caratheodory approximations and stability of solutions to non lipschitz stochastic fractional differential equations of ito doob type abstract the existence and uniqueness theorem of solutions provides an effective tool for the model validation of both deterministic and stochastic equations the objective of this paper is to establish the existence and uniqueness of solutions for a class of ito doob stochastic fractional differential equations under non lipschitz condition which is weaker than lipschitz one and contains it as a special case the solution is constructed with the aid of caratheodory approximation moreover the continuous dependence of solutions on the initial value is investigated in view of the stability of solutions in the sense of mean square finally an example is given to illustrate the theory

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 the micromagnetic energy of a uniaxial ferromagnet we identify the optimal scaling law for a nonconvex nonlocal variational problem representing the magnetic energy of a uniaxial ferromagnet our analysis is restricted to a certain parameter regime in which the surface tension is sufficiently small relative to the other parameters of the problem 1998 john wiley sons inc

ito type stochastic fuzzy differential equations with delay abstract in the paper we give some foundations for the studies of stochastic fuzzy delayed differential equations we prove the existence and uniqueness of solutions to such the equations to obtain our result we assume that the coefficients of the equation satisfy the lipschitz condition together with linear growth condition we estimate the distance between approximate solution and exact solution also the stability of solution with respect to the initial history is shown an application of stochastic fuzzy delayed differential equations in the modeling of population growth is indicated

almost sure stability of linear stochastic differential equations with jumps under the nondegenerate condition as in the diffusion case see 14 21 6 the linear stochastic jump diffusion process projected on the unit sphere is a strong feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump diffusions and the corresponding diffusions due to 22 the unique deterministic lyapunov exponent can be represented by the furstenberg khas minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign the critical case of zero lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated several examples are treated for illustration

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

properties of solutions of stochastic differential equations with continuous state dependent switching abstract this work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions the word hybrid highlights the coexistence of continuous dynamics and discrete events the underlying process has two components one component describes the continuous dynamics whereas the other is a switching process representing discrete events one of the main features is the switching component depending on the continuous dynamics in this paper weak continuity is proved first then continuous and smooth dependence on initial data are demonstrated in addition it is shown that certain functions of the solutions verify a system of kolmogorov s backward differential equations moreover rates of convergence of numerical approximation algorithms are dealt with

on the approximations of solutions to neutral sdes with markovian switching and jumps under non lipschitz conditions abstract in this paper we investigate the existence and uniqueness of solutions to neutral stochastic differential equations with markovian switching and jumps nsdewmsjs under non lipschitz conditions on the other hand we present the euler approximate solutions for nsdewmsjs and show that the convergence of the euler approximate solutions to the true solutions by applying it o formula bihari s lemma and burkholder davis gundy s lemma some examples are provided to illustrate the main results

convergence of numerical solutions to neutral stochastic delay differential equations with markovian switching recently numerical solutions of stochastic differential equations have received a great deal of attention it is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet in the paper the euler maruyama method for neutral stochastic delay differential equations is developed the key aim is to show that the numerical solutions will converge to the true solutions under the local lipschitz condition

remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay in this paper we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm f sup q 0 f q under non lipschitz condition with lipschitz condition being considered as a special case and a weakened linear growth condition the solution is constructed by the successive approximation

efficient numerical techniques for burgers equation this paper presents new efficient numerical techniques for solving one dimensional quasi linear burgers equation burgers equation is used as a model problem in the study of turbulence boundary layer behavior shock waves convection dominated diffusion phenomena gas dynamics acoustic attenuation in fog and continuum traffic simulation using a non linear cole hopf transformation the burgers equation is reduced to one dimensional diffusion equation the linearized diffusion equation is semi discretized by using method of lines mol which leads to a system of ordinary differential equations in time resulting system of ordinary differential equations is solved by backward differentiation formulas bdf of order one two and three and the analysis of numerical errors are presented numerical results for modest values of kinematic viscosity are compared with the exact solution to demonstrate the efficiency of proposed numerical methods

