An Adaptive Finite-Difference Method for Solving the Sturm-Liouville Problem.
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OHIO
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The Sturm-Liouville S-L boundary value problem arises frequently in the study of mathematical physics. Most problems cannot be solved analytically and therefore must be solved with one of the many numerical techniques available. The majority of the current methods use either a uniform mesh or a predetermined non-uniform mesh on which to approximate the problem. It is widely accepted that such problems can be solved more accurately and efficiently with an adaptive method. In this thesis, primarily, the finite-difference methods are discussed for solving the S-L problem. An adaptive finite-difference method is developed in which the final mesh of grid points depends on the solution. The non-uniform discretization and the resulting matrix problem are explored in detail by establishing all the important properties, selecting a solution method, deriving convergence results and error estimates, and examining mesh refinement criteria.
- Theoretical Mathematics