Mathematical modelling of rock bolt systems II

The main goal of the paper is to describe a reinforcement consisting of fully grouted bolts, which is applied to stabilizing underground openings and tunnels. After a variational formulation is given, the existence and uniqueness is proved. Some asymptotic results that make it possible to replace the real system with a continuous one more suitable for discretization are presented. Some other types of reinforcements and properties are studied.     

Mathematical modelling of rock bolt systems I

The main goal of the paper is to give a variational formulation of the behaviour of bolt systems in rock mass. The problem arises in geomechanics where bolt systems are applied to reinforce underground openings by inserting steel bars or cables. After giving a variational formulation, we prove the existence and uniqueness and some other properties.     

Mathematical Model of Ice Melting on Transmission Lines

During ice storms, ice forms on high voltage electrical lines. This ice formation often results in downed lines and has been responsible for considerable damage to life and property as was evidenced in the catastrophic ice storm of Quebec recently. There are two main aspects, viz., the formation of ice and its timely mitigation. In this paper, we mathematically model the melting of ice due to a higher current applied to the transmission wire. The two dimensional cross-section contains four layers consisting of the transmission wire, water due to melting of ice, ice, and the atmosphere. The model includes heat equations for the various regions with suitable boundary conditions. Heat propagation and ice melting are expressed as a Stefan-like problem for the moving boundary between the layers of ice and water. The model takes into account gravity which leads to downward motion of ice and to forced convection of heat in the water layer. In this paper, the results are applied to the case when the cross-sections are concentric circles to yield melting times for ice dependent on the increase in intensity of the electrical flow in the line. This research has been supported in part by Manitoba Hydro and NSERC.     

Dynamic analysis of a non-conservative band-wheel system with a moving boundary Part 1: Mathematical modelling

The dynamic band/wheel system with a moving boundary is very different from that with a fixed length, it is a non-conservative mass system and the boundary position is not fixed. In this paper, the moving boundary is one unknown to be determined associated with the string displacement, and the partial differential equation of the transverse vibrations of the band and the transversality condition of the moving boundary are derived by the calculus of variation and Hamilton’s principle. In order to verify the dynamic formulation is correct, we reduce the governing equation and separation point to coincide with the previous studies. It is found that the physical properties of the moving boundary could be obtained from the geometric constraint of the band/wheel system.     

Mathematical modelling of heat transfer in a catalytic reformer

In this work we analyse heat transfer in a system of channelsconnected by thin conducting walls. The channels are packedwith catalytic pellets that promote exothermic catalytic combustionreactions and endothermic reforming reactions in adjacent channels.A model is developed in which the thermal conductivity and thethickness of the interconnecting wall can be used as controlparameters characterizing the heat exchange between the neighbouringchannels. The model is to be used as a mathematical tool toanalyse design alternatives and develop accurate numerical techniques.Our objective is to study how the heat is transferred acrossthe conducting walls and how this influences the temperaturedistribution in the channels. We use an asymptotic techniqueto do this. The structure of the walls is then examined in detail,focusing on the case when we have layered walls.     

Mathematical modelling of the catalyst layer of a polymer electrolyte fuel cell

In this paper, we derive a mathematical model for the cathodecatalyst layer of a polymer electrolyte fuel cell. The modelexplicitly incorporates the restriction placed on oxygen inreaching the reaction sites, capturing the experimentally observedfall in the current density to a limiting value at low cellvoltages. Temperature variations and interfacial transfer ofO₂ between the dissolved and gas phases are also included. Boundson the solutions are derived from which we provide a rigourousproof that the model admits a solution. Of particular interestare the maximum and minimum attainable values. We perform anasymptotic analysis in several limits inherent in the problemby identifying important groupings of parameters. This analysisreveals a number of key relationships between the solutions,including the current density, and the composition of the layer.A comparison of numerically computed solutions and asymptoticsolutions shows very good agreement. Implications of the resultsare discussed and future work is outlined.     

