oscillation of certain neutral delay parabolic equations abstract sufficient conditions are given to ensure the solutions of a class of neutral parabolic equations with continuous distributed deviating arguments to oscillate

dirichlet inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following complex ginzburg landau equation with cubic nonlinearity on r n for t 0 u t a i u b i u 2 u a b 0 under initial and dirichlet boundary conditions u x 0 h x for x u x t q x t on where h q are given smooth functions under suitable conditions we prove the existence of a global solution in h1 further this solution approaches to the solution of the nls limit under identical initial and boundary data as a b 0

streamwise upwind algorithm for computing unsteady transonic flows past oscillating wings a streamwise upwind algorithm has been extended to compute unsteady flows with a moving grid system and applied to compute flows over oscillating wings at transonic mach numbers comparisons have been made between results obtained from this upwind algorithm using both temporally nonconservative and conservativeimplicit methods with the results obtained from a central difference method and also with experimental data the efficiency and practicality of the temporally nonconservative implicit solver are illustrated results to show the robustness and accuracy of the upwind method for unsteady computations compared to the central difference method are presented

neumann inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following neumann inhomogeneous boundary value problem for the complex ginzburg landau equation on r n n 3 u t a i u b i u 2 u a b t 0 under initial condition u x 0 h x for x and neumann boundary condition u n k x t on where h k are given functions under suitable conditions we prove the existence of a global solution in h1

integral average method for oscillation of second order partial differential equations with delays using a generalized riccati transformation some new oscillation criteria for second order partial differential equations with delays are found through the method of integral average these results can be considered as generalizations and improvements of the results due to kamenev in ordinary differential equations cases

a dirichlet inhomogeneous boundary value problem for a generalized ginzburg landau equation abstract the ginzburg landau equation has been used as a simplified mathematical model for various pattern formation systems in mechanics physics and chemistry in this paper we study a generalized complex ginzburg landau equation in two spatial dimension with a fifth order nonlinear term and cubic terms containing spatial derivatives we prove a global existence and uniqueness theorem

second order accurate spatial differencing for the transonic small disturbance equation current methods for calculating transonic flows with the small disturbance potential equation are typically only first order accurate in the supersonic regions of the flow however calculations using the full potential equation show significant improvements in accuracy when second order methods are used instead of first order methods in this paper algorithms for the small disturbance equations are presented for both steady and unsteady flows with spatial differencing that is second order accurate in both subsonic and supersonic regions of the flow these algorithms are stable simple extensions of implicit monotone algorithms the former are only first order accurate spatially in the supersonic regions several calculations in two dimensions have been made and the first and second order calculations are compared the comparisons include steady flow over a korn airfoil and unsteady flow over a pitching airfoil all of the comparisons show that the new methods are just as stable numerically as the old methods but improve the accuracy of the solutions the algorithm improvements can be implemented in present computer codes by making minor coding modifications i introduction c urrent methods1 for calculating transonic flows with the small disturbance potential equation are typically only first order accurate spatially in the supersonic regions of the flow however calculations using the full potential equation2 3 show significant improvements in accuracy when second order methods are used in ref 3 the results show significant improvements in capturing the shock waves and in the resulting values of the lift and drag coefficients in this paper algorithms are presented that have second orderaccurate spatial differencing for the small disturbance equation these algorithms are stable simple extensions of implicit monotone algorithms4 the previous algorithms are only first order spatially accurate in supersonic regions the stability of the second order methods is achieved by a judicious combining of monotone computational modules including the time derivatives the simplicity of the firstorder methods is retained by using only the simple switches from those monotone methods in this paper several calculations in two dimensions are presented to demonstrate the properties of the new methods comparisons are made with first order methods to demonstrate the improved accuracy there are several comparisons of calculations for steady and unsteady transonic flows the comparison of flow over a korn airfoil shows that the second order method captures a double shock whereas the results of the first order method show only one shock the calculations demonstrate that the new methods are just as stable as the old ones with regard to the convergence rates obtained and to the acceleration parameters and time steps employed these improvements in accuracy can be implemented in current computer codes by making only minor coding changes