existence uniqueness and stability of mild solutions for second order neutral stochastic evolution equations with infinite delay and poisson jumps in this paper we study a class of second order neutral stochastic evolution equations with infinite delay and poisson jumps snseeips in which the initial value belongs to the abstract space b we establish the existence and uniqueness of mild solutions for snseeips under non lipschitz condition with lipschitz condition being considered as a special case by means of the successive approximation furthermore we give the continuous dependence of solutions on the initial data by means of a corollary of the bihari inequality an application to the stochastic nonlinear wave equation with infinite delay and poisson jumps is given to illustrate the theory

successive approximations to solutions of stochastic differential equations abstract in the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions we note that lemma 3 in this paper is a generalization of gard s lemma and guarantees the existence of functions which satisfy the conditions of theorems 2 and 3 in this paper respectively theorem 3 includes as a special case a generalization of yamada s theorem which is proved by the method of the successive approximations

pth moment asymptotic stability for neutral stochastic functional differential equations with levy processes in this paper we will consider a class of neutral stochastic functional differential equations with levy processes levy processes contain a number of very important processes as special cases such as brownian motion the poisson process stable and self decomposable processes and subordinators and so on but its sample paths are discontinuity which makes the analysis more difficult in this paper we try to get over this difficulty the contributions of this paper are as follows a we will use lyapunov functional method to study the pth moment asymptotic stability and almost sure asymptotic stability of neutral stochastic functional differential equations with levy processes b under the result of a we will investigate two types of continuity of the solution continuous in the pth moment and continuous in probability finally we provide an example to illustrate the usefulness of the obtained results

interpolation inequalities in pattern formation abstract we prove some interpolation inequalities which arise in the analysis of pattern formation in physics they are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts coarsening and branching in micromagnetics and superconductors the main ingredient in the proof of our inequalities is a geometric construction which was first used by choksi conti kohn and one of the authors in 4 in the study of branching in superconductors

robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay this paper establishes the existence and uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate this result may be used to examine almost sure robust stability to illustrate our idea more carefully we carefully discuss a scalar stochastic integro differential equation with neutral type and its asymptotic stability including the exponential stability and the polynomial stability

stability of degenerate diffusions with state dependent switching this paper deals with the ergodic properties of hybrid systems modelled by diffusion processes with state dependent switching we investigate the sufficient conditions expressed in terms of the parameters of the underlying process which would ensure the existence of a unique invariant probability and stability in distribution of the flow it turns out that the conditions would depend on certain averaging mechanisms over the states of the discrete component of the hybrid system

biorthogonal multiwavelets on the interval for numerical solutions of burgers equation to numerically solve the burgers equation in this paper we propose a general method for constructing wavelet bases on the interval 0 1 derived from symmetric biorthogonal multiwavelets on the real line in particular we obtain wavelet bases with simple structures on the interval 0 1 from the hermite cubic splines in comparison with all other known constructed wavelets on the interval 0 1 our constructed wavelet bases on the interval 0 1 from the hermite cubic splines not only have good approximation and symmetry properties with extremely short supports but also employ a minimum number of boundary wavelets with a very simple structure these desirable properties make them to be of particular interest in numerical algorithms our constructed wavelet bases on the interval 0 1 are then used to solve the nonlinear burgers equation our method is based on the finite difference formula combined with the collocation method therefore our proposed numerical scheme in this paper is abbreviated as mfdcm mixed finite difference and collocation method some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy

exponential stability in mean square of neutral stochastic differential functional equations abstract the aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d x t g xt f t x t g t xt dt t xt dw t where xt x t s s 0 with 0 is the past history of the solution several interesting examples are a given for illustration

