This course expands on the ideas of calculus and the techniques introduced in ordinary differential equations. The student successfully completing this course will be able to combine analytic, graphical, and numeric methods to solve problems that model a variety of physical phenomena. A tentative course description is included below:
As a part of this course, students can expect to cover a wide variety of material with emphasis on analytic techniques and modeling. Student’s should expect most classes to revolve around a lecture format with the occasional reading quiz to begin class. Occasionally, we may have need to hold class in the computer lab when working on specific topics, but advance notice will be given for the change of venue. A list of topics and lecture summaries are included below:
Unit 1: Series Solutions, Fourier Series Representations
Lecture 1: Reduction of Order Method
This lecture revolves around a recap of major results from ODE including the structure of solutions to second order equations and the integrating factor method for solving first order equations. Remind students how painful it was to come up with a second linearly independent solution when there was a second root. Using this we introduce the method of reduction of order as a simple means of finding a second linearly independent solution.
Lecture 2: Euler Equations
This lecture introduces Euler equations as a simplest case of 2nd order linear equations with non-constant coefficients. The terms indicial equation and indicial roots are introduced. The three cases for solutions of Euler equations are proven and used in a few examples.
Lecture 3: Power Series Solution Methods
In this lecture we introduce the idea of using a power series to approximate the solution to a differential equation by showing a Taylor series approximation is very possible, though computationally intensive. We use this as a motivation to suggest a more general approach. A first order example is used to show that the occurrence of a recurrence relation is a necessary step in the process. The lecture ends with a solution of Airy’s equation and a discussion of the prevalence of special functions that arise as solutions to differential equations.
Lecture 4: Ordinary vs. Singular Point Equations
This lecture begins by discussing power series and convergence including radius of convergence and whether we need to be worried about such things in our method. This leads to a definition of the terms ordinary point and singular point. Singular points are then further refined as being regular or irregular singular points. An example is worked which shows the consequence of ignoring this distinction. Finally, the lecture closes by exploring the solution to a special differential equation called Legendre’s equation with a promise that we will see the Legendre functions generated by this equation later in the semester.
Lecture 5: Method of Frobenius
This lecture begins by introducing a Frobenius series and a generalization of the indicial equation introduced in Lecture 2. A theorem describing all the possible cases is shown (but not proved) with emphasis placed on coming up with the “easy” solution and using reduction of order to come up with the other solution. The lecture is concluded by showing the relationship with Euler’s equations in Lecture 2.
Lecture 6: Bessel’s Equation
This lecture is devoted to solutions of Bessel’s equation of order p for both indicial roots r = p, r=-p. We further discuss properties of Bessel functions including Bessel functions of the second kind as solutions of Bessel’s equation, with a promise that we will see these functions later in the semester. A reminder of properties of the Gamma function and their relationship to Bessel functions closes the lecture.
Lecture 7: Expansions Involving Trigonometric Functions
In this lecture we take a step back and ask the question why power series are special. On some level, it seems like trying to make a square peg fit into a round hole to use power functions to approximate something periodic since we’ll need many, many terms to have a usable approximation. Why not approximate using expansions of $latex \cos(x)$ and $latex \sin(x)$? Is this even possible? The answer, of course, is yes and involves the Fourier series. We derive formulas for the Fourier coefficients for a function with period $latex 2\pi$ and demonstrate the expansion including remarks about convergence at possible points of discontinuity.
Lecture 8: Fourier Series of Functions with Arbitrary Periods
In this lecture we show how a Fourier series representation can be thought of as two representations, one for the even part of a function, and one for the odd part of a function. This leads to two series – called the Fourier sine and Fourier cosine series. Examples showing the periodic extension technique are discussed.
Lecture 9: Even and Odd Extensions
In this lecture we show how a Fourier series representation can be thought of as two representations, one for the even part of a function, and one for the odd part of a function. This leads to two series – called the Fourier sine and Fourier cosine series. Examples showing the periodic extension technique are discussed.
Lecture 10: Fourier Series Convergence
In this lecture we discuss the convergence of the Fourier series in the mean square sense by studying the partial sums of the Fourier series. This leads to the derivation of the concepts of square integrability, square summable, Bessel’s inequality, and Parseval’s identity. We also define the complex Fourier series.
Lecture 11: Generalized Fourier Series and Orthogonality
Quite possibly the most important lecture of the semester we take the ideas we have developed thus far in the semester and apply them to developing a general notion of Fourier series in terms of expansions of orthogonal functions. Verifications of the orthogonality of trigonometric functions, Legendre functions, and Bessel functions are discussed with special emphasis on connecting these concepts with the ideas of orthogonality and norms as developed in a first course on linear algebra.
