oscillation for solutions of nonlinear neutral differential equations with impulses abstract this paper is concerned with nonlinear neutral differential equations with impulses of the form some oscillation criteria for solutions of this equation are established an interesting example is also given which illustrates that impulses play an important role in giving rise to the oscillation of equations

oscillation for first order nonlinear delay differential equations abstract almost sharp oscillation and nonoscillation criteria are obtained for the first order nonlinear delay differential equation x t p t j 1 m x t j j sign x t 1 0 t t 0 where j 1m j 1 some applications are given

an adaptive strategy for elliptic problems including a posteriori controlled boundary approximation we derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary the numerical solution is assumed to be defined on a finite element mesh whose boundary vertices are located on the boundary of the continuous problem no assumption is made on a geometrically fitting shape a posteriori error estimates are given in the energy norm and the l 2 norm and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation furthermore a strategy is presented to compute the effect of the non discretized part of the domain on the error starting from a coarse mesh this especially implies that parts of the domain where the measured error is small stay non discretized the presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary the reliable computation of the a posteriori estimates and a mesh adaptation based on delaunay techniques numerical examples illustrate that errors outside the initial discretization will be detected

oscillation in neutral delay difference equations with variable coefficients in this paper we consider the neutral delay difference equation yn pnyn k qnyn l 0 n 0 1 2 where pn qn n 0 1 2 are real numbers with pn 0 qn 0 k and l are nonnegative integers we establish some sufficient conditions for the oscillation of all solutions of 1 without the condition n 0 qn

oscillation analysis of numerical solutions for delay differential equations with real coefficients abstract the paper is mainly concerned with the oscillations of numerical solutions by the linear methods for a kind of linear delay differential equations with positive and negative coefficients some conditions under which the numerical solutions are oscillatory are obtained and it is proved that oscillations of the analytical solutions are preserved by the numerical solutions under mild conditions numerical experiments are given to demonstrate our results

implementation of a fragmentation coagulation scattering model for the dynamics of stirred liquid liquid dispersions abstract the model for the evolution of a liquid liquid dispersion including the so called volume scattering effect introduced by the authors in a fasano f rosso a new model for the dynamics of dispersions in a batch reactor in h j bungartz r h w hoppe c zenger eds proceedings of the symposium organized by the sonderforschungsbereich 438 on the occasion of karl heinz hoffman s 60th birthday in lectures in applied mathematics springer verlag berlin 2000 pp 123 141 and incorporating multiple breakage see a fasano f rosso dynamics of droplets in an agitated dispersion with multiple breakage i formulation of the model and physical consistency math methods appl sci 28 6 2005 631 659 a fasano f rosso dynamics of droplets in an agitated dispersion with multiple breakage ii uniqueness and global existence math methods appl sci 28 9 2005 1061 1088 is considered from a viewpoint closer to applications the choice of kernels is discussed on the basis of the technical literature not just using very abstract kernels like in most mathematical papers and simulations are performed utilizing data from different sources

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

oscillation for higher order nonlinear delay differential equations in this paper we establish some oscillation criteria for n order non linear delay differential equationx n t email protected i 1 mp i t x email protected i a i 1x email protected i 0 t t 0 and the super linear n order neutral delay differential equation x t px email protected n q t x email protected a 1x email protected 0 t t 0 in the case i 1 email protected i m and a 1 respectively where n 1 is an odd integer q p email protected c t 0 0 t s t i 0

new oscillation criteria for first order nonlinear neutral delay differential equations some new oscillation criteria of first order nonlinear neutral delay differential equations with variable coefficients are given our results extend and improve some of the well known previously results in the literature some examples are considered to illustrate our main results

oscillation and nonoscillation for neutral differential equations abstract we obtain respectively sufficient conditions for the existence of a positive solution and the oscillation of all solutions of the neutral differential equation formula our results extend and improve some recent results in the literature

the existence of positive solution and bounded oscillation for even order neutral differential equations with unstable type abstract in this paper we consider the n order neutral delay differential equation with unstable type x t p t x t n q t x t 1 x t t t 0 where n 2 is an even integer 1 0 0 and p q c t 0 r we first prove that this equation always has an unbounded positive solution then some new bounded oscillation and nonoscillation criteria are obtained

oscillation theorems for first order nonlinear neutral functional differential equations abstract this paper discusses a class of first order nonlinear neutral differential equations with variable coefficients and variable deviations sharp conditions are established for all solutions of the equations to be oscillatory linearized oscillation criteria of the equations are also given

