random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations abstract this paper first introduces the so called quasi continuous random dynamical system rds on a separable banach space the quasi continuity is weaker than all the usual continuities and thus is easier to check in practice we then establish a necessary and sufficient condition for the existence of random attractors for the quasi continuous rds we also give a general method to obtain the random attractors for the rds on the banach space l q d for q 2 as an application it is shown that the rds generated by the stochastic reaction diffusion equation possesses a finite dimensional random attractor in l q d for any q 2 a comparison result of fractal dimensions under the different l q norms is also obtained

the method of fundamental solutions for the numerical solution of the biharmonic equation abstract the method of fundamental solutions mfs is a relatively new technique for the numerical solution of certain elliptic boundary value problems it falls in the class of methods generally called boundary methods and like the well known boundary integral equation method is applicable when a fundamental solution of the differential equation is known in the mfs the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem the locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible in many applications these quantities are determined by a least squares fit of the boundary conditions a nonlinear problem which is solved using standard software in this paper the mfs is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow

the method of regularized stokeslets a numerical method for computing stokes flows in the presence of immersed boundaries and obstacles is presented the method is based on the smoothing of the forces leading to regularized stokeslets the resulting expressions provide the pressure and velocity field as functions of the forcing the latter expression can also be inverted to find the forces that impose a given velocity boundary condition the numerical examples presented demonstrate the wide applicability of the method and its properties solutions converge with second order accuracy when forces are exerted along smooth boundaries examples of segmented boundaries and forcing at random points are also presented

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

pullback attractors for the non autonomous fitzhugh nagumo system on unbounded domains abstract the existence of a pullback attractor is established for the singularly perturbed fitzhugh nagumo system defined on the entire space r n when external terms are unbounded in a phase space the pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far field values of solutions although the limiting system has no global attractor we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded and hence do not blow up as a small parameter approaches zero

probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion the proliferation and migration dichotomy of the tumor cell invasion is examined within a two component continuous time random walk ctrw model the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration are derived the transport of tumor cells is formulated in terms of the ctrw with an arbitrary waiting time distribution law while proliferation is modeled by a logistic growth the overall rate of tumor cell invasion for normal diffusion and subdiffusion is determined

on the sources placement in the method of fundamental solutions for time dependent heat conduction problems abstract the method of fundamental solutions is a more and more popular meshless method for solving boundary or initial boundary value problems the most important issue in this method is the determination of the positions of the source points the accuracy of the method depends strongly on the distribution of the source points in this paper placement of these points for transient heat conduction problems is studied the problems are initial boundary value problems and they are considered in a time space domain because of that the placement of the source points differs from the classical distribution of the source points for boundary values problems in the paper four different possible sources distributions are considered for 1d 2d and 3d transient heat conduction problems the results show very good accuracy in case of the source points placed in a space much bigger than the considered region additionally with the negative time coordinate

numerical computations of fractional nonlinear hartmann flow with revised heat flux model abstract in this communication we have examined the magneto hydrodynamic flow of viscoelastic liquid with revise version of thermal flux thermal conductivity of the fluid is variable effects of body forces such as gravity are also examined in terms of convection non integer time derivatives are used for the better understanding of viscoelastic flow characteristics in literature these derivatives have been proved suitable for the predictions of hereditary systems such as viscoelastic flows heat transfer is analyzed in the flow field via modified fractional thermal gradient temperature gradient exists between the flow boundaries and increases with the passage of time flow is generated by the variable movement of flow boundary all the segments of numerical modeling are completely addressed in this paper i e physical system mathematical model simulation prediction validation and verification unsteady motion of in compressible fluid is directed by differential equations with fractional partial derivatives for the solutions of these equations numerical techniques are necessary to use governing equations with appropriate flow conditions have been discretized by finite difference along with finite element algorithms theoretical error appraisals are hypothesized and created numerical plan has been approved by doing error investigations numerically regarding spatial directions we have shown the estimate error and plotted the error curves in logarithmic scales of l 2 and h 1 norms by taking absolute approximations of the space discretization local nusselt number and coefficients of skin friction are determined for non integer viscoelastic model flow field is simulated for various values of physical parameters the obtained results showed that fractional exponent has opposite effects on the velocity temperature profiles for time t 2 it is also noted that temperature gradient increases by the increase of fractional number and thermal relaxation parameter viscoelastic flows in polymer and plastics industries can be addressed by similar technique

dynamics for a stochastic reaction diffusion equation with additive noise abstract in this paper we present a new scheme to study the dynamics of a stochastic reaction diffusion equation with the nonlinearity satisfying a dissipative condition with polynomial growth of arbitrary order p 2 firstly we use this scheme to establish some new estimates higher order integrability of the difference of the solutions near the initial time instead of using the usual estimates about higher regularities and higher order integrability of solutions secondly we verify that the attraction in the usual l 2 q l 2 q d pullback random attractor indeed can be l 2 norm for any 0 and the solutions of the equation are continuous in h 0 1 q with respect to initial data thirdly we obtain the existence of pullback random attractor in h 0 1 q as an application of the continuity even for the corresponding deterministic cases the results and methods introduced in this paper are new

