generalized chapman kolmogorov equation a unifying approach to the description of anomalous transport in external fields the generalized chapman kolmogorov equation v m kenkre e w montroll and m f shlesinger j stat phys 9 45 1973 is discussed it is demonstrated that this equation unifies recently proposed kinetic equations of fractional order that describe anomalous transport in external fields as well as continuous time random walks the conditions under which the individual models can be established are discussed

newton method for stationary and quasi stationary problems for smoluchowski type equations abstract an efficient implementation of the newton krylov iterative method finding numerical solutions of aggregation fragmentation equations without spontaneous fragmentation is reported it is based on application of fast numerical schemes for evaluation of the smoluchowski operator associated with these equations and structure of jacobi matrix allowing implementation of fast multiplication by a vector we also propose an efficient preconditioner based on lower triangular band matrix extracted from a jacobi matrix resulting in a dramatic speedup of internal gmres based iterations between the newton steps we compare the efficiency of the novel method with the existing ones and demonstrate a noticeable superiority of our approach moreover we show that the new method may be utilized for modeling of a wider class of aggregation fragmentation phenomena and derive a new class of quasi stationary solutions for aggregation fragmentation equations source of monomers

random tree type partitions as a model for acyclic polymerization holtsmark 3 2 stable distribution of the supercritical gel we study the asymptotic behavior in the thermodynamic limit as n of a particular probability distribution on the set of all forests of trees on a set of n elements monomers we establish the existence of three stages of polymerization dependent upon the ration of association and dissociation rates of monomers and we analyse the near critical and supercritical stages in the supercritical stage we discover that the molecular weight of the largest connected component gel has the holtsmark distribution

an introduction to mathematical models of coagulation fragmentation processes a discrete deterministic mean field approach abstract we summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity an integer multiple of some basic unit size in particular we discuss smoluchowski s equation for aggregation the becker doring model of simultaneous aggregation and fragmentation and more general models involving coagulation and fragmentation

the asymptotic distributions of the size of the largest length of a reversible markov process of polymerization in this paper the subcritical near critical and supercritical asymptotic behaviour of a reversible markov process as a chemical model for polymerization was studied we establish the existence of three distinct stages subcritical near critical and supercritical stages of polymerization in the thermodynamic limit as n depending on the value of strength of the fragmentation reaction these three stages correspond to the size of the largest length of polymers of size n to be itself of order log n na 1 2 a 1 and n respectively especially a 2 2 1 when the coagulation reaction rate constant rij of the polymers satisfies rij i j with 1 2 3 2

two species coagulation approach to consensus by group level interactions we explore the self organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole to this end we employ the widespread example of coagulation kinetics and characterize which interaction types lead to consensus formation and which do not as well as the corresponding different macroscopic patterns the crucial technical point is extending the usual one species coagulation dynamics to the two species one this is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning the corresponding solutions are calculated in the long time limit in which consensus may or may not be reached the lack of consensus is characterized by means of scaling limits of the solutions the possible applications of our results to some topics in which consensus reaching is fundamental such as collective animal motion and opinion spreading dynamics are also outlined

asymptotic behavior of solutions to the coagulation fragmentation equations ii weak fragmentation the discrete coagulation fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones the assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important our results on the asymptotic behavior of solutions generalize the corresponding results of ball carr and penrose for the becker doring equation

pre and post gel size distributions in ir reversible polymerisation a class of irreversible coagulation processes can be modelled by smoluchowski s coagulation equation with rate constants kij a b i j cij non negative a b and c for c not 0 a gelation transition occurs the authors obtain explicit solutions for the size distribution ck t with ck 0 delta k1 next they construct and solve the equations for reversible polymerisation by incorporating break up processes in the kinetic equation with a unimolecular fragmentation rate fij lambda ninjkij ni j the degeneracy factors nk obey k 1 nk 1 2 sigma kijninj with i j k and n1 1 and the strength parameter lambda exp g kbt where the binding energy g to infinity for irreversible coagulation explicit results are only given for flory s polymerisation models raf and braf 1 in the vicinity of the gel point the authors verify the scaling hypothesis and calculate critical exponents

