rotation has a different influence on the rescaled optimum fields of velocity w 1 temperature ensuremath theta 1 and the vertical component of the vorticity f 1 the increasing ta for fixed r leads to vanishing of the boundary layers of w 1 and ensuremath theta 1 opposite to this the increasing ta leads first to a formation of boundary layers of the field f 1 but further increasing the rotation causes vanishing of these boundary layers we obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying doering constantin method to the problems of rotating convection the wave number ensuremath alpha 1 corresponding to the optimum fields follows the asymptotic relationship ensuremath alpha 1 r 5 1 4 for intermediate rayleigh numbers however when r becomes large with respect to ta after a transition region the power law for ensuremath alpha 1 becomes close to the power law for the case without rotation the nusselt number nu is close to the nonrotational bound 0 32r 1 3 for the case of large r and small ta nu decreases with increasing taylor number thus the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed rayleigh number for the regions of rayleigh and taylor numbers where the numerical and asymptotic bounds on nu can be compared the numerical bounds are about 70 lower than the asymptotic bounds

upper bounds on nusselt number at finite prandtl number abstract we study rayleigh benard convection based on the boussinesq approximation we are interested in upper bounds on the nusselt number nu the upwards heat transport in terms of the rayleigh number ra that characterizes the relative strength of the driving mechanism and the prandtl number pr that characterizes the strength of the inertial effects we show that up to logarithmic corrections the upper bound nu ra 1 3 from 1 persists as long as pr ra 1 3 and then crosses over to nu pr 1 2 ra 1 2 this result improves the one of wang 2 by going beyond the perturbative regime pr ra the proof uses a new way to estimate the transport nonlinearity in the navier stokes equations capitalizing on the no slip boundary condition it relies on a new calderon zygmund estimate for the non stationary stokes equations in l 1 with a borderline muckenhoupt weight

transport bounds for a truncated model of rayleigh benard convection abstract we investigate absolute limits on heat transport in a truncated model of rayleigh benard convection two complementary mathematical approaches a background method analysis and an optimal control formulation are used to derive upper bounds in a distinguished eight ode model proposed by gluhovsky tong and agee in the optimal control approach the flow no longer obeys an equation of motion but is instead a control variable both methods produce the same estimate but in contrast to the analogous result for the seminal three ode lorenz system the best upper bound apparently does not always correspond to an exact solution of the equations of motion

rigorous bound on the plane shear flow dissipation rate the doering constantin approach phys rev lett 69 1648 1992 to derive an upper bound on the dissipation rate in terms of a properly chosen stationary flow profile can be slightly improved by casting it in the form of a variational principle for the profile only schwarz s inequality is needed to prove this its solution can be evaluated analytically the mean dissipation bound is var epsilon le ital u sup 3 ital l sup minus 1 9 radical 2 provided the reynolds number re ital ul nu exceeds 12 radical 2 and the mean transverse momentum flux is bounded by ital j sub 31 le nu ital l sup minus 1 sup 2 re sup 2 9 radical 2

analysis of thick sandwich construction by a 3 2 order theory abstract this study is an extension of the 1 2 order plate theory to a higher order 3 2 theory based on the equivalent single layer assumptions the in plane and transverse displacement components are expressed as cubic and quadratic expansions through the thickness of the sandwich construction also the transverse stress component is assumed to vary as a cubic function through the thickness utilizing reissner s definitions for kinematics of thick plates the displacement components at any point on the plate are approximated in terms of weighted average quantities displacements and rotations that are functions of the in plane coordinates the undetermined coefficients defining the in plane and transverse displacement fields are then expressed in terms of the weighted average displacements and rotations and their derivatives by directly employing reissner s definitions and enforcing the zero transverse shear stress conditions on the upper and lower surfaces of the sandwich panel the coefficients defining the transverse stress component are obtained by requiring the transverse strain component which is expressed in terms of the unknown coefficients of the transverse stress component from a mixed constitutive relation to be the least squares equivalent of the kinematic definition of the transverse strain component the resulting expressions for the unknown coefficients of the transverse stress component are related to resultant strains and curvatures defined from kinematic relations the equations of equilibrium and boundary conditions of the sandwich plate based on the 3 2 higher order theory are derived by employing the principles of virtual displacements the robustness and accuracy of this 3 2 order plate theory are established through comparisons with exact solutions available in the literature the finite element implementation of the present 3 2 order plate theory is also discussed