a neumann boundary value problem for a generalized ginzburg landau equation we study the following generalized 1d ginzburg landau equation on w 0 x 0 u t 1 i m u x x a 1 ib 1 u 2u x a 2 ib 2 u 2u x 1 i n u 4u with initial and neumann boundary conditions u x 0 h x u x 0 t p t under suitable conditions we prove that there is a unique weak solution that exists for all time

finite dimensional behaviour for the derivative ginzburg landau equation in two spatial dimensions abstract in this paper we study the long time behaviour of the solutions for the generalized derivative ginzburg landau equation in two spatial dimensions we show the existence of the global attractor for this equation and establish the estimates of the upper bounds of hausdorff and fractal dimensions for the global attractor

inviscid limits of the complex ginzburg landau equation in the inviscid limit the generalized complex ginzburg landau equation reduces to the nonlinear schrodinger equation this limit is proved rigorously with h1 data in the whole space for the cauchy problem and in the torus with periodic boundary conditions the results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of l2 in the focusing case in any spatial dimension furthermore optimal convergence rates are proved the proofs are based on estimates of the schrodinger energy functional and on gagliardo nirenberg inequalities

forced oscillation of solutions of certain hyperbolic equations of neutral type some oscillation properties of solutions of certain hyperbolic equations of neutral type are studied and sufficient conditions for the oscillation of the solutions of a boundary value problem in a cylindrical domain are given

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

oscillation of solutions of neutral partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of neutral partial functional differential equations of the form t p t t u x t i 1 l i t u x t i a t u x t k 1 s a k t u x t k t q x t u x t j 1 m q j x t f j u x t j x t 0 g where is a bounded domain inrnwith a piecewise smooth boundary and is the laplacian in the euclideann spacern

on the initial value problem for the generalized two dimensional ginzburg landau equation abstract the ginzburg landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics physics and chemistry most work so far concentrates on ginzburg landau type equations with one spatial dimension 1d in this paper the authors study a complex generalized ginzburg landau equation with two spatial dimensions 2d sufficient conditions for the existence and uniqueness of global solutions for the initial value problem of the generalized 2d ginzburg landau equation are obtained this rigorously establishes the foundation for further investigation of this type of model

adomian s decomposition method for eigenvalue problems we extend the adomian s decomposition method to work for the general eigenvalue problems in addition to the existing applications of the method to boundary and initial value problems with nonlinearity we develop the hamiltonian inverse iteration method which will provide the ground state eigenvalue and the explicit form eigenfunction within a few iterations the method for finding the excited states is also proposed we present a space partition method for the case that the usual way of series expansion failed to converge

note on loss of regularity for solutions of the 3 d incompressible euler and related equations one of the central problems in the mathematical theory of turbulence is that of breakdown of smooth indefinitely differentiable solutions to the equations of motion in 1934 j leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3 d incompressible navier stokes equations the problem is still open we show in this report that breakdown of smooth solutions to the 3 d incompressible slightly viscous i e corresponding to high reynolds numbers or highly turbulent navier stokes equations cannot occur without breakdown in the corresponding solution of the incompressible euler ideal fluid equation we prove then that solutions of distorted euler equations which are equations closely related to the euler equations for short term intervals do breakdown

higher order accuracy for upwind methods by using the compatibility equations a new algorithm has been developed for obtaining higher order accuracy in upwind schemes for the euler and navier stokes equations in this method the compatibility relations for the euler equations are used to construct formulas for the higher order interpolates by using these formulas computed results are obtained for steady inviscid flow through a nozzle and also steady in viscid and viscous flow over an airfoil this approach provides an alternative to total variati on diminishing and essentially nonoscillatory schemes