oscillations caused by several retarded and advanced arguments in this paper we study the oscillatory behavior of equations of the forms y t i 1npiy t i 0 and y t i 1npiy t i 0 where piand i i 1 2 n are positive constants we prove that each one of the following conditions 1 pi i1 e for some i i 1 2 n 2 i 1npi 1 e where min 1 2 n 3 i 1npi 1 n i 1n i 1 e or 4 1 n i 1n pi i 1 2 21 e implies that every solution of and oscillates a generalization in the case where the coefficientspi i 1 2 n are positive and continuous functions is also presented

on the oscillation of solutions and existence of positive solutions of neutral difference equations abstract we obtain sufficient conditions for the oscillation of all solutions and existence of positive solutions of the neutral difference equation x n cx n m p n x n k 0 n 0 1 2 where c and p n are real numbers m and k are integers and p n m and k are nonnegative

a time and spaceadaptive algorithm for the linear time dependent schrodinger equation we prove an a posteriori error estimate for the linear time dependent schrodinger equation in bbb r n from this we derive a residual based local error estimator that allows us to adjust the mesh and the time step size in order to obtain a numerical solution with a prescribed accuracy as a special feature the error estimator controls localization and size of the finite computational domain in each time step an algorithm is described to compute this solution and numerical results in one space dimension are included

necessary and sufficient conditions for oscillation of first order nonlinear neutral differential equations abstract in this paper we prove that every solution of the first order nonlinear neutral differential equation x t p x t q t j 1 m x t j j sign x t 1 0 t t 0 oscillates if and only if t 0 q s exp 1 ln p j 1 m j 1 s d s when j 1 m j 1 ln p 0 and t 0 q s d s when j 1 m j 1 ln p 0 where p 0 j 0 j 0 j 1 2 m q c t 0 0

local bisection refinement for n simplicial grids generated by reflection a simple local bisection refinement algorithm for the adaptive refinement of n simplicial grids is presented the algorithm requires that the vertices of each simplex be ordered in a special way re

oscillation of solutions for first order neutral differential equations with distributed deviating arguments some new sufficient conditions for oscillation and nonoscillation are obtained for first order neutral differential equations with distributed deviating arguments our conditions are sharp in the sense that when the coefficients are constants the conditions are also necessary

a robust finite element method for nonhomogeneous dirichlet problems in domains with curved boundaries in this paper the authors consider a simple finite element method on an approximately polygonal domain using linear elements the dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques their analysis takes into account the change in domain and data transfer and optimal error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable it is proved that the natural extension of this finite element approximation to the original domain is optimal order accurate

exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition the coagulation fragmentation equation describes the concentration f i t of particles of size i in nn 0 at time t geq 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break up we show that when the rate of fragmentation is sufficiently stronger than that of coalescence f i t i in nn 0 tends to an unique equilibrium as t tends to infinity although we suppose that the initial datum is sufficiently small we do not assume a detailed balance or reversibility condition the rate of convergence we obtain is furthermore exponential

asymptotic behaviour of solutions to the becker doring equations for arbitrary initial data on determine le comportement asymptotique des solutions des equations d amas de becker doring sous certaines hypotheses de decroissances sur les donnees initiales faites dans un article anterieur

metastable states for the becker doring cluster equations the becker doring equations in whichc l t can represent the concentration ofl particle clusters or droplets in say a condensing vapour at timet are begin array 20 c dc l t mathord left vphantom dc l t dt right kern nulldelimiterspace dt j l 1 t j l t l 2 3 end array with j l t a l c 1 t c l t b l 1 c l 1 t and eitherc1 const case a or rho sum limits 1 infty lc l const case b the equilibrium solutions arec l q l z l where q l prod limits 2 l a r 1 mathord left vphantom a r 1 b r right kern nulldelimiterspace b r the density of the saturated vapour defined as rho s sum limits 1 infty lq l z s l wherez s is the radius of convergence of the series is assumed finite it is proved here that subject to some further plausible conditions on the kinetic coefficientsa l andb l there is a class of metastable solutions of the equations withc1 z s small and positive which take an exponentially long time to decay to their asymptotic steady states an exponentially long time means one that increases more rapidly than any negative power of the given value ofc1 z s or in caseb s as the latter tends to zero the main ingredients in the proof are i a time independent upper bound on the solution of the kinetic equations this upper bound is a steady state solution of case a of the equations of the type used in the becker doring theory of nucleation and ii an upper bound on the total concentration of particles in clusters greater than a certain critical size which with suitable initial conditions remains exponentially small until the time becomes exponentially large