an averaging principle for stochastic dynamical systems with levy noise abstract the purpose of this paper is to establish an averaging principle for stochastic differential equations with non gaussian levy noise the solutions to stochastic systems with levy noise can be approximated by solutions to averaged stochastic differential equations in the sense of both convergence in mean square and convergence in probability the convergence order is also estimated in terms of noise intensity two examples are presented to demonstrate the applications of the averaging principle and a numerical simulation is carried out to establish the good agreement

on the sobolev space of isometric immersions we prove that every w 2 2 isometric immersion from a convex regular domain of r 2 into r3 can be approximated in w 2 2 norm by smooth isometric immersions from the same domain into r3

impulsive neutral stochastic functional integro differential equations with infinite delay driven by fbm in this paper we study a class of impulsive neutral stochastic functional integro differential equations with infinite delay driven by a standard cylindrical wiener process and an independent cylindrical fractional brownian motion fbm with hurst parameter h 1 2 1 in the hilbert space we prove the existence and uniqueness of the mild solution for this kind of equations with the coefficients satisfying some non lipschitz conditions which include the classical lipschitz conditions as special case an example is provided to illustrate the theory some well known results are generalized and extended

mean square convergence of one step methods for neutral stochastic differential delay equations this paper deals with strong approximations of the solutions of neutral stochastic differential delay equations nsddes in ito sense a general framework for the strong convergence of a class of drift implicit one step schemes to the solutions of nsddes is established two examples to illustrate the applicability of our results are provided

an accurate approximation algorithm for burgers equation in the presence of small viscosity abstract most of the existing numerical schemes developed to solve burgers equation cannot exhibit its correct physical behavior for very small values of viscosity this difficulty can be overcome by using splitting methods derived for near integrable system this class of methods has positive real coefficients and can be used for non reversible systems such as burgers equation it also has the advantage of being able to account small viscosity in the accuracy the algorithm is based on the combination of implicit explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability the resulting algorithm is accurate efficient and easy to implement the new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity

neutral stochastic functional differential equations with additive perturbations the paper deals with the solution to the neutral stochastic functional differential equation whose coefficients depend on small perturbations by comparing it with the solution to the corresponding unperturbed equation of the equal type we give conditions under which these solutions are close in the 2m th mean on finite time intervals and on intervals whose length tends to infinity as small perturbations tend to zero

asymptotic properties of consensus type algorithms for networked systems with regime switching topologies this paper is concerned with asymptotic properties of consensus type algorithms for networked systems whose topologies switch randomly the regime switching process is modeled as a discrete time markov chain with a finite state space the consensus control is achieved by using stochastic approximation methods in the setup the regime switching process the markov chain contains a rate parameter e 0 in the transition probability matrix that characterizes how frequently the topology switches on the other hand the consensus control algorithm uses a stepsize m that defines how fast the network states are updated depending on their relative values three distinct scenarios emerge under suitable conditions we show that when 0 e o m a continuous time interpolation of the iterates converges weakly to a system of randomly switching ordinary differential equations modulated by a continuous time markov chain in this case a scaled sequence of tracking errors converges to a system of switching diffusion when 0 email protected email protected the network topology is almost non switching during consensus control transient intervals and hence the limit dynamic system is simply an autonomous differential equation when email protected email protected the markov chain acts as a fast varying noise and only its averaged network matrices are relevant resulting in a limit differential equation that is an average with respect to the stationary measure of the markov chain simulation results are presented to demonstrate these findings

stability of hybrid dynamic systems containing singularly perturbed random processes to meet the challenge of handling dynamic systems in which continuous dynamics and discrete events coexist we develop stability analysis of such systems governed by ordinary differential equations and stochastic differential equations with regime switching modulated by a markov chain involving a small parameter the small parameter is used to reflect either the in herent two time scales in the original system or the different rates of regime switching among a large number of states of the discrete events the smaller the parameter is the more rapid switching the system will experience to reduce the complexity one attempts to effectively replace the actual systems by a limit system with simpler structure to ensure the validity of such a replacement in a longtime horizon it is crucial that the original system is stable the fast regime changes and the large state space of the markov chain make the stability analysis difficult under suitable conditions using the limit dynamic systems and the perturbed lyapunov func tion methods we show that if the limit systems are stable then so are the original systems this justifies the replacement of a complicated original system by its limit from a longtime behavior point of view index terms hybrid system lyapunov method markov chain singular perturbation stability switching diffusion