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

necessary and sufficient conditions for oscillation of neutral delay parabolic differential equations necessary and sufficient conditions are established for the oscillation of neutral delay parabolic differential equations of the form t u x t pu x t t a t du x t k 1 sa k t du x t r k t j 1 mq j t u x t s j t x t wx 0 g where w is a bounded domain in r n with a piecewise smooth boundary w and d is the laplacian in euclidean n space r n

h oscillation of solutions of certain vector hyperbolic differential equations with deviating arguments in this paper sufficient conditions are established for h oscillation of solutions of a class of vector hyperbolic differential equations with deviating arguments where h is a unit vector in r m the main results are illustrated by two examples

an inhomogeneous boundary value problem for nonlinear schrodinger equations abstract we consider a nonlinear schrodinger equation in a domain r n with the inhomegeneous dirichlet boundary condition u q where q is a given smooth function the nonlinear term contributes a positive term to the energy we prove the existence of global solutions of finite energy

asymptotics for the generalized two dimensional ginzburg landau equation abstract in this paper the authors have studied a generalized ginzburg landau equation in two spatial dimensions 2d they have shown that this equation under periodic boundary conditions has the maximal attractor with finite hausdorff dimension this rigorously establishes the foundation for further investigation of this type of model

global existence of solutions to the derivative 2d ginzburg landau equation abstract in this paper we study a complex derivative ginzburg landau equation with two spatial variables 2d we obtain sufficient conditions for the existence and uniqueness of global solutions for the initial boundary value problem of the derivative 2d ginzburg landau equation and improve the known results

forced oscillation for certain systems of hyperbolic differential equations in this paper sufficient conditions are established for the forced oscillation of certain systems of hyperbolic differential equations with deviating arguments the main result is illustrated by an example

numerical study of lorenz s equation by the adomian method abstract in this paper we use the decomposition method of adomian for solving differential systems coming from physics meteorology we also give a comparison between the runge kutta method and the decomposition technique furthermore we reconfirm the famous butterfly effect

monotone switches in implicit algorithms for potential equations applied to transonic flows numerical calculations of transonic flows by potential equations typically use algorithms that change the method of calculation for regions of subsonic and supersonic flow in this paper implicit approximatefactorization algorithms are modified to use the monotonic switch in the type of finite differencing that was developed by godunov for the euler equations calculations of flows over airfoils by these algorithms are compared with calculations by other commonly used methods for the small disturbance potential equation comparisons are made with the murman cole method for both steady and unsteady flows for the full potential equation comparisons are made with the method of hoist and ballhaus for steady flows the comparisons show that the monotone methods are more stable for steady flows converged solutions are obtained for cases where the older methods fail for unsteady flows solutions are obtained for cases where the murman cole switch requires a time step more than ten times smaller in order for the calculations to remain stable these improvements are achieved with no increase in computer storage and only minor modifications in current codes

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

the ginzburg landau equation posed in a quarter plane abstract we study the following ginzburg landau equation gl ut i uxx i u 2u u 0 0 0 for a full line problem with u x 0 u0 x h2 global existence uniqueness is established for a half line problem with u x 0 u0 x h2 0 u 0 t q t c2 0 u0 0 q 0 the following results are available 1 local existence uniqueness 2 criteria for the existence of a small amplitude solution on any finite interval by means of small initial and boundary data 3 global existence in the case 3 or 0

length scales in solutions of the complex ginzburg landau equation abstract we generalise and in certain cases improve on previous a priori estimates of sobolev norms of solutions to the generalised complex ginzburg landau equation a set of dynamic length scales based on ratios of these norms is defined we are able to derive lower bounds for time averages and long time limits of these length scales the bounds scale like the inverses of our l bounds

oscillation properties for systems of hyperbolic differential equations of neutral type abstract sufficient conditions are established for the oscillations of systems of hyperbolic differential equations of the form 2 t 2 p t u i x t r 1 d r t u i x t r a i t u i x t j 1 m k 1 s a ijk t u j x k t q i x t u i x t j 1 m h 1 l q ijh x t u j x h t x t 0 g i 1 2 m where is a bounded domain in rn with a piecewise smooth boundary and is the laplacian in euclidean n space rn