traveling wave solutions for nonlinear equations using symbolic computation abstract in this paper a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations pdes is presented and implemented in a computer algebraic system the key idea of this method is to take full advantage of a riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh method it is quite interesting that the sign of the parameter can be used to exactly judge the number and types of such travelling wave solutions in this way we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods more importantly for some equations with no extra effect we also find other new and more general solutions at the same time by introducing appropriate transformations our method is further extended to the nonlinear pdes whose balancing numbers may be any nonzero real numbers the efficiency of the method can be demonstrated on a large variety of nonlinear pdes such as those considered in this paper burgers huxley equation coupled korteweg de vries equation caudrey dodd gibbon kawada equation active dissipative dispersive media equation generalized fisher equation and nonlinear heat conduction equation

extended simplest equation method for nonlinear differential equations abstract the modified simplest equation method to look for the exact solutions of the nonlinear differential equations is presented our approach is applied to search for the exact solutions of the sharma tasso olver and the burgers huxley equations the new exact solutions of these equations are obtained

resonant soliton and complexiton solutions for 3 1 dimensional boiti leon manna pempinelli equation abstract in this letter the linear superposition principle is used to discuss the 3 1 dimensional boiti leon manna pempinelli equation with bilinear derivatives as a result we obtain new resonant soliton and complexiton solutions by discussing two different cases involved the parameters these solutions are a class of n wave solutions of linear combinations of exponential traveling waves

a note on the computation of an orthonormal basis for the null space of a matrix a highly regarded method to obtain an orthonormal basis z for the null space of a matrix a t is the qr decomposition of a where q is the product of householder matrices in several optimization contexts a x varies continuously with x and it is desirable the z x vary continuously also in this note we demonstrate that the standard implementation of the qr decomposition does not yield an orthonormal basis z x whose elements vary continuously with x we suggest three possible remedies

a projected lagrangian algorithm for semi infinite programming a globally convergent algorithm is presented for the solution of a wide class of semi infinite programming problems the method is based on the solution of a sequence of equality constrained quadratic programming problems and usually has a second order convergence rate numerical results illustrating the method are given

a fractal model for large eddy simulation of turbulent flow abstract a new class of subgrid closures for large eddy simulation les of turbulence is developed based on the construction of synthetic fractal subgrid scale fields the relevant mathematical tool fractal interpolation allows to interpolate the resolved velocity with fields that have fluctuations down to much smaller scales and to compute the required stresses explicitly in one dimension the approach is used in the context of the coarse grained burgers equation then fractal interpolation is extended to three dimensions and is used to formulate a subgrid model for the filtered navier stokes equations the model is applied to les of both steady and freely decaying isotropic turbulence we find that the assumption of fractality per se is not enough to yield physically meaningful results and we explore several variants of the model in which the rules to generate the synthetic fields explicitly incorporate the condition that energy dissipation take place in one dimension this is accomplished by means of an additional transport equation that allows to dynamically determine the fractal dimension in three dimensions good results are obtained only once the fractal dimension is allowed to vary in different eigendirections of the resolved strain rate tensor so as to nearly maximize energy dissipation

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

frequency maximization of laminated sandwich plates under general boundary conditions using layerwise optimization method with refined zigzag theory abstract the present paper extends the layerwise optimization lo procedure to the maximization problem of the fundamental frequencies of sandwich plates with fibrous composites and low stiffness core layers frequencies are calculated by the ritz method based on a refined zigzag theory rzt polynomial functions which satisfy at least geometrical boundary conditions with boundary indexes are employed as displacement functions and they enable satisfying arbitrary sets of boundary conditions for rectangular plates results of the experimental modal analysis validate the accuracy of the present calculations and a comparison with results of the classical laminated theory clpt and the first order share deformation theory fsdt supports the effectiveness of the present method optimized results are compared with other typical sets of lay up configurations and this shows the lo method as suitable means to the optimization problem for sandwich plates

