This page contains some links to pages with material relevant to the
introductory partial differential equations course.
You will also find links to *Mathematica* notebooks which you
can download and modify for further exploration.

- Section 01 current semester syllabus
- Preliminary definitions and examples of partial differential equations, boundary value problems, and initial boundary value problems.
- Derivations of some common partial differential equations including gas flow through a pipe, flow of heat energy through a one-dimensional medium, and the vibration of a flexible string. A derivation of the heat equation in three dimensions is also presented.
- An introduction to the main technique to be used for solving initial boundary value problems, separation of variables.
- An introduction to first-order linear partial differential equations and methods for solving them.
- Quasilinear partial differential equations and a generalization of the method of characteristics for solving them.
- A few applications of first-order partial differential equations with emphasis on mathematical biology and models of traffic flow.
- An introduction to Fourier series. Properties of periodic functions, even and odd functions, and orthogonal functions are included. Orthogonality of the trigonometric functions and their use in Fourier Series is explored.
- Issues surrounding the convergence of Fourier series are explored. A discussion of the Gibbs phenomenon is also included.
- Examples of Fourier series of some piecewise smooth functions. Mathematica notebook: FourierSeries.nb.
- Differentiation and integration of Fourier series is discussed along with Parseval's identity and Bessel's inequality.
- Separation of variables and Fourier series are used to solve the one-dimensional heat equation. Various boundary conditions are explored. An example of solving the heat equation subject to nonhomogeneous boundary conditions is included. Non-dimensionalizing the heat equation is also presented.
- The solution of the heat equation on an unbounded domain using the fundamental solution to the heat equation is explored.
- The Maximum Principle for the one-dimensional heat equation is presented. The accompanying Minimum Principle and uniqueness of solutions to the heat equations is discussed.
- Separation of variables and Fourier series are used to solve the one-dimensional wave equation. Various boundary conditions are explored.
- D'Alembert's solution to the wave equation on finite and infinite strings is presented.
- The energy integral and uniqueness of solutions to the one-dimensional wave equation is explored.
- A decomposition approach to solving nonhomogeneous wave equations is discussed. Miscellaneous examples of the wave equation with various initial conditions.
- An introduction to Laplace's and Poisson's equations. Solution techniques are presented as well.
- Laplace's equation on a disk and its solution technique are introduced. A derivation of the Laplacian operator in polar and spherical coordinates.
- Laplace's equation on a rectangle with Neumann boundary conditions is explored. The case for the disk is also covered here.
- Solving Laplace's equation on a rectangle with mixed Dirichlet and Neumann boundary conditions.
- A closed form solution for Laplace's equation on a disk in the form of Poisson's formula.
- An introduction to Sturm-Liouville boundary value problems.
- Some properties of eigenvalues and eigenfunctions of regular Sturm-Liouville boundary vlaue problems.
- A brief review of solution techniques for Euler equations.
- An overview of the solution of Bessel's equation of order one.
- Final examination review.