some analytical and numerical solutions for colloidal aggregation with fragmentation we study the kinetic equation which describes the evolution of cluster size distribution of the aggregation fragmentation problem the so called generalized smoluchowski equation for several dynamics of cluster growth by using the method of the generating function we find analytical solutions for some cases besides numerical solutions are found for the time evolution of clusters where the aggregation and fragmentation kernels do not allow a complete analytical solution

the smoluchowski coagulation equations with continuous injection we study a system of equations which models the formation of clusters by coagulation with particles of unit size being injected at a time dependent rate we observe that the criteria under which gelation occurs are the same as for the constant mass and constant monomer cases which have been studied previously we identify a variety of types of behaviour in the large time limit depending on the coagulation kernel and on the rate at which monomer is introduced into the system the results are obtained by means of exact generating function techniques matched asymptotic expansions and numerical simulations

on a class of continuous coagulation fragmentation equations abstract a model for the dynamics of a system of particles undergoing simultaneously coalescence and breakup is considered each particle being assumed to be fully identified by its size existence of solutions to the corresponding evolution integral partial differential equation is shown for product type coagulation kernels with a weak fragmentation the failure of density conservation or gelation is also investigated in some particular cases

strongly differentiable solutions of the discrete coagulation fragmentation equation we examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters in contrast to previous investigations our analysis does not involve finite dimensional truncations of the system instead we treat the problem as an infinite dimensional differential equation posed in an appropriate banach space and apply perturbation results from the theory of strongly continuous semigroups of operators the existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates

weak solutions to the cauchy problem for the diffusive discrete coagulation fragmentation system abstract the initial value problem for the discrete coagulation fragmentation system with diffusion is studied this is an infinite countable system of reaction diffusion equations describing coagulation and fragmentation of discrete clusters moving by spatial diffusion in all space r d the model considered in this work is a generalization of smoluchowski s discrete coagulation equations existence of global in time weak solutions to the cauchy problem is proved under natural assumptions on initial data for unbounded coagulation and fragmentation coefficients this work extends existence theory for this system from the case of clusters distribution on bounded domain subject to no flux boundary condition to the case of all r d

convergence to equilibrium for the discrete coagulation fragmentation equations with detailed balance under the condition of detailed balance and some additional restrictions on the size of the coefficients we identify the equilibrium distribution to which solutions of the discrete coagulation fragmentation system of equations converge for large times thus showing that there is a critical mass which marks a change in the behavior of the solutions this was previously known only for particular cases as the generalized becker doring equations our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

application of wavelet collocation method for hyperbolic partial differential equations via matrices abstract in this work we developed an efficient computational method based on legendre and chebyshev wavelets to find an approximate solution of one dimensional hyperbolic partial differential equations hpdes with the given initial conditions the operational matrices of integration for legendre and chebyshev wavelets are derived and utilized to transform the given pde into the linear system of equations by combining collocation method convergence analysis and error estimation associated to the presented idea are also investigated under several mild conditions numerical experiments confirm that the proposed method has good accuracy and efficiency moreover the use of legendre and chebyshev wavelets are found to be accurate simple and fast

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

a generalized laguerre spectral petrov galerkin method for the time fractional subdiffusion equation on the semi infinite domain abstract in this paper a generalized laguerre spectral petrov galerkin method is proposed to solve a class of time fractional subdiffusion equations on the semi infinite domain we use the generalized associated laguerre functions of the first kind and the generalized laguerre functions as basis functions in time and space directions separately the respective projection error estimates can be obtained as well we derive the projection error estimates of the solutions in time space directions the approximation results of the fully discrete spectral scheme are introduced in this paper some numerical results are presented to illustrate the efficiency of this method

