on the propagation of binary signals in damped mechanical systems of oscillators in the present work we explore efficient ways to transmit binary information in discrete semi infinite chains of coupled oscillators using the process of nonlinear supratransmission a previous work showed that such transmission is possible and indeed reliable under the idealistic condition when weak or no damping is present in this paper we study a more realistic case and propose the design of mechanical devices in order to avoid the loss of information consisting on the linear concatenation of several such mechanical systems our results demonstrate that the loss of information can be minimized or avoided using such physical structures

anomalous continuous time random walk spectral diffusion in coherent third order optical response recursive relations are derived for nonlinear optical response functions of a two level chromophore with stochastic frequency fluctuations described by a continuous time random walk stationary ensembles are constructed and signatures of anomalous relaxation in the photon echo signal are discussed for a two state jump modulation with a power law waiting time density function psi t approximately t alpha 1 stretched exponential decay of the photon echo signal is predicted for 0 alpha 1 and power law asymptotics for 1 alpha 2

boundary control problem and optimality conditions for the cahn hilliard equation with dynamic boundary conditions this paper is concerned with a boundary control problem for the cahn hilliard equation coupled with dynamic boundary conditions in order to handle the control problem we restrict our analysis to

optimal control of an allen cahn equation with singular potentials and dynamic boundary condition in this paper we investigate optimal control problems for allen cahn equations with differentiable singular nonlinearities and a dynamic boundary condition involving differentiable singular nonlinearities and the laplace beltrami operator the approach covers both the cases of distributed controls and of boundary controls the cost functional is of standard tracking type and box constraints for the controls are prescribed parabolic problems with nonlinear dynamic boundary conditions involving the laplace beltrami operator have recently drawn increasing attention due to their importance in applications while their optimal control was apparently never studied before in this paper we first extend known well posedness and regularity results for the state equation and then show the existence of optimal controls and that the control to state mapping is twice continuously frechet differentiable between appropriate function spaces based on these results we establish the first order necessary optimality

existence of bounded solutions for a class of quasilinear elliptic systems on manifolds with boundary we consider nonlinear elliptic partial differential equations for quasilinear operators of the form a u div a x u u a0 x u u x subject to fully nonlinear boundary conditions involving boundary operators of the form for each 0 b u div b x u u b0 x u u u x the main goal of this paper is to give under suitable assumptions on a and b an explicit l estimate for bounded solutions of these elliptic boundary value problems then we establish the existence of at least one solution to such problems extending the author s previous work our methods rely on the definition of approximate problems deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit these methods can be applied to a large class of equations involving operators of leray lions type on suitable banach spaces for a general class of boundary operators b which are possibly of the same order as a as examples these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea level temperatures for relatively long time scales

subordinated stochastic processes with aged operational time in this paper subordinated stochastic processes are considered where the renewal process acting as the operational time it is assumed that the observation of the process begins at a certain time after the start of the renewal process a recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process two point correlation functions for certain subordinated stochastic processes particularly for the generalized ornstein uhlenbeck process were calculated a model of relaxation in a disordered medium with traps and obstacles is proposed

a modified bhattacharya exponential method to approximate positive and bounded solutions of the burgers fisher equation in this work we investigate numerically the classical burgers fisher equation using a modified bhattacharya method the partial differential equation under investigation possesses nonnegative and bounded solutions and under some suitable parameter conditions these solutions are traveling waves it is well known that the use of the bhattacharya approach leads to the design of numerical techniques that are sensitive to zero solutions however in this manuscript we provide a correction of that technique in order to approximate solutions of the burgers fisher equation that are bounded in 0 1 the proposed methodology is explicit and we establish thoroughly the capability of the technique to preserve the non negativity the boundedness and the monotonicity of the numerical approximations as well as the constant solutions of the continuous model the new class of methods introduced in this work considers the presence of a free parameter and we show that this family tends to an explicit and standard discretization of the burgers fisher equation when the free parameter tends to infinity some simulations illustrate the main features of the method

multipoint correlation functions for continuous time random walk models of anomalous diffusion recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous time random walk ctrw in an external potential two and three point correlation functions are calculated for waiting time distributions with an anomalous power law profile t 1 0 1 on intermediate time scales with a crossover to an exponential long time decay comparison of the ctrw with the brownian harmonic oscillator model gaussian process illustrates how higher order correlation functions may be used to distinguish between dynamical models that have the same two point correlation function

