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:<\/p>\n
As a part of this course, students can expect to cover a wide variety of material with emphasis on analytic techniques and modeling. \u00a0Student’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. \u00a0A list of topics and lecture summaries are included below:<\/p>\n
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This lecture revolves around a recap of major results from ODE including the structure of\u00a0solutions to second order equations and the integrating factor method for solving first order\u00a0equations. Remind students how painful it was to come up with a second linearly\u00a0independent 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.<\/p>\n
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This lecture introduces Euler equations as a simplest case of 2nd order linear equations with non-constant coefficients. \u00a0The terms indicial equation and indicial roots are introduced. \u00a0The three cases for solutions of Euler equations are proven and used in a few examples.<\/p>\n
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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. \u00a0We use this as a motivation to suggest a more general approach. \u00a0A first order example is used to show that the occurrence of a recurrence relation is a necessary step in the process. \u00a0The 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.<\/p>\n
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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. \u00a0This leads to a definition of the terms ordinary point and singular point. \u00a0Singular points are then further refined as being regular or irregular singular points. \u00a0An example is worked which shows the consequence of ignoring this distinction. \u00a0Finally, 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.<\/p>\n
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This lecture begins by introducing a Frobenius series and a generalization of the indicial equation introduced in Lecture 2. \u00a0 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.<\/p>\n
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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. \u00a0A reminder of properties of the Gamma function and their relationship to Bessel functions closes the lecture.<\/p>\n
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In this lecture we take a step back and ask the question why power series are special. \u00a0On 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. \u00a0Why not approximate using expansions of $latex \\cos(x)$ and $latex \\sin(x)$? \u00a0Is this even possible? \u00a0The answer, of course, is yes and involves the Fourier series. \u00a0We 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.<\/p>\n
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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. \u00a0This leads to two series – called the Fourier sine and Fourier cosine series. \u00a0Examples showing the periodic extension technique are discussed.<\/p>\n
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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. \u00a0This leads to two series – called the Fourier sine and Fourier cosine series. \u00a0Examples showing the periodic extension technique are discussed.<\/p>\n
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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. \u00a0This leads to the derivation of the concepts of square integrability, square summable, Bessel’s inequality, and Parseval’s identity. \u00a0We also define the complex Fourier series.<\/p>\n
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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. \u00a0Verifications 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.<\/p>\n
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In this lecture, we introduce some of the major differential equations of physics and engineering and talk about general notations and terminology. \u00a0We solve a few first order PDEs using the idea of characteristic curves. \u00a0These 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.<\/p>\n
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In this lecture, we introduce the one dimensional wave equation as a consequence of analyzing the motion of a vibrating string. \u00a0We find the normal modes of vibration of the string by introducing the separation of variables technique. \u00a0The lecture concludes with the fascinating result that the normal modes of vibration are represented by Fourier series!<\/p>\n
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Much of the fascinating geometry of the wave equation is obscured by the representation of the solution in terms of Fourier series. \u00a0In 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. \u00a0Calculations of specific string positions are made using this solution form.<\/p>\n
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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 \u00a0$latex x = 0$ and $latex x = L$. \u00a0Our 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. \u00a0A specific example is solved in Maple with associated graphs.<\/p>\n
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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. \u00a0We set up the basic problem on the rectangle and solve by separating variables. \u00a0We have a new eigenfunction! The hyperbolic sine makes an appearance.<\/p>\n
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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. \u00a0We set up the basic problem on the rectangle and solve by separating variables. \u00a0We have a new eigenfunction! The hyperbolic sine makes an appearance.<\/p>\n
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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. \u00a0We 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.<\/p>\n
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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. \u00a0This leads to a further decomposition of the rectangle and what is called a general Poisson problem. \u00a0To 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.<\/p>\n