stability of nonlinear regime switching jump diffusion abstract motivated by networked systems stochastic control optimization and a wide variety of applications this work is devoted to systems of switching jump diffusions treating such nonlinear systems we focus on stability issues one of the distinct features considered here is that the switching process depends on the jump diffusions first asymptotic stability in the large is obtained then the study on exponential p stability is carried out connection between almost surely exponential stability and exponential p stability is exploited also presented are smooth dependence on the initial data using the smooth dependence necessary conditions for exponential p stability are derived then criteria for asymptotic stability in distribution are provided a couple of examples are given to illustrate our results

a high order compact finite difference scheme and precise integration method based on modified hopf cole transformation for numerical simulation of n dimensional burgers system abstract this paper modifies a n dimensional hopf cole transformation to the n dimensional burgers system we obtain the n dimensional heat conduction equation through the modification of the hopf cole transformation then the fourth order precise integration method pim in combination with a spatially global sixth order compact finite difference cfd scheme is presented to solve the equation with high accuracy moreover coupling with the strang splitting method the scheme is extended to multi dimensional two three dimensional burgers system numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method moreover the two dimensional and three dimensional examples demonstrate excellent adaptability and the numerical simulation results also have very high accuracy in medium reynolds numbers

existence of global solutions and invariant measures for stochastic differential equations driven by poisson type noise with non lipschitz coefficients the purpose of this paper is twofold firstly we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi martingales secondly we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent

distributed consensus algorithms in sensor networks with imperfect communication link failures and channel noise the paper studies average consensus with random topologies intermittent links and noisy channels consensus with noise in the network links leads to the bias variance dilemma running consensus for long reduces the bias of the final average estimate but increases its variance we present two different compromises to this tradeoff the a nd algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition slowly decaying to zero and the a nc algorithm where the weights are constant but consensus is run for a fixed number of iterations iota then it is restarted and rerun for a total of p runs and at the end averages the final states of the p runs monte carlo averaging we use controlled markov processes and stochastic approximation arguments to prove almost sure convergence of a nd to a finite consensus limit and compute explicitly the mean square error mse variance of the consensus limit we show that a nd represents the best of both worlds zero bias and low variance at the cost of a slow convergence rate rescaling the weights balances the variance versus the rate of bias reduction convergence rate in contrast a nc because of its constant weights converges fast but presents a different bias variance tradeoff for the same number of iterations iota p shorter runs smaller iota lead to high bias but smaller variance larger number p of runs to average over for a static nonrandom network with gaussian noise we compute the optimal gain for a nc to reach in the shortest number of iterations iota p with high probability 1 delta epsiv delta consensus epsiv residual bias our results hold under fairly general assumptions on the random link failures and communication noise

on the class of high order time stepping schemes based on pade approximations for the numerical solution of burgers equation abstract numerical solution of burgers equation is presented using finite difference methods in space and positivity preserving pade approximations in time a class of high order time stepping schemes is introduced for practical purposes first second third and fourth order schemes are constructed efficient parallel versions of these schemes are given using a splitting technique of rational functions accuracy of the schemes is demonstrated by solving a test problem and comparing numerical results with the exact solution time evolution graphs show the physical phenomenon of the problem convergence tables are given to verify the theoretical order of convergence

existence uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay in this paper we obtain the existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay at phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm email protected sup q 0 f q under non lipschitz condition with lipschitz condition being considered as a special case and a weakened linear growth condition the solution is constructed by the successive approximation furthermore we give the continuous dependence of solutions on the initial value by means of the corollary of bihari inequality

stabilization of stochastic functional differential systems with delayed impulses abstract this paper is concerned with the stabilization problem for a class of stochastic functional differential system with delayed impulses by applying a generalized comparison principle and the average dwell time method some sufficient criteria for input to state stable iss type and exponential stability properties are derived the coefficients of the upper bound for the diffusion operator of lyapunov functions are allowed to be time varying which improves the existing results greatly two numerical examples are presented to illustrate the effectiveness of the proposed results

