Fragmentation Kernel Research Articles

Explicit and approximate solutions for a classical hyperbolic fragmentation equation using a hybrid projected differential transform method

Population balance equations are widely used to study the evolution of aerosols, colloids, liquid–liquid dispersion, raindrop fragmentation, and pharmaceutical granulation. However, these equations are difficult to solve due to the complexity of the kernel structures and initial conditions. The hyperbolic fragmentation equation, in particular, is further complicated by the inclusion of double integrals. These challenges hinder the analytical solutions of number density functions for basic kernel classes with exponential initial distributions. To address these issues, this study introduces a new approach combining the projected differential transform method with Laplace transform and Padé approximants to solve the hyperbolic fragmentation equation. This method aims to provide accurate and efficient explicit solutions to this challenging problem. The approach's applicability is demonstrated through rigorous mathematical derivation and convergence analysis using the Banach contraction principle. Additionally, several numerical examples illustrate the accuracy and robustness of this new method. For the first time, new analytical solutions for number density functions are presented for various fragmentation kernels with gamma and other initial distributions. This method significantly enhances solution quality over extended periods using fewer terms in the truncated series. The solutions are compared and verified against the finite volume method and the homotopy perturbation method, showing that the coupled approach not only estimates number density functions accurately but also captures integral moments with high precision. This research advances computational methods for particle breakage phenomena, offering potential applications in various industrial processes and scientific disciplines.

Local mass-conserving solution for a critical coagulation-fragmentation equation

The critical coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels is known to not have global mass-conserving solutions when the initial mass is greater than 1. We show that for any given positive initial mass with finite second moment, there is a time T⁎>0 such that the equation possesses a unique mass-conserving solution up to T⁎. The novel idea is to singularly perturb the constant fragmentation kernel by small additive terms and study the limiting behavior of the solutions of the perturbed system via the Bernstein transform.

Stochastic Theory of Discrete Binary Fragmentation-Kinetics and Thermodynamics.

We formulate binary fragmentation as a discrete stochastic process in which an integer mass k splits into two integer fragments j, , with rate proportional to the fragmentation kernel . We construct the ensemble of all distributions that can form in fixed number of steps from initial mass M and obtain their probabilities in terms of the fragmentation kernel. We obtain its partition function, the mean distribution and its evolution in time, and determine its stability using standard thermodynamic tools. We show that shattering is a phase transition that takes place when the stability conditions of the partition function are violated. We further discuss the close analogy between shattering and gelation, and between fragmentation and aggregation in general.

Finite volume approach for fragmentation equation and its mathematical analysis

This work is focused on developing a finite volume scheme for approximating a fragmentation equation. The mathematical analysis is discussed in detail by examining thoroughly the consistency and convergence of the numerical scheme. The idea of the proposed scheme is based on conserving the total mass and preserving the total number of particles in the system. The proposed scheme is free from the trait that the particles are concentrated at the representative of the cells. The verification of the scheme is done against the analytical solutions for several combinations of standard fragmentation kernel and selection functions. The numerical testing shows that the proposed scheme is highly accurate in predicting the number distribution function and various moments. The scheme has the tendency to capture the higher order moments even though no measure has been taken for their accuracy. It is also shown that the scheme is second-order convergent on both uniform and nonuniform grids. Experimental order of convergence is used to validate the theoretical observations of convergence.

Accurate and efficient approximations for generalized population balances incorporating coagulation and fragmentation

This study focuses on development of two approaches based on finite volume schemes for solving both one-dimensional and multidimensional nonlinear simultaneous coagulation-fragmentation population balance equations (PBEs). Existing finite volume schemes and sectional methods such as fixed pivot technique and cell average technique have many issues related to accuracy and efficiency. To resolve these challenges, two finite volume schemes are developed and compared with the cell average technique along with the exact solutions. The new schemes have features such as simpler mathematical formulations, easy to code and robust to apply on nonuniform grids. The numerical testing shows that both new finite volume schemes compute the number density functions and their corresponding integral moments with higher precision on a coarse grid by consuming lesser CPU time. In addition, both schemes are extended to approximate generalized simultaneous coagulation-fragmentation problems and retains the numerical accuracy and efficiency. For the higher dimensional PBEs (2D and 3D), the investigation and verification of the numerical schemes is done by deriving new exact integral moments for various combinations of coagulation kernels, selection functions and fragmentation kernels.

