Generalized Trapezoidal Formulas For The Symmetric Heat Equation In Polar Coordinates II. The Case Of Point Sources

The present paper is in continuation of our previous paper Chawla et al. [4]. An important class of applications of the radially symmetric heat equation in polar coordinates: u_{t}=v ( u_{rr}\,+\; ( a/r ) u_{r} ), involve the presence of a continuous point source of heat at the centre of the sphere ( a=2 ) or on the axis of the cylinder ( a=1 ). This necessitates a modification of the radial grid used in [4]; our modification of the radial grid in the present paper accommodates a point source of heat at r=0 in a natural way. We then extend generalized trapezoidal formulas GTF( \alpha ) for the one-step time integration of these problems. Again, with the help of the generalized finite Hankel transforms introduced in [4] we are able to obtain, in a natural way, analytical solutions of the heat equation in the presence of point sources of heat for both the cases a=1 and a=2 . Numerical experiments are provided to compare the performance of the GTF( \alpha ) time integration scheme with the schemes based on the backward Euler and the classical arithmetic-mean trapezoidal formula.     

Tridiagonal solver-based L-stable one-step schemes for nonlinear parabolic equations

An attractive feature of the widely used Crank-Nicolson (C-N) scheme for parabolic equations is that it is a tridiagonal solver-based (TSB) scheme. But, in case of inconsistencies in the initial and boundary conditions or when the ratio of temporal to spatial steplengths is large, it can produce unwanted oscillations or an unacceptable solution. As alternative to C-N, Chawla et al. [2, 3] introduced L-stable generalized trapezoidal formulas (GTF(α)) which can give a more acceptable solution by a judicious choice of the parameter α; however, GTF are not TSB schemes. It is natural to ask for L-stable TSB schemes. In the present paper, we first introduce a one-parameter family of generalized midpoint formulas (GMF(α)); again GMF are not TSB schemes. We then introduce a two-parameter family through a linear combination of the GMF and the classical trapezoidal formula, and show the existence of a one-parameter subfamily of L-stable TSB schemes; these schemes are unconditionally stable. The computational performance of the obtained schemes is compared with the C-N scheme by considering a nonlinear reaction-diffusion equation.     

An alternating direction generalized trapezoidal formula scheme for parabolic differential equations in two space dimensions

A well-known ADI scheme for parabolic differential equations in two space dimensions is the Peaceman-Rachford scheme; this scheme employs the backward Euler formula for integration in time and is unconditionally stable. An ADI Crank -Nicolson scheme, which employs the classical trapezoidal formula for integration in time, is unconditionally unstable. We investigaan ADI implementation of the generalized trapezoidal formula GTF(α) for integration in time. The obtained ADI-GTF(α) scheme is unconditionally stable for all α ≥ 1; interestingly, ADIGTF(α) scheme includes the Peaceman-Rachford scheme for α→∞. Numerical experiments demonstrate that while the Peaceman-Rachford scheme can give quite pronounced unwanted oscillations in the computed solution, an ADI-GTF(α) scheme can provide a more stable and accurate approximation for the true solution.     

广义预测控制在火电厂单元机组协调控制中的应用研究

设计有效的火电厂单元机组机炉协调控制系统是提高热工自动化水平的关键问题。本文采用了广义预测控制算法，它具有算法简单，在线计算量小，鲁棒性强等特点。将其应用于火电厂单元机组的协调控制系统进行仿真研究，对被控对象的不确定性具有良好的适应性和鲁棒性。仿真结果表明该算法是有效的。     

On the Convergence of a Spline Method for Singular Two Point Boundary Value Problems Arising in Physiology

The second order spline method developed by Iyengar et al. [11] for the problems $$eqalign{ x^{-alpha}(x^alpha y^prime)^prime & = f(x,y),quad 0