on the approximate solutions of fragmentation equations a numerical model based on the finite volume scheme is proposed to approximate the binary breakage problems initially it is considered that the particle fragments are characterized by a single property i e particle s volume we then investigate the extension of the proposed model for solving breakage problems considering two properties of particles the efficiency to estimate the different moments with good accuracy and simple extension for multi variable problems are the key features of the proposed method moreover the mathematical convergence analysis is performed for one dimensional problems all mathematical findings and numerical results are validated over several test problems for numerical validation we propose the extension of bourgade filbet 2008 math comput 77 851 882 doi 10 1090 s0025 5718 07 02054 6 model for solving two dimensional pure breakage problems in this aspect numerical treatment of the two dimensional binary breakage models using finite volume methods can be treated to be the first instance in the literature

extension of moment projection method to the fragmentation process the method of moments is a simple but efficient method of solving the population balance equation which describes particle dynamics recently the moment projection method mpm was proposed and validated for particle inception coagulation growth and more importantly shrinkage here the method is extended to include the fragmentation process the performance of mpm is tested for 13 different test cases for different fragmentation kernels fragment distribution functions and initial conditions comparisons are made with the quadrature method of moments qmom hybrid method of moments hmom and a high precision stochastic solution calculated using the established direct simulation algorithm dsa and advantages of mpm are drawn

new solutions to the fragmentation equation a new general class of exact explicit scaling solutions to the fragmentation equation is given this class is described by a breakage rate a x xlambda and the daughter distribution function yb x mod y a gamma xy gamma 2 1 a delta x y delta 2 and includes as special cases all previously known scaling solutions to the fragmentation equation for a subset of this class with gamma lambda and a delta delta lambda the complete time dependent solution for a monodisperse initial condition is also given

kinetics of fragmentation a general discussion of the kinetics of continuous irreversible fragmentation processes is presented for a linear process where particle breakup is driven by an external force the authors develop a scaling theory to describe the evolution of the cluster size distribution they treat the general case where the breakup rate of a cluster of mass x varies as xlambda when lambda 0 corresponding to larger clusters more likely to break up the scaled cluster size distribution phi x decays with the scaled mass x as x 2 exp axlambda as x for small mass phi x has the log normal form exp a ln2 x if the breakup kernel has a small size cutoff while phi x has a power law tail in the absence of a cutoff they also show that a conventional scaling picture applies only for the case lambda 0 for lambda 0 they develop an alternative formulation for the cluster size distribution in which the typical mass scale is determined by the initial condition in this regime they also investigate the nature of a shattering transition where mass is lost to a dust phase of zero mass particles they also study the kinetics of a nonlinear collision induced fragmentation process they analyse the asymptotic behaviour of a simple minded class of models in which a two particle collision results in either 1 both particles splitting into two equal pieces 2 only the larger particle splitting in two or 3 only the smaller particle splitting they map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations scaling is found to hold for different ranges of homogeneity index for the three models

distribution of the number of fragmentations in continuous fragmentation the continuous fragmentation equation is generalized to include the number of fragmentations as a variable the fragmentation rate and the probability for the production of progeny size y from parent size x are both made dependent on the number n of fragmentations experienced by the fragment during its history an interpretation where n is the number of impacts experienced by the fragment is possible by also allowing a single progeny equalling the parent in size solutions for the cases where does not depend on x are studied the theory is adapted to model clustering in fragmentation generated cellular structures

correlations between fragment sizes in sequential fragmentation size correlations in a sequential fragmentation process originate from the correlations the progeny of a single fragmentation event necessarily have a multidimensional equation describing the binary fragmentation of a unit segment is formulated the fragments are assumed to keep their positions with respect to the original segment if the fragmentation rate is not dependent on fragment size the distribution for n successive adjacent fragments is shown to asymptotically approach an n dimensional log normal distribution explicit forms of the asymptotic distribution and the correlations are given in terms of the parameters of the probability that a fragmentation event with parent size y produces progeny size x

numerical calculation of the moments of the population balance equation the combined cfd pbm population balance models are computationally intensive so a possibility is to calculate only a few moments of the probability density function pdf of the pbm minimizing the computational costs however this formulation results in an excess of unknowns with respect to equations which is referred to as a closure problem one approach for dealing with this closure problem is to apply a numerical quadrature approximation on the other hand a different possibility is to compute the pdf and from this the moments of interest if required in this work the two mentioned approaches are discussed and numerical experiments are used to show the capability of the methods for predicting the moments of the pbe in particular the quadrature method of moments and a time space least squares spectral method will be discussed

transient and asymptotic behaviour of the binary breakage problem the general binary breakage problem with power law breakage functions and two families of symmetric and asymmetric breakage kernels is studied in this work a useful transformation leads to an equation that predicts self similar solutions in its asymptotic limit and offers explicit knowledge of the mean size and particle density at each point in dimensionless time a novel moving boundary algorithm in the transformed coordinate system is developed allowing the accurate prediction of the full transient behaviour of the system from the initial condition up to the point where self similarity is achieved and beyond if necessary the numerical algorithm is very rapid and its results are in excellent agreement with known analytical solutions in the case of the symmetric breakage kernels only unimodal self similar number density functions are obtained asymptotically for all parameter values and independent of the initial conditions while in the case of asymmetric breakage kernels bimodality appears for high degrees of asymmetry and sharp breakage functions for symmetric and discrete breakage kernels self similarity is not achieved the solution exhibits sustained oscillations with amplitude that depends on the initial condition and the sharpness of the breakage mechanism while the period is always fixed and equal to ln 2 with respect to dimensionless time

