The Numerical Solution of the Fourth Boundary Value Problem for Parabolic Partial Differential Equations

A direct method for the numerical solution of the implicit finitedifference equations derived from a parabolic differential equationwith periodic spatial boundary conditions is presented in algorithmicfrom. Consideration is given to the stability of the roundingerrors involved in the solution process and numerical resultsare derived which compare favourably with those obtained fromthe analytical solution and a matrix spectral resolution methodwhich is closely allied to the method of lines.     

Configurational temperature thermostat for fluids undergoing shear flow: application to liquid chlorine

Non-equilibrium molecular dynamics (NEMD) simulations play a major role in characterizing the rheological properties of fluids undergoing shear flow. However, all previous studies of flows in molecular fluids either use an ‘atomic’ thermostat which makes incorrect assumptions concerning the streaming velocity of atoms within their constituent molecules, or they employ a centre of mass kinetic (COM) thermostat which only controls the temperature of relatively few degrees of freedom (3) in complex high molecular weight compounds. In the present paper we show how recently developed configurational expressions for the thermodynamic temperature can be used to develop thermostatting mechanisms which avoid both of these problems. In this work, we propose a thermostat based on a configurational expression for the temperature and apply it to NEMD simulations of chlorine undergoing Couette flow. The results so obtained are compared with those obtained using a COM kinetic thermostat. At equilibrium the properties of systems thermostatted in the two different ways are of course equivalent. We show that the two responses only differ far from equilibrium. In particular, we show that the formation of a string phase for extremely high shear rates is an artefact of the COM thermostat. At the largest shear rates studied with the configurational thermostat, no string phase is observed.     

The Successive Peripheral Block Over-relaxation Method

In the finite difference representation of a partial differentialequation, the mesh points on a two dimensional grid are orderedin a new manner, viz. around successive peripherals of the regionof integration. The resulting coefficient matrix is such that the theory ofsuccessive block over-relaxation is valid and this techniqueis used to solve Dirichlet problems in regions particularlysuited to the new ordering viz. the unit square with a squarehole removed from the centre and a circular annular region.Comparisons are made between this new ordering and standardmethods and improved rates of convergence are obtained.     

Cu-Ni-Mg-Al-CO3四元水滑石的合成及结构分析

以Cu(NO3)2·2H2O,Ni(NO3)2·6H2O,Mg(NO3)2·6H2O和Al(NO3)3·9H2O为原料 ，以NaOH和Na2CO3为沉淀剂，采用成核／晶化隔离法合成了Cu-Ni-Mg-Al-CO3四元 水滑石，通过XRD，FT－IR，TG－DTA，ICP等手段对样品进行测试和表征，并且对 水滑石结构的层板和层间阴离子相互作用进行分析和讨论。结果表明，控制Cu^2+ 离子与其它二价金属离子的配比，可制备晶体结构规整的M(Ⅱ）4M(Ⅲ)2(OH) 12CO3·4H2O型水滑石；Cu^2+离子的引入导致了明显的Jahn-Teller效应；且随 Mg^2+量的增加，使得这种效应更加显著，另一方面Mg^2+量的增加有利于Cu-Ni- Mg-Al-CO3四元水滑石晶体在a方向上生长，增强其热稳定性。     

Accurate Numerical Determination of the Intersection Point of the Solution of a Differential Equation with a Given Algebraic Relation

In this paper, the authors present an adaptation of the simplerational extrapolation scheme (Bulirsch & Stoer, 1966) tofind the point of intersection of the solution curve of an ordinarydifferential equation with a given curve or algebraic relationship.This point could also possibly be a point of discontinuity ofthe derivative. The integration algorithm is illustrated witha simple example.     

Piecewise Polynomial Collocation for Volterra-type Integral Equations of the Second Kind

The exact solution of a given integral equation of the secondkind of Volterra type(with regular or weakly singular kernel)is projected into the space of (continuous) piecewise polynomialsof degree m 1 and with prescribed knots by using collocationtechniques. It is shown that a number of discrete methods forthe numerical solution of such equations based on product integrationtechniques or on finite-difference methods are particular discreteversions of collocation methods of the above type. The errorbehaviour of approximate solutions obtained by collocation (includingtheir discretizations) is discussed.     

The Solution of a Two-dimensional Time-dependent Diffusion Problem Concerned with Oxygen Metabolism in Tissues

An approximate numerical solution of the diffusion equationin a semi-infinite two-dimensional domain is ibtained usingfa hopscotch technique. The solution describes the movement ofoxygen in a metabolizing medium covered by a finite layer ofanother non-metabolizing material. The initial condition isdetermined by the steady state resulting from a uniform concertrationof oxygen concerntration within a circular area r R in thebounding surface, and zero oxygen gradient normal to the surfaceforr > R. The time-dependence of oxygen flux into the circularares is obtained by an interpolation and integration method.