similarity solution of coagulation equation with an inverse kernel abstract we analyse a model for the aggregation of polystyrene particles which arises in an electrorheology system in which linear clusters grow upon the application of an alternating electromagnetic field we consider a coagulation kernel involving negative powers of cluster sizes and investigate the reduction of the governing equations to a similarity solution in the large time limit comparison between the experimental results and the theory presented here shows a good collapse of the data onto a single curve which matches the theoretical results particularly well at the larger cluster sizes

gelation time in the discrete coagulation fragmentation equations with a bilinear coagulation kernel the discrete coagulation fragmentation equations with a bilinear coagulation kernel describing the evolution of clusters of particles undergoing binary reactions are studied we compute the gelation time and post gelation mass for the pure coagulation equation a method of characteristics is developed to solve numerically the partial differential equation satisfied by a moment generating function for a product coagulation kernel and a constant fragmentation rate the accuracy of the numerical method is verified by comparison of the numerical results with an exact solution for the number density of monomers for a given coagulation rate a critical value of the fragmentation rate values greater than which lead to mass conservation is identified

kinetics of random aggregation fragmentation processes with multiple components a computationally efficient algorithm is presented for exact simulation of the stochastic time evolution of spatially homogeneous aggregation fragmentation processes featuring multiple components or conservation laws the algorithm can predict the average size and composition distributions of aggregating particles as well as their fluctuations regardless of the functional form e g composition dependence of the aggregation or fragmentation kernels furthermore it accurately predicts the complete time evolutions of all moments of the size and composition distributions even for systems that exhibit gel transitions we demonstrate the robustness and utility of the algorithm in case studies of linear and branched polymerization processes the last of which is a two component process these simulation results provide the stochastic description of these processes and give new insights into their gel transitions fluctuations and long time behavior when deterministic approaches to aggregation kinetics may not be reliable

critical behavior of the heterogeneous random coagulation fragmentation processes in this paper we study the asymptotic distributions of the heterogeneous random coagulation fragmentation processes hcfp which model the coagulation fragmentation and diffusion of clusters of particles on a lattice based on a closed form of the stationary distribution for the hcfp we prove that the mutually dependent clusters with a finite size finite particles in the sub critical stage will become independent in the critical and super critical stages and the asymptotic distributions of the number of clusters may converge to gaussian or poisson distribution according to its size to be small or large in the critical and super critical stages

asymptotic behaviour of solutions to the diffusive fragmentation coagulation system abstract we study the time asymptotics of solutions to the diffusive fragmentationdashcoagulation system which we derived in a previous work j f collet f poupaud transp theory and stat phys 25 3 5 1996 503 513 in the case where a detailed balance equilibrium exists we derive convergence to such a state by using a careful analysis of the free energy dissipation rate of the system in the case where no detailed balance equilibrium exists we show the existence of an attractor by using classical energy inequalities and some estimates on higher moments of the solution

self attracting poisson clouds in an expanding universe we consider the following elementary model for clustering by ballistic aggregation in an expanding universe at the initial time there is a doubly infinite sequence of particles lying in a one dimensional universe that is expanding at constant rate we suppose that each particle p attracts points at a certain rate a p 2 depending only on p and when two particles say p and q collide by the effect of attraction they merge as a single particle p q the main purpose of this work is to point at the following remarkable property of poisson clouds in these dynamics under certain technical conditions if at the initial time the system is distributed according to a spatially stationary poisson cloud with intensity 0 then at any time t 0 the system will again have a poissonian distribution now with intensity t where the family solves a generalization of smoluchowski s coagulation equation

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

levy meets boltzmann strange initial conditions for brownian and fractional fokker planck equations abstract we study normal and anomalous diffusion processes with initial conditions of the broad levy type i e with such initial conditions which per se exhibit a diverging variance in the force free case the behaviour of the associated probability density function features distinct shoulders which can be related to the probability current flowing away from the origin in the presence of an external potential which eventually leads to the emergence of a non trivial normalisable equilibrium probability density function the initially diverging variance becomes finite in particular the effects of strange initial conditions for the harmonic ornstein uhlenbeck potential are explored to some detail methods to quantify the dynamics related to such kinds of processes are investigated

