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.