Coagulation‐Fragmentation Equations with Multiplicative Coagulation Kernel and Constant Fragmentation Kernel

AbstractWe study a critical case of coagulation‐fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton‐Jacobi equation, which results from applying the Bernstein transform to the original coagulation‐fragmentation equation. Our results include well‐posedness, regularity, and long‐time behaviors of viscosity solutions to the Hamilton‐Jacobi equation in certain regimes, which have implications to well‐posedness and long‐time behaviors of mass‐conserving solutions to the coagulation‐fragmentation equation. © 2021 Wiley Periodicals LLC.

Large time behavior for a Hamilton–Jacobi equation in a critical coagulation-fragmentation model

We study the large time behavior of the sublinear viscosity solution to a singular Hamilton-Jacobi equation that appears in a critical Coagulation-Fragmentation model with multiplicative coagulation and constant fragmentation kernels. Our results include complete characterizations of stationary solutions and optimal conditions to guarantee large time convergence. In particular, we obtain convergence results under certain natural conditions on the initial data, and a nonconvergence result when such conditions fail.

Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation

AbstractWe consider a stochastic individual‐based model in continuous time to describe a size‐structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We here address the problem of nonparametric estimation of the kernel ruling the divisions based on the eigenvalue problem related to the asymptotic behavior in large population. This inverse problem involves a multiplicative deconvolution operator. Using Fourier techniques we derive a nonparametric estimator whose consistency is studied. The main difficulty comes from the nonstandard equations connecting the Fourier transforms of the kernel and the parameters of the model. A numerical study is carried out and we pay special attention to the derivation of bandwidths by using resampling.

Solutions with peaks for a coagulation-fragmentation equation. Part II: Aggregation in peaks

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. In a companion paper we constructed a two-parameter family of stationary solutions concentrated in Dirac masses, and we carefully studied the asymptotic decay of the tails of these solutions, showing that this behaviour is stable. In this paper we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.

Existence and uniqueness of steady-state solution to a singular coagulation–fragmentation equation

In this article, we prove the existence of an equilibrium solution to the coagulation–fragmentation equation with •[(i)] a singular coagulation kernel in the form K(x,y)=(1+xλ+yλ)(xy)σ,where σ∈0,12 and λ−σ∈[0,1], and•[(ii)] a fragmentation kernel in the form F(x,y)=b0+b(x+y)m,where b0>0 and b,m≥0. Importantly, in the entire study, the detailed balance condition is not assumed. It is shown that the equilibrium solution of the problem under consideration is unique in the space of all continuous functions on (0,∞). The proposed results are demonstrated by a numerical illustration.

Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.

Numerical approach for stochastic differential equations of fragmentation; application to avalanches

We build and develop a unifying method for the construction of a continuous time branching–fragmentation processes on the space of all fragmentation sizes, induced either by continuous fragmentation kernels or by discontinuous ones. This construction leads to a stochastic model for the fragmentation phase of an avalanche. We introduce an approximation scheme for the process which solves the corresponding stochastic differential equations of fragmentation. One of the main achievements of the paper is to compute the distributions of the branching processes approximating the forthcoming branching–fragmentation process. Finally, we present numerical results that confirm the validity of the fractal property which was emphasized by our model for an avalanche.

Existence of mass conserving solution for the coagulation–fragmentation equation with singular kernel

In this article, we investigate the existence of a solution to the initial value problem associated to a singular coagulation–fragmentation equation. We analyze the problem with singular coagulation kernel and multiple fragmentation kernel. It is also shown that the solution satisfies the mass conservation law.

