Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

The role of thermal relaxation in nanoparticle melting is studied using a mathematical model based on the Maxwell–Cattaneo equation for heat conduction. The model is formulated in terms of a two-phase Stefan problem. We consider the cases of the temperature profile being continuous or having a jump across the solid–liquid interface. The jump conditions are derived from the sharp-interface limit of a phase-field model that accounts for variations in the thermal properties between the solid and liquid. The Stefan problem is solved using asymptotic and numerical methods. The analysis reveals that the Fourier-based solution can be recovered from the classical limit of zero relaxation time when either boundary condition is used. However, only the jump condition avoids the onset of unphysical “supersonic” melting, where the speed of the melt front exceeds the finite speed of heat propagation. These results conclusively demonstrate that the jump condition, not the continuity condition, is the most suitable for use in models of phase change based on the Maxwell–Cattaneo equation. Numerical investigations show that thermal relaxation can increase the time required to melt a nanoparticle by more than a factor of ten. Thus, thermal relaxation is an important process to include in models of nanoparticle melting and is expected to be relevant in other rapid phase-change processes.     

Decision tree search methods in fuzzy modeling and classification

This paper proposes input selection methods for fuzzy modeling, which are based on decision tree search approaches. The branching decision at each node of the tree is made based on the accuracy of the model available at the node. We propose two different approaches of decision tree search algorithms: bottom-up and top-down and four different measures for selecting the most appropriate set of inputs at every branching node (or decision node). Both decision tree approaches are tested using real-world application examples. These methods are applied to fuzzy modeling of two different classification problems and to fuzzy modeling of two dynamic processes. The models accuracy of the four different examples are compared in terms of several performance measures. Moreover, the advantages and drawbacks of using bottom-up or top-down approaches are discussed.     

Individual-based modelling and simulation of microbial processes: yeast fermentation and multi-species composting

Controlled microbial activity is the core of many industrial processes. Such dynamic microbial processes must be carefully studied to optimize their application. They are usually tackled by means of continuous mathematical modelling at the population level (top-down). An alternative approach is individual-based modelling (IbM) (bottom-up). INDISIM is a discrete and spatially explicit IbM. It sets the rules that govern each microbe and its interaction with its local environment, as well as the significant environmental processes. Then it performs simulations that include a large number of microbes, and the behaviour of the whole system emerges. The rules are changed to reproduce the behaviour of microbes depending on the system to be studied. Two adaptations of INDISIM to study yeast fermentations and multi-species composting are presented in this article (INDISIM-YEAST and INDISIM-COMP), proof of INDISIM's versatility. A few representative results are also shown.     

Statistical properties of several models of fractional random point processes

Statistical properties of several models of fractional random point processes have been analyzed from the counting and time interval statistics points of view. Based on the criterion of the reduced variance, it is seen that such processes exhibit nonclassical properties. The conditions for these processes to be treated as conditional Poisson processes are examined. Numerical simulations illustrate part of the theoretical calculations.     

Discrete time controllable processes with bounded drift and their application in queueing systems

Recursive algorithms are considered for construction of optimal policies for some special classes of controllable stochastic processes, which are called discrete-time controllable semi-regenerative processes. By means of these processes many controllable queueing systems can be modelled. Numerical examples are given for the investigation of optimal policies in such systems.     

Numerical schemes for random ODEs with affine noise

Numerical schemes for random ordinary differential equations, abbreviated RODEs, with an affine structure can be derived in a similar way as for affine control systems using Taylor expansions that resemble stochastic Taylor expansions for Stratonovich stochastic differential equations. The driving noise processes can be quite general, such as Wiener processes or fractional Brownian motions with continuous sample paths or compound Poisson processes with piecewise constant sample paths, and even more general noises. Such affine-Taylor schemes of arbitrarily high order are constructed here. It is shown how their structure simplifies when the noise terms are additive or commutative. A derivative free counterpart is given and multi-step schemes are derived too. Numerical comparisons are provided for various explicit one-step and multi-step schemes in the context of a toggle switch model from systems biology.     

A queue with a pair of instantaneous independent bernoulli feedback processes

In this paper we are concerned with several random processes that occur in M/G₂/l queue with instantaneous feedback in which the feedback decision process is a pair of independent Bernoulli processes. The stationary distribution of the output process has been obtained. Results for particular queues with feedback and without feedback are obtained. Some operating characteristics are derived for this queue. Some interesting results are obtained for departure processes. Optimum service rate is obtained. Numerical examples are provided to test the feasibility of the queueing model     

A variable step-size control algorithm for the weak approximation of stochastic differential equations

In this paper, we propose two local error estimates based on drift and diffusion terms of the stochastic differential equations in order to determine the optimal step-size for the next stage in an adaptive variable step-size algorithm. These local error estimates are based on the weak approximation solution of stochastic differential equations with one-dimensional and multi-dimensional Wiener processes. Numerical experiments are presented to illustrate the effectiveness of this approach in the weak approximation of several standard test problems including SDEs with small noise and scalar and multi-dimensional Wiener processes.     

