Volume: 60 lecture periods of 90 minutes each
Textbook: Pinchover, Y. and Rubinstein, J., An Introduction to Partial Differential Equations, Cambridge University Press,
Link to publisher's description
Some parts of the course will include excerpts from Evans, G., Blackledge, J. and Yardley, P., Analytic Methods for Partial Differential Equations, Springer,
Link to publisher's description
Prerequisites: Vv286 or Vv256
Last Taught: Spring Term 2015
Background and Goals: This course gives an introduction to classical methods for solving partial differential equations (PDEs). PDEs lie at the heart of all analytical problems of engineering and physics. They occur in the modelling of such diverse subjects as traffic jams, transmission of signals through telegraph lines, electromagnetic problems, fluid flow, heat transfer, bending and buckling of beams and mechanical vibrations of any kind.
The focus of this course is on learning new methods and adapting methods from the solution of ordinary differential equations (ODEs) to the solution of PDEs. A recurring theme is the reduction of PDEs to ODEs, e.g., using the Laplace or Fourier transforms, separation of variables or analysis of the solution curves (method of characteristics). Therefore, close familiarity with the material of Vv256 or Vv286 will be very helpful. As the course focuses on methods and not proofs, it is very much application-oriented. (The figures to the left of this text accompany actual examples in the course.)
Key Words: Conservation laws and the derivation of PDEs from physical models; quasilinear first-order PDEs and the method of characteristics; Burgers’s equation and weak solutions; shock waves; the eikonal equation and other nonlinear first-order PDEs; classification of quasilinear second-order PDEs and their transformation into normal form; boundary value problems of various kinds; the wave equation on an infinite string and d’Alembert’s method; the heat equation in a finite bar and its solution through separation of variables; Fourier-Euler series and their convergence; spaces of weighted square-integrable functions and the problem of best approximation; Sturm-Liouville boundary value problems; separation of variables for nonhomogeneous one-dimensional evolution equations; problems on infinite and semi-infinite bars and the Fourier transform; dispersive solutions; analysis of the telegraph equation; separation of variables in higher dimensions; Bessel functions and Legendre polynomials; multipole expansions in electromagnetics. If time permits: the Poisson equation and properties of harmonic functions.
Detailed Content: We first review some basic material from vector calculus and give an introduction to concepts and notation in the theory of partial differential equations. We then discuss partial differential equations of first order, focusing on the method of characteristics for solving linear and quasilinear equations. Special attention will be given to Burgers's equation, shock waves and traffic problems.
Next, we turn to quasilinear PDEs of second order, with special emphasis on the classical evolution equations (the heat and wave equation) and the potential equation. These are derived from physical models, and related equations (such as the cable and beam equations) are also discussed. The classification and the normal forms for second-order PDEs in two variables are also discussed in some detail.
As a first example, we consider the wave equation for the infinite string, proving its well-posedness for finite time intervals and deriving d'Alembert's solution. The treatment includes weak solutions as natural results from d'Alembert's solution formula.
This is followed by an extensive treatment of the method of separation of variables for the classical evolution equations on finite one-dimensional spatial domains. As part of this discussion, an introduction to the theory of square-integrable functions, Fourier series and Sturm-Liouville theory is given.
The separation-of-variables approach for the heat equation on the infinite bar leads naturally to the introduction of the Fourier transform. We give a brief account of the classical theory, with emphasis on practical applications.
Next, we turn to elliptic equations, in particular, the Laplace equation. There are marked differences in the properties of the solutions compared with those of the evolution equations, and we discuss some of them (such as the maximum principle). We also show how the separation of variables can be applied to certain problems.
Finally, we give an account of the separation-of-variables technique applied to some equations in higher dimensions, including a brief introduction to Bessel functions and Legendre polynomials.
Overview:
Lecture | Subject | Textbook Sections |
---|---|---|
1 | Introduction | 1.1 - 1.3 |
2 | Method of Characteristics | 2.1 - 2.3; 2.6 |
3 | The Cauchy Problem | 2.4 - 2.5 |
4 | The Cauchy Problem | 2.4 - 2.5 |
5 | Burgers's Equation | 2.7 |
6 | The Eikonal Equation | 2.8 |
7 | Nonlinear PDEs | 2.9 |
8 | Classification of 2nd order PDEs | 3 |
9 | Classical PDEs of Physics and Engineering | 1.4 - 1.6, 5.6.1 |
10 | Classical PDEs of Physics and Engineering | 1.4 - 1.6, 5.6.1 |
11 | First Midterm Exam | 1-3 |
12 | The Wave Equation for an Infinite String | 4 |
13 | Separation of Variables | 5 |
14 | Square Integrable Functions | --- |
15 | Fourier Series | 5 |
16 | Sturm-Liouville Problems | 6 |
17 | The Wave and Heat Equations | 5 |
18 | The Laplace Equation | 7.7 |
19 | The Fourier Transform | Evans excerpt |
20 | The Fourier Transform | Evans excerpt |
21 | The Eigenvalue Problem for the Laplacian | 9.5 |
22 | Second Midterm Exam | 4-6; 7.7 |
23 | The Eigenvalue Problem for the Laplacian | 9.5 |
24 | Bessel Functions and the Problem for a Disk | 9.6-9.8 |
25 | Legendre polynomials and the Problem for a Sphere | 9.6-9.8 |
26 | Legendre polynomials and the Problem for a Sphere | 9.6-9.8 |
27 | Elliptic Equations | 7 |
28 | Elliptic Equations | 7 |
29 | Elliptic Equations | 7 |
30 | Final Exam | 7, 9.5-9.8, Evans |
Lecture Slides (2015 version; PDF):
(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, it is not advisable to print out all these slides at the beginning of the term.)
Assignments (2014 version; PDF files):
(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, your assignments may differ from these.)