MATH 542 NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS
This web site is in support of Math 542 (Numerical Solutions of Differential Equations) taught by Dr. Don Short. MATLAB is used extensively in this course as the calculational software for all homework assignments and for all class demonstrations. The MATLAB code demonstrated in class may be found on this site. A book has not been selected for this course. A survey of possible texts found that either the coverage of a major topic was omitted or the level of the coverage was wrong.
Introduction:(2 weeks) A brief survey of the theory and analytic solution of both ordinary and partial differential equations. A standard set of initial value and boundary value problems will be developed to be used throughout the course.
Much of this material may be found in texts titled Introduction to Differential Equations or Introduction to Partial Differential Equations. The one topic that we will cover which is most often not found in these texts is first order partial differential equations.
Initial Value Problems:(4 weeks) Runge-Kutta and Multi-Step methods will be developed together with error estimates. The numerical solution of stiff equations will also be covered.
The book that you used for Math 541 will have some material on this topic, but it will probably not be at the level of sophistication that we need.
Finite Difference Approximations:(4 weeks) Finite Difference Methods will be developed for boundary value, evolution and wave equation problems.
Again, your Math 541 book will probably have some discussion of this topic, but we will need to develop it in more depth.
Finite Element Method:(4 weeks) The Finite Element Method will be developed and applied to boundary value problems over non-rectangular regions.
Most introductory numerical analysis text will not have coverage of the finite element method.
POSSIBLE REFERENCE BOOKS:
Your Math 541 book
Numerical Solution of Ordinary Differential Equations by Lawrence Shampine
This book covers only ordinary differential equations in more depth than this course. However, it is a good reference for the ODE portion of this course.
A First Course in Numerical Analysis of Differential Equations by Arieh Iserles
This book is written from an advanced mathematical viewpoint. The coverage of the Finite Element Method is limited to the theory.
Numerical Methods For Differential Equations by Michael Celia and William Gray The Chapter 3 coverage of the Finite Element Method is well done and will serve as a guide for our course.
By Don Short, email@example.com