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

generalized cahn hilliard equation for biological applications recently we considered a stochastic discrete model which describes fronts of cells invading a wound e khain l m sander and c m schneider mizell j stat phys 128 209 2007 in the model cells can move proliferate and experience cell cell adhesion in this work we focus on a continuum description of this phenomenon by means of a generalized cahn hilliard equation gch with a proliferation term as in the discrete model there are two interesting regimes for subcritical adhesion there are propagating pulled fronts similar to those of the fisher kolmogorov equation the problem of front velocity selection is examined and our theoretical predictions are in the good agreement with a numerical solution of the gch equation for supercritical adhesion there is a nontrivial transient behavior where density profile exhibits a secondary peak to analyze this regime we investigated relaxation dynamics for the cahn hilliard equation without proliferation we found that the relaxation process exhibits self similar behavior the results of continuum and discrete models are in good agreement with each other for the different regimes we analyzed

kinetic equations for reaction subdiffusion systems derivation and stability analysis we derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by vlad and ross phys rev e 66 061908 2002 reaction and diffusion processes are separable in a typical reaction diffusion system and their combined influence on the evolution of the density of a species is a simple sum our derivation shows that this is no longer true for subdiffusive entities undergoing reactions the strong memory effects in the transport process i e the non markovian nature of subdiffusion results in a nontrivial combination of reactions and spatial dispersal which we discuss in detail we carry out a linear stability analysis of the derived reaction subdiffusion system to understand the effects of memory on pattern formation we find that the turing instability persists in the subdiffusive system however the memory modifies the turing threshold and the characteristics of the band of unstable modes close to this threshold

toy model of fractional transport of cancer cells due to self entrapping a simple mathematical model is proposed to study the influence of cell fission on transport the model describes fractional in time tumor development which is a one dimensional continuous time random walk the model is relevant for consideration of both solid and diffusive cancers

pullback attractors of non autonomous stochastic degenerate parabolic equations on unbounded domains abstract this paper is concerned with pullback attractors of the stochastic p laplace equation defined on the entire space r n we first establish the asymptotic compactness of the equation in l 2 r n and then prove the existence and uniqueness of non autonomous random attractors this attractor is pathwise periodic if the non autonomous deterministic forcing is time periodic the difficulty of non compactness of sobolev embeddings on r n is overcome by the uniform smallness of solutions outside a bounded domain

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

non markovian random processes and traveling fronts in a reaction transport system with memory and long range interactions the problem of finding the propagation rate for traveling waves in reaction transport systems with memory and long range interactions has been considered our approach makes use of the generalized master equation with logistic growth hyperbolic scaling and hamilton jacobi theory we consider the case when the waiting time distribution for the underlying microscopic random walk is modeled by the family of gamma distributions which in turn leads to non markovian random processes and corresponding memory effects on mesoscopic scales we derive formulas that enable us to determine the front propagation rate and understand how the memory and long range interactions influence the propagation rate for traveling fronts several examples involving the gaussian and discrete distributions for jump densities are presented

the method of fundamental solutions for axisymmetric potential problems in this paper we investigate the application of the method of fundamental solutions mfs to two classes of axisymmetric potential problems in the first the boundary conditions as well as the domain of the problem are axisymmetric and in the second the boundary conditions are arbitrary in both cases the fundamental solutions of the governing equations and their normal derivatives which are required in the formulation of the mfs can be expressed in terms of complete elliptic integrals the method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem copyright 1999 john wiley sons ltd

pullback attractors for the norm to weak continuous process and application to the nonautonomous reaction diffusion equations abstract in this paper we introduce the concept of norm to weak continuous process in a banach space and obtain the existence of pullback attractors for this kind of process then we give a new method for proving the existence of the pullback attractors as an application we obtain the existence of pullback attractors for nonautonomous reaction diffusion equation in h 0 1 with exponential growth of the external force

numerical method for oseen s linearized equations in three dimensional exterior domains in this paper we consider oseen s linearized equations which appear in fluid mechanics in three dimensional exterior domains and propose a new numerical method based on fundamental solution method coupled with collocation method to solve this problem we show the effectiveness of our method by numerical experiments

