MATH 401 - 501: Advanced Engineering Mathematics, Fall 2001

General information

• Instructor: Dr. Raytcho Lazarov, Blocker 505C, 845 7578
• Teaching Assistant: none
• Time: MWF 12:40 -- 1:30 pm
• Classroom: CE 136
• Office Hours: MWF 11:00 pm -- 12:00 am or by appointment
• Text: (suggested) A.W. Bush, Perturbation methods for engineers and scientists, CRC Press, 1992; (suggested) M.R. Spiegel, Fourier Analysis with Applications to BVPs, Schaum's outlines, McGraw-Hill, 1974; (suggested) S. Fulling, Math 401 Lecture Notes, on sale at Copy Corner.

Exam Schedule

• Monday, October 1, 2001 Test # 1
• Friday, November 5, 2001, Test # 2
• Monday, December 10, 2001, Final Exam, 10:30 - 12:30 am

Grading Policy

• Your grade for the course will be computed as follows:
  • (a) 1/3 will be determined by your homework assignments
  • (b) 1/3 will be determined by the two midterm tests
  • (c) 1/3 will be determined by your final exam (the final exam is comprehensive)
• your MINIMUM grade will be A, B, C, or D, for averages of 90%, 80%, 70%, or 55%, respectively.

Course description

General Information

• This is a one-semester course on solving problems in physics and engineering by differential equations. The main goal of the course is to provide knowledge, basic technique, and experience in solving various applied and engineering problems by using expansions in the form of asymptotic series. The course will emphasize on two basic analytical techniques: (1) perturbation methods and asymptotic expansions; (2) Fourier series and separation of variables applied to partial differential equations.
• The techniques, being analytical rather than numerical, provide an alternative to a direct computer solution. An awareness and knowledge of the analytical approach is often essential even when direct numerical approach is adopted. Examples of such problems are various flows with boundary layers, concentrated sources and sinks, or/and singularities due to corners or jump discontinuities of the coefficients, etc.
• In both, perturbation methods and Fourier methods, approximate expressions are generated from asymptotic series. These may not and often converge but, in a truncated form provide a useful approximation to the original problem. Since these expressions often contain a parameter, they might be used to derive approximate formulas for some important physical quantities.

  Course Outline

• I. Perturbation theory and asymptotic approximations (~5 weeks): (1) Perturbation theory for algebraic equations; Regular perturbation theory (power series) and its shortcomings; (2) Asymptotics and uniformity; stretched time; (3) Boundary-layer problems.
• II. Partial differential equations and Fourier methods (~10 weeks): (1) Introduction to PDEs and boundary-value problems: The heat equation; (2) Basic PDE concepts; linearity and homogeneity; (3) Separation of variables and Fourier series; (4) Sturm--Liouville problems and special functions - a quick survey; (5) The linear wave equation (if time permits); (6) Types of PDEs (parabolic, hyperbolic, elliptic); well-posed problems (if time permits).

Some supplementary books you might find useful

Perturbation theory:

• E. J. Hinch, Perturbation Methods;
• J. D. Cole, Perturbation Methods in Applied Mathematics;
• A. S. Cakmak et al., Computational and Applied Mathematics for Engineering Analysis

Fourier analysis:

• An on-line textbook from Georgia Tech: Evans Harrell and James Herod, Linear Methods of Applied Mathematics;
• G. P. Tolstov, Fourier Series
• D. Bleecker and G. Csordas, Basic Partial Differential Equations;
• J. D. Logan, Applied Mathematics: A Contemporary Approach.