Unit 2: Partial Differential Equations and BVPs
Lecture 12: Introduction to Partial Differential Equations
In this lecture, we introduce some of the major differential equations of physics and engineering and talk about general notations and terminology. We solve a few first order PDEs using the idea of characteristic curves. These examples are designed to demonstrate some of the geometry involved with solving partial differential equations and the fact that most PDEs (unlike ODEs) have solutions involving arbitrary functions rather than arbitrary constants.
Lecture 13: Normal Modes of Vibration, Separation of Variables (Dirichlet BCs)
In this lecture, we introduce the one dimensional wave equation as a consequence of analyzing the motion of a vibrating string. We find the normal modes of vibration of the string by introducing the separation of variables technique. The lecture concludes with the fascinating result that the normal modes of vibration are represented by Fourier series!
Lecture 14: d’Alembert’s Method
Much of the fascinating geometry of the wave equation is obscured by the representation of the solution in terms of Fourier series. In this lecture, we re-represent the solution to the vibrating string problem as a more general case of the transport equation solved in lecture 12 via characteristics. Calculations of specific string positions are made using this solution form.
Lecture 15: Models of Diffusion (Neumann BCs)
Continuing with our investigation of physical models that result in PDEs we introduce the one dimensional heat equation as describing the distribution of heat in a one-dimensional rod with ends at $latex x = 0$ and $latex x = L$. Our example revolves around a boundary value problem having Neumann boundary conditions to illustrate that a solutions of Dirichlet problems involve sine expansions while Neumann problems involve cosine expansions. A specific example is solved in Maple with associated graphs.
Lecture 16: Vibrating Membrane (Double Fourier Sine Series)
In this lecture, we show how Laplace’s equation arises naturally as a condition to describe steady-state temperature distributions of the 1D and 2D heat equations. We set up the basic problem on the rectangle and solve by separating variables. We have a new eigenfunction! The hyperbolic sine makes an appearance.
Lecture 17: Steady State Temperature Distributions, Laplace’s Equation
In this lecture, we show how Laplace’s equation arises naturally as a condition to describe steady-state temperature distributions of the 1D and 2D heat equations. We set up the basic problem on the rectangle and solve by separating variables. We have a new eigenfunction! The hyperbolic sine makes an appearance.
Lecture 18: Decomposing the Rectangle, Non-homogeneous Boundary Conditions
In this lecture we use our solution of Laplace’s equation to allow us to solve a more general problem in which the edges of the rectangle are all held at temperatures governed by arbitrary functions of x or y. We conclude the lecture with a curious observation that the center of the rectangle is the average value of the edge temperatures and the maxima, minima of the solutions occur at the boundary.
Lecture 19: Poisson’s Equation, Method of Eigenfunction Expansions
In this lecture, we show that Laplace’s equation on the rectangle can be solved in the case of the presence of a time independent heat source. This leads to a further decomposition of the rectangle and what is called a general Poisson problem. To solve this problem we turn to the method of eigenfunction expansions and note the presence of a new boundary value problem called the Helmholtz equation.
Unit 3: Eigenfunction Expansions in Other Coordinates
Lecture 20: Eigenfunction Expansions, Sturm-Liouville Problems
In this lecture, we further explore the relationships between orthogonal functions, orthogonal expansions and differential equations. We consider two general types of boundary value problems: regular Sturm-Liouville and singular Sturm-Liouville problems. We establish conditions on the eigenvalues and eigenfunctions associated with these problems including uniqueness and existence results that involve our old friend the Wronskian. The lecture concludes with our first formal definition of an eigenfunction expansion.
Lecture 21: The Hanging Chain Problem and Bessel Expansions
As a sort of “proof of concept” application of our previous lecture we consider the oscillations of a hanging chain. This problem is of some historical significance as it is often credited as the problem for which Daniel Bernoulli first identified the inclusion of Bessel functions. Our approach involves separation of variables and the use of properties of Bessel functions established previously in the semester.
Lecture 23: The Laplacian in Several Coordinates (Polar, Cylindrical, Spherical)
In this lecture, we review the geometries associated with polar, cylindrical, and spherical coordinates. We derive a formula for the Laplacian in each of these coordinate systems for use in future lectures in this unit. Some basic properties of the Laplacian related to harmonic functions are discussed.
Lecture 24: Vibrations of a Circular Membrane via Bessel Function Expansions
In this lecture, we consider an application of the two dimensional wave equation on a circular domain. The circular domain suggests the use of the polar form of the Laplacian. We begin by studying initial configurations of a vibrating circular membrane fixed at its circumference that are radially symmetric. Our analysis leads to a solution in terms of Bessel expansions.