necessary and sufficient conditions for oscillation of first order nonlinear neutral difference equations in this paper we prove that every solution of the first order nonlinear neutral difference equation x n px n t q n j 1m x n s j b jsign x n s 1 0 n n 0 oscillates if and only if s n 0 q sexp t 1lnp j 1m b j 1 s when j 1 m b j 1 lnp 0 where p b j 0 t 0 and s j 0 are integers j 1 2 m q n 0 n 0

oscillation for higher order superlinear delay differential equations with unstable type abstract we first show that the n order superlinear delay differential equation with unstable type x n t p t x t 1 x t t t 0 always has an unbounded and nonoscillatory solution where n 1 is an even integer p c t0 0 0 and 1 then an almost sharp bounded oscillation and nonoscillation criterion is obtained

kinetics of helium bubble formation in nuclear materials the formation and growth of helium bubbles due to self irradiation in plutonium has been modelled by discrete kinetic equations for the number densities of bubbles having k atoms analysis of these equations shows that the bubble size distribution function can be approximated by a composite of i the solution of partial differential equations describing the continuum limit of the theory but corrected to take into account the effects of discreteness and ii a local expansion about the advancing leading edge of the distribution function in size space both approximations contribute to the memory term in a close integrodifferential equation for the monomer concentration of single helium atoms the present boundary layer theory for discrete equations is compared to the numerical solution of the full kinetic model and to the previous approximation of schalclach and wolfer involving a truncated system of moment equations

development and convergence analysis of a finite volume scheme for solving breakage equation in this paper we develop a new discretization method based on a finite volume approach for solving the breakage population balance equation the scheme is designed to conserve the total mass of the system and predict the total number of particles as well as the complete particle size distribution very accurately the new method is computationally very efficient easy to code and robust to apply on uniform and nonuniform meshes development of the method is completed by providing a detailed mathematical analysis that includes consistency and convergence of the numerical solution it is proved that the scheme is second order convergent independently of the type of mesh moreover numerical results are compared against several test cases including analytically tractable and practically oriented problems it is found that the new method predicts the complete particle size distribution and its moments with high accuracy the mathematical results of convergence analysis are also validated numerically

oscillatory properties of the first order nonlinear advanced and delayed differential inequalities on donne des conditions suffisantes pour que des inegalites differentielles retardees ou avancees aient des solutions oscillatoires

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

adaptive finite elements for exterior domain problems we present an adaptive finite element method for solving elliptic problems in exterior domains that is for problems in the exterior of a bounded closed domain in bbb r d d in 2 3 we describe a procedure to generate a sequence of bounded computational domains omega h k k 1 2 more precisely a sequence of successively finer and larger grids until the desired accuracy of the solution u h is reached to this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator furthermore we prove convergence of the adaptive algorithm numerical examples show the optimal order of convergence

oscillation of second order nonlinear neutral differential equations abstract in this paper some sufficient conditions for oscillation and nonoscillation are obtained for the second order nonlinear neutral differential equation x t p t x t q t f x t 0 t 0 where 0 p t 1 q t 0 0

the cascadic multigrid method for elliptic problems the paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of 10 from the multigrid point of view no correction cycles are needed from the cascade principle view a basic iteration method without any preconditioner is used at successive refinement levels for a prescribed error tolerance on the final level more iterations must be spent on coarser grids in order to allow for less iterations on finer grids a first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of 9 in short ccg method whichused the cg method as basic iteration method on each level in 18 it has been proven that the ccg method is accurate with optimal complexity for elliptic problems in 2d and quasi uniform triangulations the present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers moreover an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non uniform grids is worked out on the basis of a posteriori estimates numerical tests confirm the efficiency and robustness of the cascadic multigrid method

two preconditioners based on the multi level splitting of finite element spaces the hierarchical basis preconditioner and the recent preconditioner of bramble pasciak and xu are derived and analyzed within a joint framework this discussion elucidates the close relationship between both methods special care is devoted to highly nonuniform meshes exclusively local properties like the shape regularity of the finite elements are utilized

existence and uniqueness of steady state solution to a singular coagulation fragmentation equation abstract in this article we prove the existence of an equilibrium solution to the coagulation fragmentation equation with i a singular coagulation kernel in the form k x y 1 x y x y where 0 1 2 and 0 1 and ii a fragmentation kernel in the form f x y b 0 b x y m where b 0 0 and b m 0 importantly in the entire study the detailed balance condition is not assumed it is shown that the equilibrium solution of the problem under consideration is unique in the space of all continuous functions on 0 the proposed results are demonstrated by a numerical illustration

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques