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.

The expression of plasminogen activator system in a rat model of periodontal wound healing

Background: The plasminogen activator system has been proposed to play a role in proteolytic degradation of extracellular matrices in tissue remodeling, including wound healing. The aim of this study was to elucidate the presence of components of the plasminogen activator system during different stages of periodontal wound healing. Methods: Periodontal wounds were created around the molars of adult rats and healing was followed for 28 days. Immunohistochemical analyses of the healing tissues and an analysis of the periodontal wound healing fluid by ELISA were carried out for the detection of tissue-type plasminogen activator (t-PA), urokinase-type plasminogen activator (u-PA), and 2 plasminogen activator inhibitors (PAI-1 and PAI-2). Results: During the early stages (days 1 to 3) of periodontal wound healing, PAI-1 and PAI-2 were found to be closely associated with the deposition of a fibrin clot in the gingival sulcus. These components were strongly associated with the infiltrating inflammatory cells around the fibrin clot. During days 3 to 7, u-PA, PAI-1, and PAI-2 were associated with cells (particularly monocytes/macrophages, fibroblasts, and endothelial cells) in the newly formed granulation tissue. During days 7 to 14, a new attachment apparatus was formed during which PAI-1, PAI-2, and u-PA were localized in both periodontal ligament fibroblasts (PDL) and epithelial cells at sites where these cells were attaching to the root surface. In the periodontal wound healing fluid, the concentration for t-PA increased and peaked during the first week. PAI-2 had a similar expression to t-PA, but at a lower level over the entire wound-healing period. Conclusions: These findings indicate that the plasminogen activator system is involved in the entire process of periodontal wound healing, in particular with the formation of fibrin matrix on the root surface and its replacement by granulation tissue, as well as the subsequent formation of the attachment of soft tissue to the root surface during the later stages of wound repair.