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

an explicit positivity preserving finite difference scheme for the classical fisher kolmogorov petrovsky piscounov equation abstract in this work we introduce a family of explicit finite difference schemes to approximate positive solutions of the classical fisher kolmogorov petrovsky piscounov equation from population dynamics a neumann stability analysis reveals that our methods are indeed stable under certain choices of the model and numerical parameters and numerical comparisons with analytical solutions are carried out in some detail

on the stochastic cahn hilliard equation with a singular double well potential abstract we prove well posedness and regularity for the stochastic pure cahn hilliard equation under homogeneous neumann boundary conditions with both additive and multiplicative wiener noise in contrast with great part of the literature the double well potential is treated as generally as possible its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed the regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem the proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework

a bounded linear integrator for some diffusive nonlinear time dependent partial differential equations we propose a numerical method to approximate the solutions of generalized forms of two bi dimensional models of mathematical physics namely the burgers fisher and the burgers huxley equations in one dimensional form the literature in the area gives account of the existence of analytical solutions for both models in the form of traveling wave fronts bounded within an interval i of the real numbers motivated by this fact we propose a finite difference methodology that guarantees that under certain analytical conditions on the model and computer parameters estimates within i will evolve discretely into new estimates which are likewise bounded within i additionally we establish the preservation in the discrete domain of the skew symmetry of the solutions of the models under study our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations

well posedness for a class of doubly nonlinear stochastic pdes of divergence type abstract we prove well posedness for doubly nonlinear parabolic stochastic partial differential equations of the form d x t div x t d t x t d t b t x t d w t where and are the two nonlinearities assumed to be multivalued maximal monotone operators everywhere defined on r d and r respectively and w is a cylindrical wiener process using variational techniques suitable uniform estimates both pathwise and in expectation and some compactness results well posedness is proved under the classical leray lions conditions on and with no restrictive smoothness or growth assumptions on the operator b is assumed to be hilbert schmidt and to satisfy some classical lipschitz conditions in the second variable

numerical analysis of the allen cahn equation and approximation for mean curvature flows we propose and analyze a semi discrete in time scheme and a fully discrete scheme for the allen cahn equation u t u 2 f u 0 arising from phase transition in materials science where is a small parameter known as an interaction length the primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods in particular by focusing on the dependence of the error bounds on optimal order and quasi optimal order error bounds are shown for the semi discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u 0 in particular all our error bounds depend on only in some lower polynomial order for small the cruxes of the analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of de mottoni and schatzman 18 19 and chen 12 and to establish a discrete counterpart of it for a linearized allen cahn operator to handle the nonlinear term finally as a nontrivial byproduct the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow

langevin formulation of a subdiffusive continuous time random walk in physical time systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power law scaling of the mean square displacement one of the most common models to describe such subdiffusive dynamics is the continuous time random walk ctrw stochastic trajectories of a ctrw can be described in terms of the subordination of a normal diffusive process by an inverse levy stable process here we propose an equivalent langevin formulation of a force free ctrw without subordination by introducing a different type of non gaussian noise we are able to express the ctrw dynamics in terms of a single langevin equation in physical time with additive noise we derive the full multipoint statistics of this noise and compare it with the scaled brownian motion sbm an alternative stochastic model describing subdiffusive dynamics interestingly these two noises are identical up to the second order correlation functions but different in the higher order statistics we extend our formalism to general waiting time distributions and force fields and compare our results with those of the sbm in the presence of external forces our proposed noise generates a different class of stochastic processes resembling a ctrw but with forces acting at all times

non markovian stochastic liouville equation and its markovian representation the non markovian variant of the stochastic liouville equation sle is studied within the continuous time random walk approach ctrwa the ctrwa based non markovian sle is shown to be equivalently represented by the corresponding conventional markovian sle this markovian representation provides a rigorous method for deriving the non markovian sle and allows for a physically clear interpretation of the specific features of this sle it also enables one to develop convenient non markovian models useful for applications some of which are discussed in detail special attention is given to the discussion of anomalous long tailed ctrw processes and non markovian sle the obtained results are applied to the analysis of the effect of rate fluctuations on chemical reaction kinetics it is shown in particular that the anomalous fluctuations not only influence the reaction rate but also change the reaction kinetics itself

