Mathematics, Statistics & Computer Science
University of Wisconsin-Stout
Jarvis Hall Science Wing 231
Menomonie, WI 54751-0790
COURSE NUMBER/TITLE:
Differential Equations: Initial and Boundary Value Problems
CREDITS:
3
COURSE DESCRIPTION:
Methods used in solving initial and boundary value differential equation
problems that arise in applied mathematics, physics, engineering,
economics, and statistics.
TEXTBOOK:
Applied Partial Differential Equations, 1st Ed., by DuChateau and Zachmann (adopted Spring
2014)
COURSE OBJECTIVES:
Upon successful completion of this course, the student will be able to:
1. Demonstrate an understanding of the theoretical background of continuous modeling with ordinary
and partial differential equations (ODEs and PDEs).
2. Analyze, distinguish, and apply mathematical tools of multivariable calculus, real and complex
analysis, and linear algebra to analyze and interpret PDEs.
3. Solve linear ODEs and PDEs using a variety of classical techniques such as separation of variables
and Laplace and Fourier transforms.
4. Solve nonlinear ODEs and PDEs using a variety of techniques such as the method of characteristics
and similarity transforms.
5. Perform dimensional and small / large parameter analysis on mathematical models as a means for
their interpretation.
6. Analyze, distinguish, and apply numerical methods for the solution of linear and nonlinear ODEs
and PDEs.
COURSE OUTLINE:
1. Continuous Modeling and ODEs (Objs 1, 3)
a. Mathematical Models
b. Linear First-Order Equations, Linear Second-Order Equations
c. Systems of ODEs
2. Numerical Methods for ODEs (Obj 6)
a. Taylor Series Methods (Euler's Methods)
b. Stability
c. Multistep Methods and Runge-Kutta Methods
d. Nonlinear Equations
e. Qualitative Analysis
3. Essential Analysis Review (Obj 2)
a. Differentiation and Integration in 1D, 2D, 3D, nD
b. Div, Grad, Curl
c. The Divergence (Green’s) Theorems
4. Continuous Modeling and PDEs (Objs 1, 3, 4, 5)
a. Mathematical Models (Common PDEs)
b. Dimensional Analysis and the Buckingham Pi Theorem
5.
6.
c. Small & Large Parameter Analysis
d. Similarity Solutions and Reduction to ODEs
Numerical Methods for PDEs (Obj 6)
a. Finite Differences for Spatial Discretization
b. Elliptic Equations and Iterative Methods
c. Temporal Discretization (Review of Numerical ODEs)
d. Parabolic Equations
e. Accuracy, Stability, and Convergence
f. Hyperbolic Equations
g. Accuracy, Stability, and Convergence
h. Very Brief Overview of Finite Element Analysis
i. Finite Element Solvers (Software Packages)
Potential topics for projects or as time permits: (Objs 1, 2, 3, 4, 5, 6)
a. Fourier Analysis (Series & Transform), DFTs, FFT
b. Stochastic (SDEs)
c. Perturbation Theory
d. Integral Equations
e. Specific Applications (Financial, Control-systems, Optics, Heat Transfer, Fluids, etc.)
f. More In-depth Study of Numerical Methods for PDEs
Updated 2/2016
11/2013