on lushnikov s model of gelation we characterize gelation for models of polymers built up as infinite volume limits of finite systems we establish rigorously the occurrence of gelation in lushnikov s model with reaction rater jk jk we obtain bounds on the size of the largest polymer in the system at timet

from the discrete to the continuous coagulation fragmentation equations the connection between the discrete and the continuous coagulation fragmentation models is investigated a weak stability principle relying on a priori estimates and weak compactness in l 1 is developed for the continuous model we approximate the continuous model by a sequence of discrete models and writing the discrete models as modified continuous ones we prove the convergence of the latter towards the former with the help of the above mentioned stability principle another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation fragmentation model

numerical simulation and convergence analysis of a finite volume scheme for solving general breakage population balance equations this paper presents a finite volume scheme fvs for solving general breakage population balance equations bpbes in particular the number density based bpbe is transformed to the form of a mass conservation law then it becomes easy to apply the well known fvss that have an important property of mass conservation following kumar and warnecke for fixed pivot fp method 16 and cell average technique cat 15 the stability and the convergence analysis of the semi discretized fvs are studied unlike the cat and the fp method the fvs is second order consistent independently of the type of meshes we also observe that fvs gives second order convergence rate on four different types of uniform and non uniform meshes with non decreasing behavior of mesh width nevertheless one order better accuracy than the fp method is achieved on locally uniform meshes it is also noticed that on non uniform random meshes the fvs shows one order and two orders higher accuracy than the cat and the fp method respectively the mathematical results of convergence analysis are validated numerically by taking three test problems the numerical simulations are also compared with the results obtained by the cat and the fp method

deformations of super kms functionals we investigate the stability of the super kms property under deformations we show that a family of continuous deformations of the super derivation in the quantum algebra yields a continuous family of deformed super kms functionals these functionals define a family of cohomologous entire cocycles in this paper we investigate the super kms skms property of functionals co on a quantum algebra our interest in skms functionals was inspired by work of kastler and by conversations with alain connes ci k jlo2 the skms construction relies on the existence of a super derivation d acting on a dense subalgebra of a c algebra d the square of d is the infinitesimal generator of a continuous one parameter automorphism group c of the quantum algebra the usual kms property relates the cyclicity of a state co to the analytic continuation of a group et of automorphisms the skms property also involves invariance under the super derivation d whose square generates the automorphism group er it is known that an skms functional on a quantum algebra defines an entire cyclic cocycle z this is just the chern character which jaffe lesniewski and osterwalder defined in the context ofsupertrace functionals on a quantum algebra jlo 1 the skms property ensures that the functional co and the cocycle z which is derived from it are invariant under this group action in this paper we study the stability of this structure under perturbations of d we study only bounded perturbations which arise from the graded super commutator with an odd element q of the algebra d we show that such perturbations dq of d can be used to define a deformation coq of co which satisfies the skms property furthermore the corresponding family of cocycles z are cohomologous of course more singular unbounded perturbations can lead to

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

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

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

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

asymptotic behaviour of solutions to the generalized becker doring equations for general initial data we prove the following asymptotic behaviour for solutions to the generalized becker doring system for general initial data under a detailed balance assumption and in situations where density is conserved in time there is a critical density s such that solutions with an initial density s converge strongly to the equilibrium with density and solutions with initial density s converge in a weak sense to the equilibrium with density s this extends the previous knowledge that this behaviour happens under more restrictive conditions on the initial data the main tool is a new estimate on the tail of solutions with density below the critical density

size distribution in the polymerisation model afrbg the polymer size distribution in flory s polymerisation model afrbg is in equilibrium and non equilibrium of the general form ck ank xi k both below and above the gelation transition here explicit expressions are derived for a alpha and xi alpha as a function of the extent of reaction alpha the combinatorial factors nk are calculated from the recursion relation 2 k 1 nk sigma i j kkijninj with kij si f sj g si g sj f and sk f f 1 k 1 using a generating function technique the authors express nk in terms of laguerre polynomials g to infinity and jacobi polynomials g finite whose large k behaviour is of the form nk approximately bk tau xi c k with tau 3 2 no gelation occurs if f or g equals 1 and with tau 5 2 gelation occurs if f 1 and g 1