a note on the self similar solutions to the spontaneous fragmentation equation we provide a method to compute self similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours our procedure is applied to specific cases i the case of mitosis where fragmentation results into two identical fragments ii fragmentation limited to the formation of sufficiently large fragments and iii processes with fragmentation kernel presenting a power like behaviour

finite connectivity attractor neural networks we study a family of diluted attractor neural networks with a finite average number of symmetric connections per neuron as in finite connectivity spin glasses their equilibrium properties are described by order parameter functions for which we derive an integral equation in replica symmetric approximation a bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin glass transition lines in the phase diagram the line separating the retrieval phase from the spin glass phase is calculated at zero temperature all phase transitions are found to be continuous

on the solution of the integro differential fragmentation equation with continuous mass loss in this paper we obtain a complete classification of all possible non trivial similarity solutions of the integro differential fragmentation equation with continuous mass loss rate these solutions include the effects of both continuous and discrete mass loss rates the similarity solutions are compared with the solutions found earlier the comparison shows that some previous solutions are obtained as special cases from our solutions the results reported here provide further evidence of the usefulness of the lie group method for obtaining similarity solutions for either differential or integro differential equations

on the similarity solution of the fragmentation equation an analysis of the similarity solution of the linear homogeneous fragmentation equation is given for a general volume conserving daughter fragment distribution for the special case of a polynomial daughter distribution of degree p an exact basic similarity solution for the time evolution of the particle volume distribution is derived the solution is proportional to the meijer g function which may be represented as a linear combination of generalized hypergeometric functions the properties of generalized hypergeometric functions and g functions are given in the special function literature and include the continuation of the solution to large values of the similarity variable which is given here for special cases

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

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

exact self similar solutions to the fragmentation equation with homogeneous discrete kernel abstract new large time asymptotic self similar solutions of the fragmentation breakage equation are derived in closed form for the case of certain homogeneous discrete fragmentation kernel a fragmentation kernel is called homogeneous and discrete if it allows only certain ratios between the size of the fragments and that of parent particle in this case the fragmentation kernel is not a continuous function the present solutions are the first explicit self similarity distributions for discontinuous breakage functions

theoretical analysis of the steady state particle size distribution in limited breakage processes the steady state particle size distribution is examined resulting from a breakage process with a maximum stable size and a homogeneous continuous kernel the dynamic breakage problem is transformed into one that allows direct solutions for the steady state distribution the latter depends on the breakage kernel and on the ratio of critical to initial size as this ratio goes to zero the steady state distribution approaches its limiting form obtained by the authors previously 7 a general theoretical analysis concerning the steady state distribution is presented herein the asymptotic behaviour is determined with regard to various limits perturbation analysis for nearly uniform kernels reveals several interesting features of the problem for the general continuous kernel the problem can be cast in a matrix form amenable to a conventional theoretical treatment finally comparisons of the new results with existing solutions of the dynamic problem for large times confirm their validity

statistical mechanics of the random k satisfiability model the random k satisfiability problem consisting in verifying the existence of an assignment of n boolean variables that satisfy a set of m ensuremath alpha n random logical clauses containing k variables each is studied using the replica symmetric framework of diluted disordered systems we present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest k 2 kg 3 and k ensuremath gg 1 the calculation of the number of solutions which allowed us phys rev lett 76 3881 1996 to predict a first order jump at the threshold where the boolean expressions become unsatisfiable with probability one is thoroughly displayed in the case k 2 the rigorously known critical value ensuremath alpha 1 of the number of clauses per boolean variable is recovered while for kg 3 we show that the system exhibits a replica symmetry breaking transition the annealed approximation is proven to be exact for large k

an explicit solution to a discrete fragmentation model the discrete binary fragmentation equation is solved explicitly far a model where tnenetrateoibreakupoiaparlicie oishek o equals k lj k l andthedaughter ri e distribution b lx equals 2 k 1 this system is closely related to a model of polymer degradation considered by simha in which ax i and b is as above in the continuum limit both of these models go over to a continuous fragmentation model in which all particles break with an equal rate a x i and the daughter size distribution is uniform b yl x z x which is at the borderline of the shattering transition

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

a study of the nonlinear breakage equation analytical and asymptotic solutions new solutions of the nonlinear collisional breakage equation are given using analytical and asymptotic methods the dynamic nonlinear breakage equation is transformed to a linear one for some simple forms of the collision kernel methods for treating the linear equation are employed to obtain solutions for the nonlinear case furthermore it is shown that under particular conditions the particle size distribution can take asymptotically a self similar form i e the shape of the appropriately normalized distribution is independent of time the self similar distribution is obtained from the solution of a double nonlinear integral equation the latter is solved in closed form and numerically after transformation to a boundary value problem for simple forms of the collision and breakage kernels results for the self similar distribution are presented and discussed

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

similarity solution for fragmentation with volume change the similarity form of the solution of the linear homogeneous equation for fragmentation with volume change is given the solution shows the time evolution of the particle volume distribution when a particle splits into a polynomial distribution of fragment volumes the limits for small and large values of the similarity variable are derived the long time limit for all values of the similarity variable is given and the characteristic time to approach the limit is identified the solution shows the effect of volume change on the particle volume distribution and on the time dependence of the moments of the distribution