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

energy estimates for the von karman model of thin film blistering we consider the behavior of buckling driven thin film blisterings using von karman s plate theory our focus is on the setting where the blistered region is the unit square with clamped boundary conditions at the vertical sides and periodic ones along the horizontal sides in this setting we prove rigorous upper and lower bounds for the elastic energy which are of the same order as the film thickness we also present a convincing argument for the necessity of branching of folds near the boundary as has been observed in experiments

stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks in this paper we investigate the stochastic functional differential equations with infinite delay some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using razumikhin method and lyapunov functions based on the obtained results we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays the result extends and improves the earlier publications two examples are given to illustrate the applicability of the obtained results

galerkin least square b spline approach toward advection diffusion equation the study uses weighted residual least squares method using cubic b splines a numerical scheme is developed to solve pdes involving advection diffusion type phenomenon the validity of the scheme is tested with several model problems method is shown to be highly accurate and computationally efficient stability and error analysis are presented advection diffusion equation is one of the governing equations used for modeling solute transport processes and water quality in rivers lakes oceans and groundwater etc to simulate this class of mathematical problems a computational scheme based on weighted residual least squares method using cubic b splines is developed in this paper moreover some test cases and results obtained are discussed and compared with the available data the suggested method is shown to be accurate enough through these test examples

numerical solution of one dimesional burgers equation explicit and exact explicit finite difference methods this paper presents finite difference solution and analytical solution of the finite difference approximations based on the standard explicit method to the one dimensional burgers equation which arises frequently in the mathematical modelling used to solve problems in fluid dynamics results obtained by these ways for some modest values of viscosity have been compared with the exact fourier one it is shown that they are in good agreement with each other

second order neutral impulsive stochastic evolution equations with delay in this paper we study the second order neutral stochastic evolution equations with impulsive effect and delay snseeids we establish the existence and uniqueness of mild solutions to snseeids under non lipschitz condition with lipschitz condition being considered as a special case by the successive approximation furthermore we give the continuous dependence of solutions on the initial data by means of corollary of the bihari inequality an application to the stochastic nonlinear wave equation with impulsive effect and delay is given to illustrate the theory

large deviation principle for stochastic boussinesq equations driven by levy noise abstract the current paper is devoted to dynamics of stochastic boussinesq equation driven by levy pure jump noise the large deviation principle of solution is established and the effect of the highly nonlinear and unbounded drift is stated moreover the large deviation principle for stochastic two dimensional hydrodynamical systems with levy noise is also obtained

p moment stability of stochastic differential equations with jumps in the paper p moment stability of stochastic differential equations with jumps is considered first some preliminary notes are given then several theorems on p moment stability of these equations are established using liapunov function at last an example is present to show application of the obtained results

domain branching in uniaxial ferromagnets a scaling law for the minimum energy we address the branching of magnetic domains in a uniaxial ferromagnet our thesis is that branching is required by energy minimization to show this we consider the nonlocal nonconvex variational problem of micromagnetics we identify the scaling law of the minimum energy by proving a rigorous lower bound which matches the already known upper bound we further show that any domain pattern achieving this scaling law must have average width of order l 2 3 where l is the length of the magnet in the easy direction finally we argue that branching is required by considering the constrained variational problem in which branching is prohibited and the domain structure is invariant in the easy direction its scaling law is different

