APPLIED MATHEMATICS GROUP - The University of North Carol ina at Chapel Hill
MATH 221, Numerical ODE/PDE I
Syllabus
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  1. Motivation

    The physical world is described by a small set of laws. One of the most useful mathematical statements of these physical laws is in the form of partial differential equations (PDE's) describing local changes in physical parameters. Though it is usually straightforward to write down the particular PDE's that correspond to a given problem, finding a solution is much more difficult. For the vast majority of real-world problems approximate techniques must be used. One of the most productive approaches is to use a numerical approximation of the PDE's of interest. These techniques are used in studying chemical reactions, ecological models, astrophysical phenomena, aircraft design, financial models and in many other application domains.

  2. Course Objectives

    The development of numerical methods for PDE's and the ensuing theoretical development is within the province of applied mathematics. This course introduces the basic aspects of the discipline at the graduate level. At the end of the course the student should be able to understand the basic theory associated with solving each type of PDE encountered in practice, be adept at choosing methods appropriate for a specific application and be proficient in the computer solution of PDE's.

  3. Syllabus

    • Basic equations of mathematical physics, other application domains

    • Minimal residual formulation

    • Numerical methods for ODE's

    • Linear advection diffusion equations

    • Parabolic equations

    • Hyperbolic equations

    • Equations of mixed type

    • Coupled PDE-ODE systems

  4. Course Texts

    One of the tenets of graduate study is the ability to critically examine multiple sources in order to verify theories and to enhance understanding. Lecture notes for the course shall be provided online in Postscript and PDF formats. These are meant to document the progress of material presented in class. It is expected and required that the student supplement the lecture notes with reading from other sources.

  5. General bibliography

    The following texts serve as general background material. Students are encouraged to actively read from these texts material related to the current lecture. Homework and final examination questions from this material should be expected.

    Finite Difference Schemes and Partial Differential Equations, John Strikwerda

    Numerical Methods for Conservation Laws, Randall LeVeque

    Finite Difference Schemes for Computational Fluid Dynamics, Phillip Colella & Gerry Puckett

  6. Suggested reading for specific topics

    Material specific to a certain topic shall be listed here as the course progresses. Generally the books or articles will be placed on reserve at the Mathematics Library in Phillips for the time period indicated

    P.G. Garabedian, Partial Differential Equations, Chapter 1, The Method of Power Series. (17 pp.). Questions to ponder: (1) Consider the relevance and applicability of the methods used in the Cauchy-Kowalewski theorem to practical computations; (2) Try to find a biography of Sophia Kowalewski (also transcribed as Sonja Kovalevski) and place the Cauchy-Kowalewski theorem in historical context.

    G. Dahlquist, Numerical Methods, Prentice-Hall, 1974, Sections 7.1 (Difference operators), 7.2 (Richardson extrapolation), Chapter 8 (Differential equations)











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