short note the method of fundamental solutions for 2d and 3d stokes problems a numerical scheme based on the method of fundamental solutions mfs is proposed for the solution of 2d and 3d stokes equations the fundamental solutions of the stokes equations stokeslets are adopted as the sources to obtain flow field solutions the present method is validated through other numerical schemes for lid driven flows in a square cavity and a cubic cavity test results obtained for a rectangular cavity with wave shaped bottom indicate that the mfs is computationally efficient than the finite element method fem in dealing with irregular shaped domain the paper also discusses the effects of number of source points and their locations on the numerical accuracy

continuous time random walks and traveling fronts we present a geometric approach to the problem of propagating fronts into an unstable state valid for an arbitrary continuous time random walk with a fisher kolmogorov petrovski piskunov growth reaction rate we derive an integral hamilton jacobi type equation for the action functional determining the position of reaction front and its speed our method does not rely on the explicit derivation of a differential equation for the density of particles in particular we obtain an explicit formula for the propagation speed for the case of anomalous transport involving non markovian random processes

reaction subdiffusion equations to analyze possible generalizations of reaction diffusion schemes for the case of subdiffusion we discuss a simple monomolecular conversion a ensuremath rightarrow b we derive the corresponding kinetic equations for the local a and b concentrations their form is rather unusual the parameters of the reaction influence the diffusion term in the equation for a component a a consequence of the non markovian nature of subdiffusion the equation for the product contains a term which depends on the concentration of a at all previous times our discussion shows that reaction subdiffusion equations may not resemble the corresponding reaction diffusion ones and are not obtained by a trivial change of the diffusion operator for a subdiffusion one

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

analysis of carreau fluid flow between corrugated plates the paper deals with a problem of carreau fluid flow between corrugated plates a meshless numerical procedure for solving nonlinear governing equation is constructed using the picard iteration method in combination with the method of fundamental solutions and the radial basis functions the dimensionless average velocity and the product of friction factor and reynolds number are calculated for different values of the fluid model and parameters of the considered region

continuity and random dynamics of the non autonomous stochastic fitzhugh nagumo system on rn abstract in this article we use the so called difference estimate method to investigate the continuity and random dynamics of the non autonomous stochastic fitzhugh nagumo system with a general nonlinearity firstly under weak assumptions on the noise coefficient we prove the existence of a pullback attractor in l 2 r n l 2 r n by using the tail estimate method and a certain compact embedding on bounded domains secondly although the difference of the first component of solutions possesses at most p times integrability where p is the growth exponent of the nonlinearity we overcome the absence of higher order integrability and establish the continuity of solutions in l p r n h 1 r n l 2 r n with respect to the initial values belonging to l 2 r n l 2 r n as an application of the result on the continuity the existence of a pullback attractor in l p r n h 1 r n l 2 r n is proved for arbitrary n 1 and p 2

regularity of random attractors for a degenerate parabolic equations driven by additive noises abstract we study the regularity of random attractors for a class of stochastic degenerate parabolic equations with the leading term involving a diffusion variable which many be non smooth or unbounded without any restrictions on the upper growth order p of the nonlinearity except that p 2 we show that the associated random dynamical system admits a unique compact random attractor in the space d 0 1 2 d n l d n for any 2 2 p 2 where d n is an arbitrary bounded or unbounded domain in r n n 2

persistent random walk of cells involving anomalous effects and random death the purpose of this paper is to implement a random death process into a persistent random walk model which produces sub ballistic superdiffusion levy walk we develop a stochastic two velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction it is assumed that the switching rate is a decreasing function of residence running time this assumption leads to the power law for the velocity switching time distribution this describes the anomalous persistence of cell motility the longer the cell moves in one direction the smaller the switching probability to another direction becomes we derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives we show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process in the long time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a levy jump distribution we find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death rate dependent diffusion coefficient monte carlo simulations confirm these bounds

laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions abstract the paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube cross section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations e g inside arteries the boundary value problem is described by two partial differential equations one for fluid flow problem and one for heat transfer problem and appropriate boundary conditions during solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically in the paper the nusselt number and the product of friction factor and reynolds number are presented for some selected geometrical parameters the number and amplitude of corrugations it is shown that for a given number of corrugations a minimal value of the product of friction factor and reynolds number can be found as it was expected the nusselt number increases with increasing amplitude and number of corrugations

continuity of strong solutions of the reaction diffusion equation in initial data abstract this paper is concerned with the continuity of strong solutions of the reaction diffusion equation with respect to initial conditions it is proved that the solutions of the equation are continuous in h 1 with respect to initial data provided the nonlinearity satisfies a dissipative condition with polynomial growth order p 2