Implicit Taylor methods for stiff stochastic differential equations

In this paper we discuss implicit Taylor methods for stiff Ito stochastic differential equations. Based on the relationship between Ito stochastic integrals and backward stochastic integrals, we introduce three implicit Taylor methods: the implicit Euler-Taylor method with strong order 0.5, the implicit Milstein-Taylor method with strong order 1.0 and the implicit Taylor method with strong order 1.5. The mean-square stability properties of the implicit Euler-Taylor and Milstein-Taylor methods are much better than those of the corresponding semi-implicit Euler and Milstein methods and these two implicit methods can be used to solve stochastic differential equations which are stiff in both the deterministic and the stochastic components. Numerical results are reported to show the convergence properties and the stability properties of these three implicit Taylor methods. The stability analysis and numerical results show that the implicit Euler-Taylor and Milstein-Taylor methods are very promising methods for stiff stochastic differential equations.

Topological methods for some boundary value problems

We establish existence of solutions for a finite difference approximation to y = f(x, y, y ') on [0, 1], subject to nonlinear two-point Sturm-Liouville boundary conditions of the form g(i)(y(i),y ' (i)) = 0, i = 0, 1, assuming S satisfies one-sided growth bounds with respect to y '. (C) 2001 Elsevier Science Ltd. All rights reserved.

Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations

In this paper we discuss implicit methods based on stiffly accurate Runge-Kutta methods and splitting techniques for solving Stratonovich stochastic differential equations (SDEs). Two splitting techniques: the balanced splitting technique and the deterministic splitting technique, are used in this paper. We construct a two-stage implicit Runge-Kutta method with strong order 1.0 which is corrected twice and no update is needed. The stability properties and numerical results show that this approach is suitable for solving stiff SDEs. (C) 2001 Elsevier Science B.V. All rights reserved.

Development, reliability, and concurrent validity of the modified inventory of potential reconstructive needs

Documentation of burn sequelae can be a difficult and time-consuming task. To date a reliable and systematic format for recording postburn trauma is lacking. The purpose of this research was two-fold: first, to develop a Modified Inventory of Potential Reconstructive Needs from the original Inventory of Potential Reconstructive Needs to allow methodical documentation of functional and cosmetic burn sequelae in all body surface areas of children with burns and, second, to establish interrater reliability and concurrent validity of the instrument, thus allowing its clinical application. Two raters scored the Modified Inventory of Potential Reconstructive Needs on 41 children with a range of burns types and severity. Excellent interrater reliability was demonstrated for both total (intraclass correlation coefficient = 0.996) and subsection inventory scores. Concurrent validity was also established with total scores showing strong positive correlations (0.73-0.76) with three indicators of burn severity. These findings provide initial support for the tool's clinical applicability, particularly in relation to rehabilitative planning and documentation.