Lecture 25: Heat Flow in a Cylinder via Modified Bessel Function Expansions
In this lecture, we consider a problem which models the steady-state temperature distribution inside a cylinder with lateral surfaces and base held at zero temperature. This model assumes a radially symmetric temperature distribution at the top of the cylinder. We utilize the cylindrical coordinates form of the Laplacian to obtain solutions in terms of modified Bessel functions.
Lecture 26: Vibrations on a Sphere via Legendre Function Expansions
In this lecture, we consider a radially symmetric version of Laplace’s equation in spherical coordinates. We show that the solution to the equation in this case is given by solving an associated Legendre differential equation. We conclude this lecture by setting up the classic Dirichlet problem on the ball.
Lecture 27: Spherical Harmonics I
Using the results of the previous lecture we attempt to establish a general case (like we did in Lecture 24 for the circular domain problem) for the Dirichlet problem inside a ball. These two problems are closely linked in that both have solutions expressible as generalized Fourier series of their boundary functions. Because the boundary in this problem is the surface of a sphere the eigenfunctions required are the classical spherical harmonics. We investigate orthogonality conditions of the spherical harmonics and a spherical harmonics series expansions.
Lecture 28: Spherical Harmonics II
Having a solution in hand for the problem on the sphere we attempt to solve an associated IVP. We conclude this lecture by stating the full solution of the Dirichlet problem on the ball. We make comparisons to the solution obtained on the disc in Lecture 24.
Lecture 29: Nodal Sets of Vibrations
For many of the problems analyzed in this unit there is a special set of points called the Nodal Set. These sets are the natural evolution of the nodes we considered as a part of the 1D wave equation in Unit 2. The nodal set is stationary because all points of the set do not move at all. We examine the nodal set of the rectangle, disc, and ball with associated pictures. There is some wonderful geometry discussed in this lecture.
Unit 4: Transform Methods and Green’s Functions
Lecture 30: Fourier Integral Representation, Fourier Transform Pair
In this lecture, we take a step back from some of the specific examples in the previous unit to see if there is a further abstraction of some of the techniques we’ve employed this semester. The answer is found in the study of Fourier transforms and Green’s functions which we concentrate on in this unit. We derive the Fourier transform by studying the Fourier integral and discuss terminology and notation that will be essential going forward.
Lecture 31: Operational Properties of the Fourier Transform
Now that we have established the form of the Fourier transform, what are we going to do with it? This lecture revolves around establishing several useful properties of the Fourier transform that will make applying the transform much easier. For students who have been exposed to the Laplace transform in ODE this process will see familiar.
Lecture 32: Fourier Transform Solutions of IVPs
As a proof of concept, we use the Fourier transform to solve several infinite dimensional PDEs including a fourth order PDE that would have been very difficult using separation of variables techniques from earlier in the semester. We conclude with an exercise which shows how d’Alembert’s solution of the wave equation is a natural consequence of our Fourier transform method. (Remember we had to work for this form when we solved the problem via separation of variables.)
Lecture 33: Source Functions and Kernels
Another of the very important lectures of the semester we show how the Fourier transform allows us to view most of the PDEs we have discussed this semester as essentially being the same problem mathematically, i.e. the convolution of a special function called a source function (or kernel) with the initial function $latex f$.
Lecture 34: Fourier Transforms and Generalized Functions
In this lecture, we establish properties of the Fourier transform related to generalized functions like the step function and Dirac delta function. We use these properties to show how the step function and delta function are related and find their Fourier transforms. We close with a discussion of convolutions involving delta functions.
Lecture 35: Non-homogeneous Models via the Fourier Transform
We have not considered many non-homogeneous models this semester. This is partly because the best technique for handling non-homogeneous problems is through the Fourier transform. We consider non-homogeneous examples of the heat and wave equation and conclude with a statement of Duhamel’s principle for an alternative approach to this problem.
Lecture 36: Introduction to Green’s Functions
Our last lecture of the semester is designed to provide us with one more level of abstraction this semester by considering the Green’s function approach. We derive the Green’s function for some time independent problems and talk about properties of the Green’s function. We conclude with the Green’s function relationship to the delta functions discussed in Lecture 34.
Final Day of Class Handout
On the final day of class students will be given a handout that summarizes the relationship between linear algebra involving matrices and eigenvectors, an differential equations involving operators and eigenfunctions. This is the viewed as the main theme of the ODE/PDE sequence and will be tested on the final exam.
Research Paper
Given the lack of class time devoted to numerical methods for solutions of PDEs students taking this course are required to write a minimum 10 page senior-quality research paper on a specific area of numerics applied to PDEs. Students are encouraged to focus on specific solution techniques which view the partial differential operator as a matrix. Meshes, sampling, finite difference, finite elements, and Crank-Nicholson schemes are all valid research topics.