Math 572 - Spring 2008 - Course Outline/Diary

Material will be entered here in reverse order, so future and current information will be at the top. I'll attempt to document what happened in class with references to the text, handouts, etc.

• Thursday, 2/7: Switch to two dimensions: basic formulas, systems, ordering unknowns, etc.
• Tuesday, 2/5: More topics: nonlinear BVPs, nonuniform grids, adaptive meshes, general linear BVPs. Sample code: bvp.m (Uniform Mesh) bvp2.m (Non-Uniform Mesh)
• Thursday, 1/31: Examples and extensions to nonlinear BVPs
• Tuesday, 1/29: Stability, Consistency and Convergence (the big result for solving DEs)
• Thursday, 1/24: The third approach to finite difference formulas is to use polynomial interpolant. This is a good way to derive the formulas. Also, we set up our first finite difference method problem for u'' = f with u(0) = a, u(L) = b. Replacing u'' by a finite difference formula and then replacing u(x) by appropriate U_i, we have a linear system to solve. Next we need to talk about errors.
• Tuesday, 1/22: Accuracy of finite difference formulas can be tested/determined by applying to polynomials as there is a relation between the accuracy (the power of h), the order of the derivative in the leading error term, and thus its effect on polynomials.
• Thursday, 1/17: Norms: Vector, Function, Grid Function (A.3,4,5); Finite Differences (Chapter 1) (finally!) Unless otherwise stated every norm will be a function or a grid function norm and the context should tell you which of those two. For finite differences: get used to Taylor expansions (Homework #1 Due Thurs 1/24)
• Tuesday, 1/15: More basics: Calculus Basics: Taylor's Theorem in 1 and multi-dimensions: the main tool for analyzing methods for solving DEs (Summary), Errors: Big Oh notation (A.2). If there's an important theme in these, it's that theory and/or mathematically accurate statements lead directly to results which are computable. In the case of Taylor, we get a polynomial and (with a bound on the derivative) a simple expression for the error. With Big Oh we have a precise statement of how an expression depends on a parameter (h), but can also suppose such a relation and by computing certain values, determine the exact nature of that relationship.
• Thursday, 1/10: Handout syllabus, go over course structure, goals, work, etc. Begin with the basics: Differential Equations (E.1): PDE vs. ODE, Initial Value vs. Boundary Value, the canonical problems: Poisson, Heat and Wave Equations for PDEs, link between the problems and the structure of the course: Poisson and BVPs, then ODEs (as IVPs), then link together for heat and wave-type equations.