a skew symmetry preserving computational technique for obtaining the positive and the bounded solutions of a time delayed advection diffusion reaction equation abstract in this work we consider a one dimensional time delayed advective version of the well known fisher kolmogorov petrovsky piscounov equation from population dynamics which extends several models from mathematical physics including the classical wave equation the nonlinear klein gordon equation a fitzhugh nagumo equation from electrodynamics and the burgers huxley equation and the newell whitehead segel equation from fluid mechanics we propose a skew symmetry preserving finite difference scheme for approximating the solutions of the model under investigation and establish conditions on the model coefficients and the numerical parameters under which the method provides positive or bounded approximations for initial data which are likewise positive or bounded respectively the derivation of the conditions under which the positivity and the boundedness of the approximations is guaranteed is based on the properties of the inverses of m matrices in fact the conditions obtained here assure that the iterative method is described in vector form through the multiplication by a matrix of this type we provide simulations in order to show that the technique is indeed conditionally positivity preserving and boundedness preserving

strong solutions for stochastic partial differential equations of gradient type unique existence of analytically strong solutions to stochastic partial differential equations spde with drift given by the subdifferential of a quasi convex function and with general multiplicative noise is proven the proof applies a genuinely new method of weighted galerkin approximations based on the distance defined by the quasi convex function spatial regularization of the initial condition analogous to the deterministic case is obtained the results yield a unified framework which is applied to stochastic generalized porous media equations stochastic generalized reaction diffusion equations and stochastic generalized degenerated p laplace equations in particular higher regularity for solutions of such spde is obtained

abstract approach to evolution equations of phase field type and applications a general system of abstract nonlinear parabolic equations deriving from phase field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved then applications are presented to two physical situations first a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting second a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case moreover through convex analysis techniques it is performed a mathematical study of the subdifferential type relations linking the phase fields and the temperatures to the internal energies of the fluids email schimper dragon ian pv cnr it keywords phase field models subdifferential operators abstract evolution equations

a physical interpretation of age dependent master equations abstract physical systems described by fast and slow relaxing variables are considered the underlying microscopic equations are assumed to be markovian considering that the macroscopic state of the system is characterized by the set of slow relaxing variables we show that the age state probability density obeys a system of age dependent master equations with time and age dependent transition rates if the time scales attached to the slow and fast relaxing variables are well differentiated there is a strong correlation between the age of a fluctuation state and the corresponding number of transition events an alternative approach based on haken s slaving principle is developed for time homogeneous microscopic equations the adiabatic elimination of fast relaxing variables leads to a system of age dependent master equations with time independent and age dependent transition rates these equations are equivalent to a generalized master equation with an inhomogeneous term

a note on parabolic equation with nonlinear dynamical boundary condition abstract we consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so called wentzell boundary condition first we prove the existence and uniqueness of global solutions as well as the existence of a global attractor then we derive a suitable ojasiewicz simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f g are real analytic moreover we provide an estimate for the convergence rate

a variational approach to dissipative spdes with singular drift we prove global well posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative wiener noise in particular the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line on which neither continuity nor growth assumptions are imposed the hypotheses on the diffusion coefficient are also very general in the sense that the noise does not need to take values in spaces of continuous or bounded functions in space and time our approach combines variational techniques with a priori estimates both pathwise and in expectation on solutions to regularized equations

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

on boundary detachment phenomena for the total variation flow with dynamic boundary conditions abstract we combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions we analyze the behavior of facets at the parts of the boundary where these conditions are imposed we devote particular attention to the radially symmetric data we observe that the boundary layer detachment actually can happen at concave parts of the boundary

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

global solution to the allen cahn equation with singular potentials and dynamic boundary conditions abstract we prove well posedness results for the solution to an initial and boundary value problem for an allen cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain the dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls in fischer 1997 13 kenzler 2001 14 we show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments

on the bifurcation of energy in media governed by 2 1 dimensional modified klein gordon equations abstract in the present work we develop a numerical technique to approximate solutions of a general class of two dimensional modified klein gordon equations that appears amongst other applications in the study of lattices consisting of josephson junctions coupled through superconducting wires when dissipative and dispersive effects are taken into account the basic motivation for our study is the need to posses a numerically accurate finite difference scheme to approximate the occurrence of the process of nonlinear supratransmission when the mentioned systems are submitted to dirichlet neumann or more generally boundary conditions of mixed type the computational technique that we present in this paper is nonstandard implicit nonlinear and not only consistent order o t 2 and conditionally stable on the approximation of solutions but it also yields consistent approximations of the local energies of the individual junctions of the total energy of the system and moreover of the rate of change of energy of the system with respect to time making it an ideal method in the analysis of the process of supratransmission generalizations to higher dimensional cases may be induced from the one and two dimensional versions of the method and applications of the technique are given to the determination of the occurrence of nonlinear supratransmission under the presence of several parameters our results show that the transmission of energy in the system starts at a well defined nonnegative lower threshold after which the creation of localized intrinsic modes that move away from the driving boundaries is imminent the study of the effects of nonzero mass and internal and external damping is carried out from a computational perspective