coagulation diffusion systems derivation and existence of solutions for the diffuse interface structure equations abstract this paper considers an infinite system of partial differential equations the coagulation diffusion equations which add spatial diffusion to the classical coagulation equations the main emphasis is placed on deriving an infinite system of ordinary differential equations which described the structured interface between reacting coagulation and dilute concentration existence of solutions to interfacial equaitons is proven under spatial boundary conditions

the nonlinear fragmentation equation we study the kinetics of nonlinear irreversible fragmentation here fragmentation is induced by interactions collisions between pairs of particles and modelled by general classes of interaction kernels for several types of breakage models we construct initial value and scaling solutions of the fragmentation equations and apply the non vanishing mass flux criterion for the occurrence of shattering transitions these properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters

a pure jump markov process associated with smoluchowski s coagulation equation the aim of the present paper is to construct a stochastic process whose law is the solution of the smoluchowski s coagulation equation we introduce first a modified equation dealing with the evolution of the distribu tion q t dx of the mass in the system the advantage we take on this is that we can perform an unified study for both continuous and discrete models the integro partial differential equation satisfied by q t t 0 can be interpreted as the evolution equation of the time marginals of a markov pure jump process at this end we introduce a nonlinear poisson driven stochastic differential equation related to the smoluchowski equation in the following way if x t satisfies this stochastic equation then the law of x t satisfies the modified smoluchowski equation the nonlinear process is richer than the smoluchowski equation since it provides historical information on the particles existence uniqueness and pathwise behavior for the solution of this sde are studied finally we prove that the nonlinear process x can be obtained as the limit of a marcus lushnikov procedure 1 introduction the coagulation model governs various phenomena as for example polymerization aggregation of colloidal particles formation of stars and planets behavior of fuel mixtures in engines etc smoluchowski s coagulation equation models the dynamic of such phenomena and describes the evolution of a system of clusters which coalesce in order to form bigger clusters each cluster is identified by its size the only mechanism taken into account is the coalescence of two clusters other effects as multiple coagulation are neglected we assume also that the rate of these reactions depends on the sizes of clusters involved in the coagulation denoting by n k t the nonnegative concentration of clusters of size k at time t the discrete

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

self similarity in a coarsening model in one dimension motivated from asymptotic laws of motion for transition layers in the equation u t 2 u xx u u 2 we consider the following model for coarsening of a fine partition of an interval find the shortest subinterval of the partition and joint it with its neighbours combining three into one making a random order assumption we develop and study an unusual coagulation equation for the distribution of interval lengths we establish the existence of a self similar solution of this equation by using laplace transform techniques simulation data indicate that random order does persist if present initially and the distribution approaches similarity form

hydrodynamic limit of coagulation fragmentation type models of k nary interacting particles hydrodynamic limit of general k nary mass exchange processes with discrete mass distribution is described by a system of kinetic equations that generalize classical smoluchovski s coagulation equations and many other models that are intensively studied in the current mathematical and physical literature existence and uniqueness theorems for these equations are proved at last for k nary mass exchange processes with k 2 an alternative nondeterministic measure valued limit diffusion approximation is discussed

an efficient algorithm for solving volterra integro differential equations based on alpert s multi wavelets galerkin method abstract this paper is concerned with the application of the multi wavelets galerkin method to linear and nonlinear second order volterra integro differential equations using the operational matrices of integration and wavelet transform matrix we reduce the problem to a set of algebraic equations for linear type we use thresholding to reduce the computational effort we demonstrate that our solutions may be efficiently represented in a multi wavelets basis due to flexible vanishing moments property of this type of multi wavelets the l 2 convergence of the scheme for the proposed equation has been investigated a series of numerical tests is provided to demonstrate the validity and applicability of the technique