kolmogorov theory via finite time averages abstract several relations from the kolmogorov theory of fully developed three dimensional turbulence are rigorously established for finite time averages over leray hopf weak solutions of the navier stokes equations the navier stokes equations are considered with periodic boundary conditions and an external forcing term the main parameter is the grashof number associated with the forcing term the relations rigorously proved in this article include estimates for the energy dissipation rate the kolmogorov wavenumber the taylor wavenumber the reynolds number and the energy cascade process for some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone while for others such as the kolmogorov energy dissipation law and the energy cascade the estimates depend also on the grashof number as compared with earlier works by some of the authors the more physical concept of finite time average is replacing the concept of infinite time average used before

attractor dimension estimates for two dimensional shear flows abstract we study the large time behavior of boundary and pressure gradient driven incompressible fluid flows in elongated two dimensional channels with emphasis on estimates for their degrees of freedom i e the dimension of the attractor for the solutions of the navier stokes equations for boundary driven shear flows and flux driven channel flows we present upper bounds for the degrees of freedom of the form c re 3 2 where c is a universal constant denotes the aspect ratio of the channel length width and re is the reynolds number based on the channel width and the imposed outer velocity scale for fixed pressure gradient driven channel flows we obtain an upper bound of form c re2 where c is another universal positive constant and the reynolds number is based on a velocity defined by the infimum over all possible trajectories of the time averaged mass flux per unit channel width we discuss these results in terms of physical arguments based on small length scales in turbulent flows

solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives the method of simplest equation is applied to nonlinear pdes the pdes contain monomials of odd and even order with respect to derivatives solitary wave solutions are studied numerous examples of the methodology are presented we apply the method of simplest equation for obtaining exact solitary traveling wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives we consider first the general case of presence of monomials of the both odd and even grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade the methodology is illustrated by numerous examples

solving air traffic conflict problems via local continuous optimization this paper first introduces an original trajectory model using b splines and a new semi infinite programming formulation of the separation constraint involved in air traffic conflict problems a new continuous optimization formulation of the tactical conflict resolution problem is then proposed it involves very few optimization variables in that one needs only one optimization variable to determine each aircraft trajectory encouraging numerical experiments show that this approach is viable on realistic test problems not only does one not need to rely on the traditional discretized combinatorial optimization approaches to this problem but moreover local continuous optimization methods which require relatively fewer iterations and thereby fewer costly function evaluations are shown to improve the performance of the overall global optimization of this non convex problem

dynamical consequences of adaptation of the growth rates in a system of three competing populations we investigate the nonlinear dynamics of a system of populations competing for the same limited resource for the case where each of the populations adapts its growth rate to the total number of individuals in all populations we consider regions of parameter space where chaotic motion of the shilnikov kind exists and present results for two characteristic values of the growth ratio adaptation factor r r 0 15 and 5 negative r can lead to vanishing of regions of chaotic motion and to a stabilization of a fixed point of the studied model system of differential equations positive r lead to changes of the shape of the bifurcation diagrams in comparison with the bifurcation diagrams for the case without adaptation for the case r 5 we observe transition to chaos by period doubling bifurcations windows of periodic motion between the regions of chaotic motion and a region of transient chaos after the last window of periodic motion the lyapunov dimension for the chaotic attractors is close to two and the lyapunov spectrum has a structure which allows a topological analysis of the attractors of the investigated system

variational bounds in dissipative systems abstract we present a general method for estimating the space time average dissipation in a large class of dissipative pdes the method leads to variational problems with spectral side conditions

improved variational principle for bounds on energy dissipation in turbulent shear flow we extend the doering constantin approach to upper bounds on energy dissipation in turbulent flows by introducing a balance parameter into the variational principle this parameter governs the relative weight of different contributions to the dissipation rate its optimization leads to improved bounds without entailing additional technical difficulties for plane couette flow the high re bounds obtainable with one dimensional background flows are methodically lowered by a factor of 2732