asymptotically almost periodic solutions of stochastic functional differential equations abstract in this paper we investigate a class of stochastic functional differential equations of the form dx t ax t f t x t x t dt g t x t x t dw t our main results concern the existence and exponential stability of quadratic mean asymptotically almost periodic mild solutions an example is given to illustrate our results

exponential stability of the split step θ method for neutral stochastic delay differential equations with jumps abstract the exponential mean square stability of the split step method for neutral stochastic delay differential equations nsddes with jumps is considered new conditions for jumps are proposed to ensure the exponential mean square stability of the trivial solution if the drift coefficient satisfies the linear growth condition it is shown that the split step method can reproduce the exponential mean square stability of the trivial solution for the constrained stepsize then by applying the chebyshev inequality and the borel cantelli lemma the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained since split step method covers euler maruyama em method and split step backward euler ssbe method the conclusions are valid for these two methods moreover they can adapt to the nsddes and the sddes with jumps finally a numerical example illustrates the effectiveness of the theoretical results

existence and stability of solutions to non lipschitz stochastic differential equations driven by levy noise in this paper successive approximation method is applied to investigate the existence and uniqueness of solutions to stochastic differential equations sdes driven by levy noise under non lipschitz condition which is a much weaker condition than lipschitz one the stability of solutions to non lipschitz sdes driven by levy noise is also considered and the stochastic stability is obtained in the sense of mean square

stability of regime switching jump diffusion processes abstract this work studies the stability of regime switching jump diffusion processes in a finite or a countably infinite state space some criteria with sufficient conditions for stability and instability are provided based on characterizing the stability property of the processes in any fixed state through constants under common measurements also some variational formula of these constants are given moreover some examples of nonlinear regime switching jump diffusion processes are provided to show the usefulness and sharpness of these criteria

a note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay in this note we prove the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay insfdes in short in which the initial value belongs to the phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm f sup q 0 f q under some caratheodory type conditions on the coefficients by means of the successive approximation especially we extend the results appeared in ren et al y ren s lu n xia remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay j comput appl math 220 2008 364 372 ren and xia y ren n xia existence uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay appl math comput 210 2009 72 79 and zhou and xue s zhou m xue the existence and uniqueness of the solutions for neutral stochastic functional differential equations with infinite delay math appl 21 2008 75 83

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

exponential stability for second order neutral stochastic differential equations with impulses in this paper we investigate the problem on the exponential stability of mild solution for the second order neutral stochastic partial differential equations with impulses by utilising the cosine function theory a set of novel sufficient conditions is derived by establishing an impulsive integral inequality as a final point an example is given to illustrate the effectiveness of the obtained theory

a difference scheme for burgers equation in an unbounded domain the numerical solution of the one dimensional burgers equation in an unbounded domain is considered two artificial boundaries are introduced to make the computational domain finite on both artificial boundaries two exact boundary conditions are proposed respectively to reduce the original problem to an initial boundary value problem in a finite computational domain a difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain at each time level only a strictly diagonal dominated tridiagonal system of linear algebraic equations needs to be solved it is proved that the difference scheme is uniquely solvable and unconditional convergent with the convergence order 3 2 in time and order 2 in space in an energy norm a numerical example demonstrates the theoretical results

stability of regime switching jump diffusions this work is concerned with the stability of a class of switching jump diffusion processes the processes under consideration can be thought of as a number of jump diffusion processes modulated by a random switching device the motivation of our study stems from a wide range of applications in communication systems flexible manufacturing and production planning financial engineering and economics a distinct feature of the two component process x t alpha t considered in this paper is that the switching process alpha t depends on the x t process this paper focuses on the long time behavior namely stability of the switching jump diffusions first the definitions of regularity and stability are recalled next it is shown that under suitable conditions the underlying systems are regular or have no finite explosion time to study stability of the trivial solution or the equilibrium point 0 systems that are linearizable in the x variable in a neighborhood of 0 are considered sufficient conditions for stability and instability are obtained then almost sure stability is examined by treating a lyapunov exponent the stability conditions present a gap for stability and instability owing to the maximum and minimal eigenvalues associated with the drift and diffusion coefficients to close the gap a transformation technique is used to obtain a necessary and sufficient condition for stability