coagulation fragmentation processes we study the well posedness of coagulation fragmentation models with diffusion for large systems of particles the continuous and the discrete case are considered simultaneously in the discrete situation we are concerned with a countable system of coupled reaction diffusion equations whereas the continuous case amounts to an uncountable system of such equations these problems can be handled by interpreting them as abstract vector valued parabolic evolution equations where the dependent variables take values in infinite dimensional banach spaces given suitable assumptions we prove existence and uniqueness in the class of volume preserving solutions we also derive sufficient conditions for global existence

the continuous coagulation equation with multiple fragmentation abstract we present a proof of the existence of solutions to the continuous coagulation equation with multiple fragmentation whenever the kernels satisfy certain growth conditions the proof relies on weak l 1 compactness methods applied to suitably chosen approximating equations the question of uniqueness is also considered

shifted jacobi spectral galerkin method for solving hyperbolic partial differential equations abstract herein we propose a numerical scheme to solve spectrally hyperbolic partial differential equations hpdes using galerkin method and approximate the solutions using double shifted jacobi polynomials the main characteristic behind this approach is that it reduces such problems to those of solving systems of algebraic equations which greatly simplifies the problem the validity and efficiency of the proposed method are investigated and verified through several examples

a global weak solution via transport projection scheme to a thin film growth kinetic model with nonlocal boundary conditions we examine the existence of a global weak solution associated with a numerical scheme for kinetic models of nuclei growth and coagulation of liquid droplet formation on a solid surface a nonlinear transport equation of the density with a bilinear operator and with a nonlocal boundary condition is coupled to an ode which gives the supersaturation the numerical scheme constructed is a transport projection method that is a combination of an s t operator semigroup associated with the continuous problem and a p delta projection operator on the discrete space three estimates 1 l infty bound 2 a uniform total variation bound and 3 l1 continuity in time of the approximate solution are established with a self contained treatment of the stability and convergence properties numerical calculations of thin film growth kinetics are reported and some related mathematical issues are examined

convergence analysis of sectional methods for solving breakage population balance equations ii the cell average technique this work presents the convergence of the cell average technique kumar et al in powder technol 179 205 228 2007 for solving breakage population balance equation similarly to our paper kumar and warnecke numerische math 2008 of this series we study convergence on four different types of meshes a second order convergence is proved for uniform locally uniform and non uniform smooth meshes finally the scheme is analyzed on random mesh and it is found that the scheme is only first order accurate nevertheless we obtain for locally uniform as well as for random mesh one order higher accuracy than the fixed pivot technique discussed by the authors in the first paper all mathematical observations of convergence analysis are also validated numerically and numerical results are compared with the results of the first part

time evolution of a random graph an initially empty no edges graph of the order of m is assumed to evolve by adding one edge at a time this edge can connect either two linked components and form a new component of a larger order coalescence of graphs or increase by one the number of edges in a given linked component the evolution equation for the generating functional of the probability to find in the system a given set of occupation numbers the numbers of graphs of the order of g having exactly edges at time t is formulated and solved exactly the expression for the graph composition spectrum is derived and analysed in the limit of large m

asymptotic solutions of the becker doring equations we describe the solution of various becker doring models first we analyse equilibrium and steady state solutions together with the large time behaviour of solutions for the case of constant monomer concentration then we solve a weak fragmentation constant mass case by using matched asymptotic expansions the methods are applied both to the full becker doring equations and to a coarse grained or contracted system comparison of the results show good qualitative agreement all the phenomena present in the full model are reproduced in the contracted system and no anomalous effects are introduced however the quantitative agreement depends strongly on the choice of parameter values in the contracted system

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

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

polymers and random graphs we establish a precise connection between gelation of polymers in lushnikov s model and the emergence of the giant component in random graph theory this is achieved by defining a modified version of the erdos renyi process when contracting to a polymer state space this process becomes a discrete time markov chain embedded in lushnikov s process the asymptotic distribution of the number of transitions in lushnikov s model is studied a criterion for a general markov chain to retain the markov property under the grouping of states is derived we obtain a noncombinatorial proof of a theorem of erdos renyi type