General Information

Fall. 4 credits. Letter grades only.

Covers Fourier and Laplace transforms, ordinary and partial differential equations, separation of variables, Method of Frobenius, Laplace transform techniques. Green’s functions, wave and diffusion equations, Solutions to Laplace’s Equation, Hermitian Operators, Sturm-Liouville operators, Bessel functions, Legendre Polynomials, spherical harmonics.

Prerequisites

Prerequisite: AEP 3200.

Topics Covered

  1. Laplace Transforms (Chapter 9)
    1. As a limit of the Fourier Transform
    2. Properties of the Laplace Transform
    3. Examples
    4. The Double-Sided Laplace Transform
  2. Differential Equations (Chapters 10 & 11)
    1. Terminology
    2. First Order Ordinary Differential Equations
      1. Exact Differentials
      2. Linear First Order Differential Equations
      3. Integrating Factors
    3. Linear Second Order Ordinary Differential Equations
      1. Homogeneous
        1. Series Solution
        2. Frobenius Method
      2. Nonhomogeneous
        1. Wronskian Solution Using Homogenous Solution
        2. Laplace and Fourier Techniques
    4. Quadrature Techniques for Nonlinear Second Order Equations
    5. Green's Function Solution to Linear Differential Equation
      1. Convolution and Correlation Review
      2. Time Dependent Equations with Constant Coefficients
      3. Position Dependent Equations with Constant Coefficients
      4. Nonhomogeneous Boundary Condition
      5. Solution with Nonhomogeneous Initial Conditions
      6. Solutions for Space-Time Problems
      7. Dispersion Relations
    6. Hermetian Operators, Orthogonal Functions and Sturm-Liouville Theory
    7. Partial Differential Equations
      1. Examples from Physics
        1. Laplace's Equation
        2. Poisson's Equation
        3. Wave Equation
        4. Diffusion Equation
        5. Helmholtz Equation
        6. Schrodinger Equation
      2. Separation of Variables
      3. Solutions in Different Coordinate Systems
        1. Bessel Function
        2. Legendre Polynomials
        3. Spherical Harmonics
  3. Integral Equations (Chapter 12)
    1. Classifications, Voltera vs. Friedholm
    2. Forming Integral Equations from Differential Equations
      1. Automatic Inclusion of Boundary Conditions
      2. Kernels
    3. Solutions
      1. Laplace Transform Techniques
      2. Neumann Series Solutions
      3. The Method of Separable Kernels
      4. Green's Function Solutions
  4. Complex Variables II (Chapter 13)
    1. Multivalued Functions
      1. Riemann Sheets
      2. Branch Points and Branch Cuts
    2. Branch Cut Integration
    3. Saddle Points
    4. Method of Steepest Descent

Workload

Weekly problem sets, two prelims, and a final. [Fall 2023]

General Advice

  • Read the textbook. It is very useful for doing the problem sets. Take advantage of TA office hours, as there are many helpful tricks and insights you can gain from them. [Fall 2023]

Testimonials

The problem sets are generally not that long (with the exception of one or two), so the workload is manageable. However, the lectures can be a bit slow and behind the homeworks, so the textbook is a more reliable resource for learning the material. In terms of content, AEP 4200 will introduce some pretty useful concepts to your toolbox, some of which you may be able to find in a differential equations class, but others which you will likely only see in physics classes (thus making it useful to have some prior exposure through the math phys class). [Fall 2023]

Past Offerings

Semester Professor Median Grade Syllabus
Fall 2021 B. Kusse. A-  
Fall 2023 Bruce Kusse   AEP4200_FA23.pdf