asymptotic properties of stochastic hybrid gilpin ayala system with jumps this paper focuses on studying the dynamics of the stochastic gilpin ayala model under regime switching with jumps the aim is to analyze what happens under the perturbations of regime switching and jumps some asymptotic properties are investigated and sufficient conditions for stochastic permanence extinction non persistence in the mean and weak persistence are provided the critical value among the extinction non persistence in the mean and weak persistence is obtained our results demonstrate that the dynamics of the model have close relations with the jumps and the stationary distribution of the markov chain

on a finite difference scheme for burgers equation in this paper we study properties of numerical solutions of burger s equation burgers equation is reduced to the heat equation on which we apply the douglas finite difference scheme the method is shown to be unconditionally stable fourth order accurate in space and second order accurate in time two test problems are used to validate the algorithm numerical solutions for various values of viscosity are calculated and it is concluded that the proposed method performs well

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

pth moment asymptotic stability of stochastic delayed hybrid systems with levy noise the problem of pth moment asymptotic stability analysis is considered for stochastic delayed hybrid systems with levy noise by virtue of ito s formula and m matrix theories we propose some sufficient conditions to guarantee the asymptotic stability and exponential stability of the system the criterion of mean square asymptotic stability is derived as well for delayed neural networks with levy noise a numerical example is provided to show the usefulness of the proposed asymptotic stability criterion

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

an exponentially fitted method for solving burgers equation in this paper an exponentially fitted method is used to numerically solve the one dimensional burgers equation the performance of the method is tested on the model involving moderately large reynolds numbers the obtained numerical results show that the method is efficient stable and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails copyright 2009 john wiley sons ltd

practical exponential stability of stochastic age dependent capital system with levy noise abstract it is well known that stability is one of most important topics in economy and control in this paper we focus on the practical stability problem for a class of stochastic age dependent vintage capital system with levy noise compared with the classical lyapunov stability theory practical stability can depict not only qualitative behavior but also quantitative properties such as specific trajectory bounds and specific transient behavior with the help of the stochastic analysis theory and ito s formula we establish some novel criteria for the mean square and almost sure practical exponential stability of the addressed system the obtained results improve some previous works given in the literature moreover an example is provided to illustrate the theoretical results

a numerical method based on crank nicolson scheme for burgers equation abstract in this paper we present a solution based on crank nicolson finite difference method for one dimensional burgers equation burgers equation arises frequently in mathematical modeling of problems in fluid dynamics hopf cole transformation e hopf the partial differential equation u t uu x u xx commun pure appl math 3 1950 201 230 j d cole on a quasilinear parabolic equation occurring in aerodynamics quart appl math 9 1951 225 236 is used to linearize burgers equation the resulting heat equation is discretized by using crank nicolson finite difference scheme this method is shown to be unconditionally stable and second order accurate in space and time numerical results obtained by the present method have been compared with exact solution for different values of reynolds number

the existence and uniqueness of the solution for stochastic functional differential equations with infinite delay abstract this paper is devoted to build the existence and uniqueness theorem of solutions to stochastic functional differential equations with infinite delay short for isfdes at phase space bc 0 r d under the uniform lipschitz condition the linear growth condition is weaked to obtain the moment estimate of the solution for isfdes furthermore the existence and uniqueness theorem of the solution for isfdes is derived and the estimate for the error between approximate solution and accurate solution is given on the other hand under the linear growth condition the uniform lipschitz condition is replaced by the local lipschitz condition the existence and uniqueness theorem is also valid for isfdes on t 0 t moreover the existence and uniqueness theorem still holds on interval t 0 where t 0 r is an arbitrary real number

on the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps this paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps by the ito formula stochastic inequality and semi martingale convergence theorem we study the asymptotic stability in the p th moment and almost sure exponential stability of solutions under the local lipschitz condition and nonlinear growth condition on the other hand we also show the convergence in probability of numerical schemes under nonlinear growth condition finally an example is provided to illustrate the theory

