|Abstract:||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.