supratransmission in a disordered nonlinear periodic structure abstract we study the interaction among dispersion nonlinearity and disorder effects in the context of wave transmission through a discrete periodic structure subjected to continuous harmonic excitation in its stop band we consider a damped nonlinear periodic structure of finite length with disorder disorder is introduced throughout the structure by small changes in the stiffness parameters drawn from a uniform statistical distribution dispersion effects forbid wave transmission within the stop band of the linear periodic structure however nonlinearity leads to supratransmission phenomenon by which enhanced wave transmission occurs within the stop band of the periodic structure when forced at an amplitude exceeding a certain threshold the frequency components of the transmitted waves lie within the pass band of the linear structure where disorder is known to cause anderson localization there is therefore a competition between dispersion nonlinearity and disorder in the context of supratransmission we show that supratransmission persists in the presence of disorder the influence of disorder decreases in general as the forcing frequency moves away from the pass band edge reminiscent of dispersion effects subsuming disorder effects in linear periodic structures we compute the dependence of the supratransmission force threshold on nonlinearity and strength of coupling between units we observe that nonlinear forces are confined to the driven unit for weakly coupled systems this observation together with the truncation of higher order nonlinear terms permits us to develop closed form expressions for the supratransmission force threshold in sum in the frequency range studied here disorder does not influence the supratransmission force threshold in the ensemble average sense but it does reduce the average transmitted wave energy

the numerical solution of a generalized burgers huxley equation through a conditionally bounded and symmetry preserving method in this article we propose a non standard finite difference scheme to approximate the solutions of a generalized burgers huxley equation from fluid dynamics our numerical method preserves the skew symmetry of the partial differential equation under study and under some analytical constraints of the model constants and the computational parameters involved it is capable of preserving the boundedness and the positivity of the solutions in the linear regime the scheme is consistent to first order in time due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative and to second order in space we compare the results of our computational technique against the exact solutions of some particular initial boundary value problems our simulations indicate that the method presented in this work approximates well the theoretical solutions and moreover that the method preserves the boundedness of solutions within the analytical constraints derived here in the problem of approximating solitary wave solutions of the model under consideration we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique for which a minimum relative error is achieved finally we linearly perturb a steady state solution of the partial differential equation under investigation and show that our simulations still converge to the same constant solution establishing thus robustness of our method in this sense

nonconvergent bounded solutions of semilinear heat equations on arbitrary domains abstract we consider the dirichlet problem for the semilinear heat equation 0 1 u t u g x u x where is an arbitrary bounded domain in r n n 2 with c 2 boundary we find a c function g x u such that 0 1 has a bounded solution whose limit set is a continuum of equilibria this extends and improves an earlier result of the first author with rybakowski in which is a disk in r 2 and g is of finite differentiability class we also show that 0 1 can have an infinite dimensional manifold of nonconvergent bounded trajectories

singular stochastic allen cahn equations with dynamic boundary conditions abstract we prove a well posedness result for stochastic allen cahn type equations in a bounded domain coupled with generic boundary conditions the nonlinear flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics the singular character of the drift part allows for a large class of maximal monotone operators generalizing the usual double well potentials one of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution neither on the domain nor on the boundary a well posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques a vanishing viscosity argument for the dynamic on the boundary is also presented

convergence to equilibrium for the cahn hilliard equation with dynamic boundary conditions this paper is concerned with the asymptotic behavior of solution to the cahn hilliard equation 0 1 u t u u u3 x t r subject to the following dynamic boundary conditions 0 2 s u u hs gsu 1 sut t 0 x 0 t 0 x and the initial condition 0 3 u t 0 u0 x x where is a bounded domain in rn n 3 with smooth boundary and s 0 s 0 gs 0 hs are given constants is the tangential laplacian operator and is the outward normal direction to the boundary this problem has been considered in the recent paper by racke and zheng adv differential equations 8 1 2003 83 where the global existence and uniqueness were proved in a very recent manuscript by pruss racke and zheng konstanzer schrift math inform 189 2003 the results on existence of global attractor and maximal regularity of solution have been obtained in this paper convergence of solution of this problem to an equilibrium as time goes to infinity is proved