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

successive approximation to solutions of stochastic differential equations with jumps in local non lipschitz conditions in this paper by using successive approximation the existence and uniqueness of initial value problem for stochastic differential equations driven by both wiener process and poisson process are studied under a local non lipschitz conditions moreover the numerical solutions are shown to converge uniformly to the analytical solutions of the stochastic differential equation with jumps

jump diffusion processes in random environments abstract in this paper we investigate jump diffusion processes in random environments which are given as the weak solutions of sdes we formulate conditions ensuring existence and uniqueness in law of solutions we investigate the markov property to prove uniqueness we solve a general martingale problem for cadlag processes this result is of independent interest application of our results to generalized exponential levy model are present in the last section

numerical solution of burgers equation by cubic hermite collocation method abstract in this paper numerical solution of the non linear burgers equation are obtained by using cubic hermite collocation method chcm the advantage of the method is continuity of the dependent variable and its derivative throughout the solution range a linear stability analysis shows that the numerical scheme based on crank nicolson approximation in time is unconditionally stable this method is applied on some test problems with different choice of collocation points to validate the accuracy of the method the obtained numerical results show that the method is efficient robust and reliable even for high reynolds numbers for which the exact solution fails moreover the method can be applied to a wide class of nonlinear partial differential equations

a note on the neutral stochastic functional differential equation with infinite delay and poisson jumps in an abstract space in this note we prove the existence and uniqueness of the solution to neutral stochastic functional differential equation with infinite delay and poisson jumps in the phase space b with non lipschitz coefficients the known results that appeared in the studies by ren and xia appl math comput 210 72 2009 ren and xiaappl math comput 214 457 2009 and xu and hu acta appl math unpublished are generalized and improved an example is provided to illustrate the theory

stochastic fractional differential equations driven by levy noise under caratheodory conditions in this manuscript we initiate a study on a class of stochastic fractional differential equations driven by levy noise the existence and uniqueness theorem of solutions to equations of this class is established under global and local caratheodory conditions our analysis makes use of the caratheodory approximation as well as a stopping time technique the results obtained here generalize the main results from pedjeu and ladde chaos solitons fractals 45 279 293 2012 xu et al appl math comput 263 398 409 2015 and abouagwa et al appl math comput 329 143 153 2018 finally an application to the stochastic fractional burgers differential equations is designed to validate the theory obtained in this manuscript we initiate a study on a class of stochastic fractional differential equations driven by levy noise the existence and uniqueness theorem of solutions to equations of this class is established under global and local caratheodory conditions our analysis makes use of the caratheodory approximation as well as a stopping time technique the results obtained here generalize the main results from pedjeu and ladde chaos solitons fractals 45 279 293 2012 xu et al appl math comput 263 398 409 2015 and abouagwa et al appl math comput 329 143 153 2018 finally an application to the stochastic fractional burgers differential equations is designed to validate the theory obtained

a model for stochastic hybrid systems with application to communication networks abstract we propose a model for stochastic hybrid systems shss where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuous time markov chains however the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the shs based on results available for piecewise deterministic markov process pdps we provide a formula for the extended generator of the shs which can be used to compute expectations and the overall distribution of the state as an application we construct a stochastic model for on off tcp flows that considers both the congestion avoidance and slow start modes and takes directly into account the distribution of the number of bytes transmitted using the tools derived for shss we model the dynamics of the moments of the sending rate by an infinite system of odes which can be truncated to obtain an approximate finite dimensional model this model shows that for transfer size distributions reported in the literature the standard deviation of the sending rate is much larger than its average moreover the later seems to vary little with the probability of packet drop this has significant implications for the design of congestion control mechanisms