Course Purpose:

This is the second semester of a one-year introductory graduate course on applied mathematics. This is the second time we offer this course.

The course is designed for graduate students who wish to learn some basic mathematical techniques that can be applied in their research and future career. Because of the rapid development of computer and network technologies, more and more mathematical techniques are used in more and more fields such as physics, chemistry, biology, economics, finance, engineering, meteorology, rheology, etc.. This course will cover some of the very basic mathematical techniques that are often not covered in regular undergraduate mathematical courses but are essential to the aforementioned application areas.

This course has been designed by a group of computational and applied mathematicians from the department of mathematics with close collaborations with many professors from different departments at Penn State University. The syllabus of the course has been extensively debated and carefully designed to address the special needs of students with diversely different backgrounds. It is expected this course will be prerequisite for other basic graduate courses in computational and applied mathematics offered by the department of mathematics at Penn State. It is also expected that students who have taken this course sequence will have a solid foundation to take most other graduate courses on campus that involve the use of mathematical and computational techniques.

Introduction to Applied Mathematics I (Fall 2002)

Prerequisite: Math 405 (equivalent) or consent of the instructor

• Cartesian vectors and tensor calculus (2 weeks): vector operations; Cartesian tensor operations; Kronecker delta and alternating tensor; surface and line integrals; Green's and Stokes' theorems; basic vector operations in spherical and cylindrical coordinates.
• Complex variables (1.5 weeks): analytic functions; Cauchy's theorem and integration formula; complex vectors.
• Applied functional analysis (2 weeks): Banach and Hilbert spaces; Riesz representations; Fredholm alternative theorem; spectral theory for compact operator
• Linear transforms (Fourier, Laplace) (1.5 week): Fourier and Laplace integral and properties, Fourier series vs. Fourier integrals; Sturm-Liouville operator.
• Ordinary differential equations (2 weeks): stability of first order linear systems, perturbation methods. Special functions.
• Partial differential equations (5 weeks): basic properties of solutions; separation of variables; Fourier and Laplace transform solutions; infinite and semi-infinite domain solutions; similarity solutions; special functions (Bessel, Legendre, spherical harmonics); boundary value (eigenvalue) problems.
• Homogenizations (1 week)

Introduction to Applied Mathematics II (Spring 2003, 597F)

Prerequisite: Introduction to Applied Mathematics I or consent of the instructor

• Calculus of variations (approximate 3 weeks): Euler-Lagrange equations; constraint problems; Hamilton's principle; stability and second variations; applications.
• Partial differential equations (approximate 3 weeks): Green's functions and fundamental solutions; eigenfunction expansions and Galerkin's method; Sobolev space and weak solutions.
• Asymptotic expansions (approximate 1 week): Laplace method; method of steepest descents; method of stationary phase.
• Regular perturbation theory (approximate 2 weeks): oscillations and periodic solutions; perturbation of eigenvalues; Lyapunov-Schmidt method.
• Singular perturbation theory (Approximate 2 weeks): initial value problems; boundary value problems.
• Wavelet analysis (approximate 1 week).
• Stochastic differential equations, Black-Scholes model (approximate 2 weeks).

Test and Grades

There will be one midterm exam and one final exam per course. The final course grade will be determined as follows:

30% homework+ 30% midterm exam + 40 % final exam.

Lecture Schedule: Tu Th 9:45-11:00AM,

Lecture Place: 302 Willard

Instructor: Yuxi Zheng, yzheng@math.psu.edu.

Course Texts J. Keener, Principles of Applied Mathematics: Transformation and Approximation, Perseus Books, 2000.

Reference Books

1. H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, Dover.

2. I. M. Gel'fand, S. V. Fomin, Calculus of Variations, Prentice-Hall, 1963.

More to be given later in the class.

Course Oversight Committee

• Andrew Belmonte, Department of Mathematics
• James Brasseur, Dept of Mechanical Engineering
• Jenny Li, Department of Mathematics and Department of Economics
• Chun Liu, Department of Mathematics
• Eric Mockensturm, Dept of Mechanical Engineering
• Jinchao Xu, Department of Mathematics, Chair
• Yuxi Zheng , Department of Mathematics, Instructor of the course