energy stability bounds on convective heat transport numerical study the concept of nonlinear energy stability has recently been extended to deduce bounds on energy dissipation and transport in incompressible flows even for turbulent flows in this approach an effective stability condition on background flow or temperature profiles is derived which when satisfied ensures that the profile produces a rigorous upper estimate to the bulk dissipation optimization of the test background profiles in search of the lowest upper bounds leads to nonlinear euler lagrange equations for the extremal profile in this paper in the context of convective heat transport in the boussinesq equations we describe numerical solutions of the euler lagrange equations for the optimal background temperature and present the numerical computation of the implied bounds s1063 651x 97 05706 1 pacs number s 47 27 te 03 40 gc 47 27 cn 47 27 ak the idea that notions of stability usually reserved for the characterization of static stationary states may be applied to dynamic and even turbulent phenomena has been proposed at various times and in various contexts the modern concept of spontaneously self imposed marginal stability was proposed in the context of thermal convection in the 1950s 1 later resulting in quantitative predictions for the bulk heat transport 2 more recently these sentiments have been given a rigorous formulation in the context of the dynamics of incompressible fluids building on a mathematical device invented by hopf 3 and utilizing a decomposition referred to as the background field method 4 5 it has now become possible to formulate variational principles for upper bounds on the time averaged rate of heat transport where the key constraint on the background profiles over which the variation takes place is technically identical to a nonlinear energy stability 6 condition the optimization problem produces nonlinear euler lagrange equations for the marginally stable profile yielding the lowest upper bound interestingly in some cases these euler lagrange equations are of the same functional form as those found in howard s theory of bounds on flow quantities in statistically stationary states 7 and with regard to the functional geometry of the constraints further connection has been established with busse s multiple boundary layer theory 8 of howard s bounds in this brief report we present the results of a numerical study of the solutions of the euler lagrange equations for the optimal background profile this study represents a practical implementation of the optimal methods developed in ref 5 and it illustrates some of the features that are expected in this kind of analysis in particular we observe that the optimal marginally stable profile may exist on different branches of solutions of the euler lagrange equations with the signal for the qualitative structural change being a change in stability of the solution consider an incompressible newtonian fluid confined to the rectangular volume between rigid noslip isothermal plates at z50 and 1 a vertical temperature gradient is imposed so in the usual nondimensional units the fluid s velocity vector field u x t 5 u 1 u 2 u 3 and temperature field t x t satisfy the boussinesq equations u t 1uiiu1ip5sdu1srakt 1

variational principle for the navier stokes equations a variational principle is presented for the navier stokes equations in the case of a contained boundary driven homogeneous incompressible viscous fluid based upon making the fluid s total viscous dissipation over a given time interval stationary subject to the constraint of the navier stokes equations the variational problem looks overconstrained and intractable however introducing a nonunique velocity decomposition mathit u mathit x t mathit ensuremath varphi mathit x t mathit ensuremath nu mathit x t opens up the variational problem so that what is presumed a single allowable point over the velocity domain mathit u corresponding to the unique solution of the navier stokes equations becomes a surface with a saddle point over the extended domain mathit ensuremath varphi mathit ensuremath nu complementary or dual variational problems can then be constructed to estimate this saddle point value strictly from above as part of a minimization process or below via a maximization procedure one of these reduced variational principles is the natural and ultimate generalization of the upper bounding problem developed by doering and constantin the other corresponds to the ultimate busse problem which now acts to lower bound the true dissipation crucially these reduced variational problems require only the solution of a series of linear problems to produce bounds even though their unique intersection is conjectured to correspond to a solution of the nonlinear navier stokes equations

a one phase algorithm for semi infinite linear programming we present an algorithm for solving a large class of semi infinite linear programming problems this algorithm has several advantages it handles feasibility and optimality together it has very weak restrictions on the constraints it allows cuts that are not near the most violated cut and it solves the primal and the dual problems simultaneously we prove the convergence of this algorithm in two steps first we show that the algorithm can find ane optimal solution after finitely many iterations then we use this result to show that it can find an optimal solution in the limit we also estimate how good ane optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding ane optimal solution under some assumptions this algorithm is generalized to solve a class of nonlinear semi infinite programming problems applications to convex programming are discussed