矩阵方程AoA-T=1/2J2的实数解的应用

相对增益阵列(RGA)大多数应用的矩阵阶数都是较小的(n=2,3或4). 我们从矩阵方程Φ(A)=1/2J2的实数解出发,应用矩阵方程Φ(A)=1/nJn的实数解在G-等价下的不变性和实数解的分块构造法,研究了Φ(A)=1/4J4的实数解的一些问题.     

On Periodic Solutions of Linear Control Systems

We prove a theorem of the alternative concerning periodic solutions of linear control systems. We show that, if the attainable sets have nonempty interior, then the main condition assumed in Theorem 4.2 of Ref. 1 (a tangential condition) can be weakened and simplified, and a stronger conclusion can be obtained.     

一类半线性积分微分方程初边值问题的爆破解和全局解

本文研究初边值问题的爆破解和全局解，证明了在ｆ的凸性假设和一定的增长性假定下解在有限时刻爆破，而在ｆ的其他假设下证明了全局解的存在性。     

二阶奇异微分方程边值问题正解的存在性

该文通过构造一个特殊的锥,利用锥压缩与锥拉伸不动点定理,研究了一类二阶奇异微分方程边值问题两个正解的存在性,改进了最近的一些结果.     

四阶奇异边值问题的正解

研究了一类四阶奇异边值问题正解的存在性,在f和g满足比超线性和次线性条件更广泛的极限条件下,利用锥压缩和拉伸不动点定理获得了正解的存在性结果,推广和包含了一些已知结果.     

The transmission problem in bending of plates with transverse shear deformation

The existence of weak solutions is proved for contact problemsin bending of elastic plates with transverse shear deformation,and for various boundary integral equations arising from potentialrepresentations of such solutions.     

四阶奇异边值问题正解的全局结构

利用上下解方法研究了一类四阶奇异边值问题正解的全局结构.证明了解集存在无界连通分支并给出了多个正解的存在性结果,推广和改进了相关文献的结果.     

三阶微分方程组边值问题常号解的存在性

利用Krasnoselskii不动点定理,结合Leray-Schauder度,研究下列三阶微分方程组边值问题{ui″′(t)=fi(t,u1(t),u2(t),u3(t)), t∈[0,1],/ui′(0)=ui″(0)=ui(1)=0, i=1,2,3. 在某些条件下,常号解的存在性和多解性.     

Mathematical modelling of the patent ductus arteriosus (PDA)

The heart is one of the most important and vital body organs, which is responsible for pumping blood to different body tissues. Abnormalities in each part of heart can lead to lack of proper blood circulation. The arterial duct is one of the most important parts of embryonic cycles which may provide the possibility of a relationship between the aorta and the pulmonary artery. The failure of this duct to close after birth causes the abnormality called patent ductus arteriosus (PDA). In this article, we propose a novel model for accurate simulation of PDA abnormality in the human heart. The model is obtained by exploiting the mathematical relation between hemodynamic parameters of heart, including pressure, blood, volume and flow. The proposed model is verified using clinical data.     

Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters

This paper deals with a nonlinear diffusion equation with double free boundaries possessing different moving parameters. We present the spreading–vanishing dichotomy and threshold between spreading and vanishing. Moreover, when spreading happens, using the zero number argument we provide sharp estimates of spreading speeds of expanding fronts, and describe how the solution approaches the semi-wave.     

关于三阶奇摄动边值问题的唯一性(英文)

本文利用上下解的构造研究了三阶奇摄动边值问题 ,建立了解的唯一性定理 .     

A remark on an integral equation via the method of moving spheres

In this paper, we study the following integral equation:

     

Mathematical Modeling to Simulate the Movement of Contaminants in Groundwater

The objectives of this paper are twofold. Firstly, we formulate a system of partial differential equations that models the contamination of groundwater due to migration of dissolved contaminants through unsaturated to saturated zone. A closed form solution using the singular perturbation techniques for the flow and solute transport equations in the unsaturated zone is obtained. Indeed, the solution can be used as a tool to verify the accuracy of numerical models of water flow and solute transport. The second part of this paper, deals with how the water level in a water reserve drops due to pumping water out of a well that is some distance away.