on the cahn hilliard equation with dynamic boundary conditions and a dominating boundary potential the cahn hilliard and viscous cahn hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well posedness and regularity results are proved

an energy based computational method in the analysis of the transmission of energy in a chain of coupled oscillators in this paper we study the phenomenon of nonlinear supratransmission in a semi infinite discrete chain of coupled oscillators described by modified sine gordon equations with constant external and internal damping and subject to harmonic external driving at the end we develop a consistent and conditionally stable finite difference scheme in order to analyze the effect of damping in the amount of energy injected in the chain of oscillators numerical bifurcation analyses to determine the dependence of the amplitude at which supratransmission first occurs with respect to the frequency of the driving oscillator are carried out in order to show the consequences of damping on harmonic phonon quenching and the delay of appearance of critical amplitude

stochastic parabolic equations with nonlinear dynamical boundary conditions abstract the existence and uniqueness of solutions to linear parabolic equations with nonlinear flux on the boundary driven by gaussian boundary noise is studied both the case of heat equation with boundary conditions of wentzell type and the case of white noise boundary conditions are considered

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

long time behavior for a nonlinear plate equation with thermal memory abstract we consider a nonlinear plate equation with thermal memory effects due to non fourier heat flux laws first we prove the existence and uniqueness of global solutions as well as the existence of a global attractor then we use a suitable ojasiewicz simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term f is real analytic moreover we provide an estimate on the convergence rate

on a positivity preserving numerical model for a linearized hyperbolic fisher kolmogorov petrovski piscounov equation abstract in this work we introduce a nonstandard finite difference scheme to approximate positive solutions of a modified hyperbolic fisher kolmogorov petrovski piscounov equation appearing in the investigation of the dynamics of some populations the technique has a consistency of second order and it provides nonnegative approximations for nonnegative initial profiles the stability analysis of the method is carried out in detail and the scheme is validated against known analytical solutions for some initial boundary value problems

a simple unified approach to some convergence theorems of l simon abstract we give a new and easier proof for the convergences result of l simon 15 concerning global and bounded solutions of a heat equation with analytic nonlinearity by constructing a new liapunov functional for the elliptic equation we are able to prove the second convergence theorem of 15 in the same way as the first theorem the results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type

numerical treatment of the spherically symmetric solutions of a generalized fisher kolmogorov petrovsky piscounov equation in the present work the connection of the generalized fisher kpp equation to physical and biological fields is noted radially symmetric solutions to the generalized fisher kpp equation are considered and analytical results for the positivity and asymptotic stability of solutions to the corresponding time independent elliptic differential equation are quoted an energy analysis of the generalized theory is carried out with further physical applications in mind and a numerical method that consistently approximates the energy of the system and its rate of change is presented the method is thoroughly tested against analytical and numerical results on the classical fisher kpp equation the heaviside equation and the generalized fisher kpp equation with logistic nonlinearity and heaviside initial profile obtaining as a result that our method is highly stable and accurate even in the presence of discontinuities as an application we establish numerically that under the presence of suitable initial conditions there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value an analytical prediction is provided for the heaviside equation against which we verify the validity of our computational code and numerical approximations are provided for several generalized fisher kpp problems

non markovian stochastic liouville equation abstract the non markovian variant of the stochastic liouville equation sle based and the continuous time random walk approach ctrwa is proposed this sle turns out be represented by the conventional markovian sle the non markovian sle is applied to the analysis of anomalous long tailed ctrw processes and in particular anomalous rate fluctuations in some chemical reactions these fluctuations are shown to strongly manifest themselves in the reaction kinetics

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

a unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces abstract in this paper we consider scalar parabolic equations in a general non smooth setting emphasizing interface conditions and mixed boundary conditions in particular we study dynamics and diffusion on a lipschitz interface and on the boundary where the diffusion coefficients are only assumed to be bounded measurable and positive semidefinite in the bulk we consider diffusion coefficients which may degenerate towards a lipschitz surface for this problem class we introduce a unified functional analytic framework based on sesquilinear forms and show maximal l p regularity and bounded h calculus for the corresponding operator providing well posedness for a large class of initial conditions and external forces