new class of running wave solutions of the 2 1 dimensional sine gordon equation a new class of running wave solutions of the 2 1 dimensional sine gordon equation is investigated the obtained waves require two spatial dimensions for their propagation i e they generalize solutions of the 2 0 dimensional sine gordon equation the parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency the obtained solutions describe a new class of josephson waves whose velocity is smaller than the swihart velocity if omega 0 the running waves are reduced to the self consistent phase current and magnetic field distributions in a large two dimensional josephson junction the self restriction coefficient for the josephson current corresponding to one of the structures is calculated

some specific mathematical constraints on 2d turbulence abstract we derive upper and lower bounds for ensemble averages of energy enstrophy and palinstrophy for the 2d periodic navier stokes equations this is carried out both in the general case and in the case where the energy power law for fully developed turbulence holds in the turbulent case the bounds are sharp up to a logarithm and provide a new lower bound on the landau lifschitz degrees of freedom we also prove two properties of the inertial term under the turbulence assumption one is that as the grashof number is increased the ensemble average of this term approaches the force the other is that an estimate of it via the ladyzhenskaya inequality is sharp on a considerable portion of the global attractor

a comparative study of several semi infinite nonlinear programming algorithms abstract a computational comparison of several recent semi infinite nonlinear programming algorithms is presented because of the importance of this area there have recently been proposed a number of algorithms having a global convergence property the algorithms used in this study include a feasible direction method and some variants of successive quadratic programming methods robustness and relative efficiency of the algorithms are examined on the basis of the results for some standard test examples chosen from earlier literature

unification of variational principles for turbulent shear flows the background method of doering constantin and the mean fluctuation formulation of howard busse abstract the recent discovery of a new variational formulation for bounding turbulent flow quantities has rekindled interest in the whole field of producing rigorous inequality results directly from the navier stokes equations initiated by howard and busse in the 1960s the new background method introduced by doering and constantin 1992 1994 1995 1996 for bounding turbulent viscous dissipation appears to differ so fundamentally from that of howard and busse that any connection between the two methods has been fascinatingly unclear recently highly suggestive results obtained asymptotically kerswell 1997 have indicated that the variational problems produced for both cases of shear flow and convection may be equivalent here we confirm this for the shear flow case by uncovering the common underlying functional which both approaches seek to make stationary busse s upper bound problem is found to be the optimal dual or complementary variational problem to that which emerges from the doering constantin approach this ensures that busse s best bound exactly coincides with that available within the doering constantin formalism for all reynolds numbers

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

an improved plate theory of 1 2 order for thick composite laminates abstract a new two dimensional laminate plate theory is developed for the linear elastostatic analysis of thick composite plates the theory employs equivalent single layer assumptions for the displacements transverse shear strains and transverse normal stress the inplane and transverse displacements are respectively linear and quadratic expansions through the laminate thickness where the low order expansion coefficients correspond to the variables of reissner s first order shear deformable theory the transverse shear strains and transverse normal stress are assumed to be quadratic and cubic respectively through the thickness they are expressed in terms of the kinematic variables of the theory by means of a least squares compatibility requirement for the transverse strains and explicit enforcement of exact traction boundary conditions on the top and bottom plate surfaces application of the virtual work principle results in the loth order equations of equilibrium and associated poisson boundary conditions a major advantage of this theory over other higher order theories lies in its perfect suitability for finite element approximation simple c 0 continuous interpolations for the kinematic variables of the first order theory and optionally c 1 interpolations for the two higher order displacements are needed to formulate effective and robust two dimensional plate elements capable of full three dimensional ply by ply strain and stress recovery within the framework of general purpose finite element codes in assessing the predictive capability of the theory analytic solutions for the problem of cylindrical bending are derived and compared with exact three dimensional elasticity results and those of the earlier version of the 1 2 order plate theory

