Null-Space Methods for Numerical Solutions of Differential Equations

A linear differential operator often has a nontrivial null space. One consequence for such an operator, L, is that solutions of equations of the form Lu=f are never unique if the null space contains more than the zero element: adding any nontrivial null function to any particular solution yields another solution. Out of the infinite set of functions satisfying the differential equation, the conventional way to select a desired solution is to require the solution (or its derivatives) to take on prescribed values at specified points, typically on the boundary of the domain over which the equation is defined. For some applications, however, it may be inconvenient or inappropriate to specify such boundary conditions. For certain problems in materials science, for example, it is more natural to specify the null-space component of the solution directly rather than indirectly via (often unknown) boundary conditions. One particular example is atomic scale simulation of stress in metals. When calculations are made on a tiny scale, the edges of the metal, which would provide a boundary for the problem, are practically an infinite distance away. In this case, the conventional method is difficult to apply, even if it were computationally feasible. Instead, a natural alternative is to specify the null-space component to single out a particular solution. In this thesis, we will develop numerical methods for computing approximate solutions to linear differential equations subject to explicit specification of the null-space component. For this purpose we will develop discretized approximations to the null spaces of relevant differential operators as well as numerical solution procedures that take advantage of such an explicit representation. To the best of our knowledge, this explicit null-space approach and our implementation of it are new. This thesis details the problem we seek to solve, discusses options for finding null bases, explains the explicit null bases we have found, and demonstrates a solution technique for solving the problem referenced above using a null space method

Null -Space Methods for Numerical Solutions of Differential Equations

141 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2007.This thesis details the problem we seek to solve, discusses options for finding null bases, explains the explicit null bases we have found, and demonstrates a solution technique for solving the problem referenced above using a null space method.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD

Patterning of RAFT Polymer Networks by Laser Deactivation

Abstract Reversible addition‐fragmentation chain‐transfer (RAFT) polymer networks grow in interest due to their living and tunable characteristics. An essential step toward exploiting these characteristics is the production of precise patterns for growth. Herein, a novel, straightforward strategy to produce patterns to control growth on RAFT‐based living polymer networks (LPNs) is presented. A 405 nm laser is used to degrade the RAFT agents in selected regions, generating a pattern. This occurs through homolytic photocleavage at the C─S bond, leading to the typical RAFT equilibrium state with the R‐group radical and a thiyl radical species, followed by irreversible loss of CS2 to produce a Z‐group derived thiol. Using this technique, both “on/off” patterns and more detailed greyscale patterns are produced using images of varying complexity. Post production growth then shows that these patterns are visually retained after iniferter growth with a fluorescent monomer. Due to the deactivation of selected RAFT agents, the authors are able to perform discriminate growth with network‐wide irradiation. This selective degradation technique serves as an alternative technique for producing patterns on RAFT LPNs