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In this lecture, we further explore the relationships between orthogonal functions, orthogonal expansions and differential equations. \u00a0We consider two general types of boundary value problems: \u00a0regular Sturm-Liouville and singular Sturm-Liouville problems. \u00a0We establish conditions on the eigenvalues and eigenfunctions associated with these problems including uniqueness and existence results that involve our old friend the Wronskian. \u00a0 \u00a0The lecture concludes with our first formal definition of an eigenfunction expansion.<\/p>\n
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As a sort of “proof of concept” application of our previous lecture we consider the oscillations of a hanging chain. \u00a0This 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. \u00a0Our approach involves separation of variables and the use of properties of Bessel functions established previously in the semester.<\/p>\n
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In this lecture, we review the geometries associated with polar, cylindrical, and spherical coordinates. \u00a0We derive a formula for the Laplacian in each of these coordinate systems \u00a0for use in future lectures in this unit. \u00a0Some basic properties of the Laplacian related to harmonic functions are discussed.<\/p>\n
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In this lecture, we consider an application of the two dimensional wave equation on a circular domain. \u00a0The circular domain suggests the use of the polar form of the Laplacian. \u00a0 \u00a0We begin by studying initial configurations of a vibrating circular membrane fixed at its circumference that are radially symmetric. \u00a0 Our analysis leads to a solution in terms of Bessel expansions.<\/p>\n
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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. \u00a0This model assumes a radially symmetric temperature distribution at the top of the cylinder. \u00a0We utilize the cylindrical coordinates form of the Laplacian to obtain solutions in terms of modified Bessel functions.<\/p>\n
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In this lecture, we consider a radially symmetric version of Laplace’s equation in spherical coordinates. \u00a0 We show that the solution to the equation in this case is given by solving an associated Legendre differential equation. \u00a0We conclude this lecture by setting up the classic Dirichlet problem on the ball.<\/p>\n
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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. \u00a0These two problems are closely linked in that both have solutions expressible as generalized Fourier series of their boundary functions. \u00a0Because the boundary in this problem is the surface of a sphere the eigenfunctions required are the classical spherical harmonics. \u00a0We investigate orthogonality conditions of the spherical harmonics and a spherical harmonics series expansions.<\/p>\n
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Having a solution in hand for the problem on the sphere we attempt to solve an associated IVP. \u00a0We conclude this lecture by stating the full solution of the Dirichlet problem on the ball. \u00a0We make comparisons to the solution obtained on the disc in Lecture 24.<\/p>\n
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For many of the problems analyzed in this unit there is a special set of points called the Nodal Set. \u00a0These sets are the natural evolution of the nodes we considered as a part of the 1D wave equation in Unit 2. \u00a0The nodal set is stationary because all points of the set do not move at all. \u00a0We examine the nodal set of the rectangle, disc, and ball with associated pictures. \u00a0There is some wonderful geometry discussed in this lecture.<\/p>\n
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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. \u00a0The answer is found in the study of Fourier transforms and Green’s functions which we concentrate on in this unit. \u00a0 We derive the Fourier transform by studying the Fourier integral and discuss terminology and notation that will be essential going forward.<\/p>\n
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Now that we have established the form of the Fourier transform, what are we going to do with it? \u00a0This 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.<\/p>\n
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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. \u00a0We conclude with an exercise which shows how d’Alembert’s solution of the wave equation is a natural consequence of our Fourier transform method. \u00a0(Remember we had to work for this form when we solved the problem via separation of variables.)<\/p>\n
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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 $latex f$.<\/p>\n
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In this lecture, we establish properties of the Fourier transform related to generalized functions like the step function and Dirac delta function. \u00a0We use these properties to show how the step function and delta function are related and find their Fourier transforms. \u00a0We close with a discussion of convolutions involving delta functions.<\/p>\n
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We have not considered many non-homogeneous models this semester. \u00a0This is partly because the best technique for handling non-homogeneous problems is through the Fourier transform. \u00a0We 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.<\/p>\n
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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. \u00a0We derive the Green’s function for some time independent problems and talk about properties of the Green’s function. \u00a0We conclude with the Green’s function relationship to the delta functions discussed in Lecture 34.<\/p>\n
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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. \u00a0This is the viewed as the main theme of the ODE\/PDE sequence and will be tested on the final exam.<\/p>\n
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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. \u00a0Students are encouraged to focus on specific solution techniques which view the partial differential operator as a matrix. \u00a0 Meshes, sampling, finite difference, finite elements, and Crank-Nicholson schemes are all valid research topics.<\/p>\n
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