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

variational bounds on energy dissipation in incompressible flows iii convection building on a method of analysis for the navier stokes equations introduced by hopf math ann 117 764 1941 a variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the boussinesq equations of motion when supplied with appropriate test background fields satisfying a spectral constraint reminiscent of an energy stability condition the variational formulation produces rigorous upper bounds on the nusselt number nu as a function of the rayleigh number ra for the case of vertical heat convection between parallel plates in the absence of sidewalls a simplified but rigorous formulation of the optimization problem yields the large rayleigh number bound nu ensuremath le 0 167 mathrm ra 1 2 1 nonlinear euler lagrange equations for the optimal background fields are also derived which allow us to make contact with the upper bound theory of howard j fluid mech 17 405 1963 for statistically stationary flows the structure of solutions of the euler lagrange equations are elucidated from the geometry of the variational constraints which sheds light on busse s j fluid mech 37 457 1969 asymptotic analysis of general solutions to howard s euler lagrange equations the results of our analysis are discussed in the context of theory recent experiments and direct numerical simulations textcopyright 1996 the american physical society

application of the method of simplest equation for obtaining exact traveling wave solutions for the extended korteweg de vries equation and generalized camassa holm equation a version of the method of the simplest equation called modified method of simplest equation is applied to the extended korteweg de vries equation and to generalized camassa holm equation exact traveling wave solutions of these two nonlinear partial differential equations are obtained the equations of bernoulli riccati and the extended tanh equation are used as simplest equations some of the obtained solutions correspond to surface water waves

kraichnan turbulence via finite time averages relations central to kraichnan s theory of fully developed two dimensional turbulence are rigorously established for finite time averages in particular we prove that if the ratio of the averages of palinstrophy to enstrophy is large then a large inertial range displaying an enstrophy cascade exists moreover if this ratio is comparable up to a logarithm to the dissipation wave number a necessary condition for turbulence then the power law for the energy spectrum until now derived only heuristically is rigorously shown to provide up to a logarithm an upper bound for the energy spectrum finally we show that deep in the dissipation range the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number the averaging times needed for these relations are bounded in terms of the generalized grashof number independent of the solution for which the time averages are taken solutions are not assumed to be on the global attractor merely within the absorbing ball an easily verified condition

optimal bounds with semidefinite programming an application to stress driven shear flows we introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows in contrast to most existing schemes we do not consider the euler lagrange equations for the minimizer instead we use series expansions to formulate a finite dimensional semidefinite program sdp whose solution converges to that of the original variational problem our formulation accounts for the influence of all modes in the expansion and the feasible set of the sdp corresponds to a subset of the feasible set of the original problem moreover sdps can be easily formulated when the fluid is subject to imposed boundary fluxes which pose a challenge for the traditional methods we apply this technique to compute rigorous and near optimal upper bounds on the dissipation coefficient for flows driven by a surface stress we improve previous analytical bounds by more than 10 times and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity we also confirm that the dissipation properties of stress driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface finally we show that sdp relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress

statistical estimates for channel flows driven by a pressure gradient we present rigorous estimates for some physical quantities related to turbulent and non turbulent channel flows driven by a uniform pressure gradient such results are based on the concept of stationary statistical solutions which is related to the notion of ensemble averages for flows in statistical equilibrium we provide a lower bound estimate for the mean skin friction coefficient and improve on a previous upper bound estimate for the same quantity both estimates are derived in terms of the reynolds number we also present lower and upper bound estimates for the mean rate of energy dissipation the mean longitudinal bulk velocity in the direction of the pressure gradient and the mean kinetic energy in terms of various physical parameters in particular we obtain an upper bound related to the energy dissipation law namely that the mean rate of energy dissipation is essentially bounded by a non dimensional universal constant times the cube of the mean longitudinal bulk velocity over a characteristic macro scale length finally we investigate the scale by scale energy injection due to the pressure gradient proving an upper bound estimate for the decrease of this energy injection as the scale length decreases