Steady oscillations in aggregation-fragmentation processes.

We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels K_{i,j}=i^{ν}j^{μ}+j^{ν}i^{μ} homogeneous in masses i and j of merging clusters and fragmentation kernels, F_{ij}=λK_{ij}, with parameter λ quantifying the intensity of the disruptive impacts. We assume a complete decomposition (shattering) of colliding partners into monomers. We show that an assumption of a steady-state distribution of cluster sizes, compatible with governing equations, yields a power law with an exponential cutoff. This prediction agrees with simulation results when θ≡ν-μ<1. For θ=ν-μ>1, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small λ, they become steady if θ is close to 2 and λ is very small. Simulation results lead to a conjecture that for θ<1 the system has a stable fixed point, corresponding to the steady-state density distribution, while for any θ>1 there exists a critical value λ_{c}, such that for λ<λ_{c}, the system has an attracting limit cycle. This is rather striking for a closed system of Smoluchowski-like equations, lacking any sinks and sources of mass.

Steady distribution of the incremental model for bacteria proliferation

We study the mathematical properties of a model of cell division structured by two variables, the size and the size increment, in the case of a linear growth rate and a self-similar fragmentation kernel. We first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue 1 from a solution of a certain one dimensional fixed point problem. Then we prove the existence and uniqueness of this fixed point in the appropriate $\mathrm{L}^1$ weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.

Estimating the division rate and kernel in the fragmentation equation

We consider the fragmentation equation∂f∂t(t,x)=−B(x)f(t,x)+∫y=xy=∞k(y,x)B(y)f(t,y)dy, and address the question of estimating the fragmentation parameters – i.e. the division rate B(x) and the fragmentation kernel k(y,x) – from measurements of the size distribution f(t,⋅) at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance Xue and Radford (2013) [26] for amyloid fibril breakage. Under the assumption of a polynomial division rate B(x)=αxγ and a self-similar fragmentation kernel k(y,x)=1yk0(xy), we use the asymptotic behavior proved in Escobedo et al. (2004) [11] to obtain uniqueness of the triplet (α,γ,k0) and a representation formula for k0. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation

In general, the non-conservative approximation of coagulation-fragmentation equations (CFEs) may lead to the occurrence of gelation phenomenon. In this article, it is shown that the non-conservative approximation of CFEs can also provide the existence of mass conserving solutions to CFEs for large classes of unbounded coagulation and fragmentation kernels.

Rigorous approach to fragmentation equation for irreversible epitaxial growth in the one-dimensional point island model

Rigorous second quantization formalism has been used to derive a fragmentation equation (FE) for the one-dimensional (1D) epitaxial growth. For simplicity, the point island model (PIM) and the case have been studied. In the continuum approximation and a mean-field decoupling the exact FE reduces to that suggested in the literature but this time with rigorous formal definitions of its terms. Furthermore, a correction factor accounting for the discrete nature of the monomers that was missing in known FEs for the 1D epitaxy has been included in the formalism. A novel technique for a numerical solution of the FEs has been suggested and illustrated in the problem of the growth following the instant deposition of the monomers. Good agreement with our earlier Monte Carlo simulations of the growth was found. In the calculations a heuristic ansatz for the fragmentation kernel appropriate to the instant deposition case has been used but the technique is also suitable for the solution of the FEs corresponding to the more conventional case of continuous monomer deposition.

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.

Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate

We are interested in the large time behavior of the solutions to the growth-fragmentation equation. We work in the space of integrable functions weighted with the principal dual eigenfunction of the growth-fragmentation operator. This space is the largest one in which we can expect convergence to the steady size distribution. Although this convergence is known to occur under fairly general conditions on the coefficients of the equation, we prove that it does not happen uniformly with respect to the initial data when the fragmentation rate in bounded. First we get the result for fragmentation kernels which do not form arbitrarily small fragments by taking advantage of the Dyson–Phillips series. Then we extend it to general kernels by using the notion of quasi-compactness and the fact that it is a topological invariant.