On top-down splaying

The splay tree is a self-adjusting binary search tree which has a good amortized performance. This paper studies some properties of top-down splay trees. Different ways to charge for the primitive operations of top-down splaying are discussed. We also give some empirical results concerning the behaviour of different top-down restructuring algorithms.This work was supported by the Academy of Finland.     

Brown凝并中两个不同直径纳米颗粒的碰撞系数

对Brown凝并中两个不同直径纳米颗粒的碰撞系数进行了研究,通过求解碰撞方程,获得了在van der Waals力和弹性变形力作用下,直径为100 nm至750 nm的邻苯二甲酸二辛酯纳米颗粒的碰撞系数．发现碰撞系数总体上随着颗粒直径和2个颗粒半径比的增加而减小;当颗粒直径为550 nm时,碰撞系数有一个突然的增加．最后给出了具有不同直径2个纳米颗粒碰撞系数的新表达式．     

Numerical solutions for asymmetric Lévy flights

Numerical Algorithms - Lévy flights are generalised random walk processes where the independent stationary increments are drawn from a long-tailed α-stable jump length distribution. We...     

Windowing Waveform Relaxation of Initial Value Problems

We present a windowing technique of waveform relaxation for dynamic systems.An effectiveestimation on window length is derived by an iterative error expression provided here.Relaxation processes canbe speeded up if one takes the windowing technique in advance.Numerical experiments are given to furtherillustrate the theoretical analysis.     

A-stable one-step methods with step-size control for stiff systems of ordinary differential equations

Two efficient third-and fourth-order processes for solving the initial value problem for ordinary differential equations are studied. Both are A-stable and so recommended for stiff systems. An economic and efficient way of step-size control is given for each of them. Numerical examples are considered.     

Fejer algorithms with an adaptive step

For Fejer processes with attractants, a general adaptive scheme for step multiplier control is proposed and the convergence of this class of algorithms to stationary points is proved. Numerical results demonstrating that the convergence rate is generally linear are presented.     

Mathematical simulation of gas–liquid mixture flow in a reservoir and a wellbore with allowance for the dynamical interactions in the reservoir–well system

Fluid dynamic processes related to mature oil field development are simulated by applying a numerical algorithm based on the gas–liquid mixture flow equations in a reservoir and a wellbore with allowance for the dynamical interaction in the reservoir–well system. Numerical experiments are performed in which well production characteristics are determined from wellhead parameters.     

Acceleration by aggregation of successive approximation methods

Methods of successive approximation for solving linear systems or minimization problems are accelerated by aggregation-disaggregation processes. These processes, which modify the iterates being produced, are characterized by a two directional flow of information between the original higher dimensional problem and a lower dimensional aggregated version. This technique is characterized by means of Galerkin approximations, and this in turn permits analysis of the method. A deterministic as well as probabilistic analysis is given of a number of specific aggregation-disaggregation examples. Numerical experiments have been performed, and these confirm the analysis and demonstrate the acceleration.     

Unsteady deformation of elastic solids in the shape of a circular cylindrical segment

The unsteady deformation of cylindrical solids is investigated using the dynamic theory of elasticity. Special cases of the general solution are pointed out. Numerical results are presented which reflect the specific feature of the stressed state of an infinitely long thick-walled cylinder, which is subjected to plane nonaxi-symmetrical loading. The method of investigating unsteady wave processes in cylindrical solids is similar to that described previously [1–3].     

Simulating weighted, directed small-world networks

Imbalanced interactions or processes and directed links are often encountered in realistic networks. In this paper, we extend the well-known small-world model of Watts and Strogatz to deal with those factors. The path length and clustering properties are analyzed. Numerical results are also presented, which indicate that the extended Watts-Strogatz model may be influenced by the diversity in interaction strength.     

Integration processes of ordinary differential equations based on Laguerre-Radau interpolations

In this paper, we propose two integration processes for ordinary differential equations based on modified Laguerre-Radau interpolations, which are very efficient for long-time numerical simulations of dynamical systems. The global convergence of proposed algorithms are proved. Numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis.

     

The computation of stability boundaries in state space for a class of biochemical engineering systems

The stability of a class of biochemical processes defined by a set of m biochemical reactions involving n components is analysed. The processes operate in a continuous mode and possess at least two stable equilibrium states: the normal operating point and a biological wash out state. Using a canonical state representation of the process dynamics the geometric structure of the operating point's stability boundary is characterized. Numerical algorithms are developed to evaluate this boundary and to visualize its extent in state space. The proposed technique is illustrated with a representative engineering example.