upper bounds on convective heat transport in a rotating fluid layer of infinite prandtl number case of intermediate taylor numbers by means of the howard busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a heated from below layer of fluid of infinite prandtl number rotating with a constant angular velocity about the vertical axis we consider the region of intermediate taylor numbers alpha41 ta alpha61 where alpha 1 is the wave number connected to the 1 alpha solution of the variational problem the studied optimum fields possess a three layer or four layer structure in addition to the internal intermediate and boundary layers ekman layers could arise between the intermediate and boundary ones for the discussed interval of taylor numbers the intermediate layers do not expand in the direction of the internal layers we present an asymptotic theory for the case of the fluid layer with rigid lower boundary and stress free upper boundary we use an improved solution of the euler lagrange equations of the variational problem for the intermediate sublayer of the optimum field this solution leads also to correction of the thicknesses of the boundary layers and to lowering of the upper bounds on the convective heat transport for the cases of fluid layer with stress free or with rigid boundaries thus the known upper bounds for these cases can be treated as upper bounds on the upper bounds on the convective heat transport for the case of the fluid layer with stress free boundaries the four layer optimum fields leads to bounds on the convective heat transport which change from r 1 3 at the lower boundary of their interval of validity to values slightly large than r 2 7 near the upper boundary of the interval of validity finally we discuss the area of application of the obtained bounds with respect to the taylor number ta and rayleigh number r

modified method of simplest equation and its application to nonlinear pdes abstract we search for traveling wave solutions of the class of pdes p 1 n 1 a p q p q t p r 2 n 2 b r q q t r s 1 n 3 c s q s q x s u 2 n 4 d u q q x u f q 0 where a p q b r q c s q d u q and f q are polynomials of q the basis of the investigation is a modification of the method of simplest equation the equations of bernoulli riccati and the extended tanh function equation are used as simplest equations the obtained general results are illustrated by obtaining exact solutions of versions of the generalized kuramoto sivashinsky equation reaction diffusion equation with density dependent diffusion and the reaction telegraph equation

on the asymptotic behavior of average energy and enstrophy in 3d turbulent flows abstract rigorous upper and lower bounds are proved for the taylor and the kolmogorov wavenumbers for the three dimensional space periodic navier stokes equations under the assumption that kolmogorov s two thirds power law holds the bounds sharpen to t gr 1 4 and gr 3 8 respectively where gr is the grashof number this provides a rigorous proof that the power law implies 1 the energy cascade 2 kolmogorov dissipation law and 3 a connection between t and the portion of phase space where a key a priori estimate on the nonlinear term is sharp is shown to be significant by means of a lower bound on any probability measure associated with an infinite time average

a fundamental limit on the heat flux in the control of incompressible channel flow this paper proves that there are no zero net wall transpiration control strategies that can sustain net heat flux below the laminar level in an incompressible channel flow with constant temperature walls the result represents a fundamental limit on the performance of a controlled nonlinear system as measured by a linear cost function over a broad class of admissible initial conditions and control inputs not a zero sum tradeoff in the frequency domain or time domain both buoyancy effects via the boussinesq approximation and viscous heating effects are accounted for and phenomenological justification for the result is also given the boundedness of solutions of the two way coupled navier stokes energy equations when both buoyancy and viscous heating are accounted for is also discussed and a new proof of existence under an appropriate small data assumption is provided

bending vibration and buckling of simply supported thick multilayered orthotropic plates an evaluation of a new displacement model abstract based upon a piecewise linear displacement field which allows the contact conditions for the displacements and the transverse shearing stresses at the interfaces to be satisfied simultaneously the non linear in the von karman sense equations of motion for thick multilayered orthotropic plates are developed successively the equations are specified to the linear boundary value problem of the bending and to the linear eigenvalue problems of the undamped vibration and buckling of rectangular plates in order to assess the accuracy of the proposed theory the sample problem of the bending free undamped vibration and buckling of a three layered symmetric cross ply square plate simply supported on all edges is investigated for purposes of comparison numerical results from the exact elasticity theory the classical lamination kirchhoff theory and the shear deformation theory timoshenko and mindlin with three different values of the shear correction factor are also presented it is found that the proposed approach is very efficient in predicting the global responses deflection natural frequencies and buckling loads of thick multilayered plates and models effects such as the distortion of the deformed normals not attainable from the classical lamination theory as well as the shear deformation theory