Introduction to Partial Differential Equations (Block 1, 2010)

Weeklies

Sept 6: Examples of partial differential equations: wave equation, transport equation, heat equation. Solutions of a homogeneous, linear PDE form a vector space; inhomogeneous equations: general solution = one particular solution + any solution of the homogeneous equation. How a Gaussian function as initial condition evolves in each of the examples. General solution of the transport equation in 2 variables. Method of characteristics to solve (here: homogeneous, linear) first-order equations. Vibrating string fixed at x=0 and x=L: Wave equation on the interval [0,L], initial conditions, boundary conditions, normal modes, more complicated solution by superposition (i.e. adding normal modes), motivating Fourier series. (This lecture corresponds roughly to Asmar, Chapter 1)

Sept 7: Motivating Fourier series for equations with constant coefficients: e^{ikx} as eigenfunctions of d/dx. Piecewise continuous and piecewise smooth functions, periodic functions, the Fourier system of trigonometric functions and their orthogonality properties. Fourier expansions of piecewise smooth functions: definition, pointwise convergence theorem, uniform convergence theorem for continuous functions. Basic examples of Fourier expansions and their use to compute sum_k (2k+1)^{-2}. The Gibbs phenomenon; more can be found at http://www.sosmath.com/fourier/fourier3/gibbs.html. (This lecture corresponds roughly to Asmar, Chapter 2, Sections 1 and 2) Further references: Good (elementary, though rather comprehensive) accounts of the classical theory of Fourier series are Edwards, "Fourier series 1,2", Springer and Zygmund, "Trigonometric Series", Cambridge University Press. The former contains, in particular, a classical example (due to Fejer) of a continuous function whose Fourier series diverges in a point.

Sept 13: We continue our discussion of Fourier series, also for periodic functions of arbitrary period. Topics include a short discussion of the proof of the pointwise convergence theorem, the Fourier series of even and odd functions, sine and cosine series.Then expansions of real and complex functions in series of complex trigonometric functions are discussed. This roughly corresponds to Asmar, sections 2.3, 2.4, the first part of 2.6 together with a somewhat simplified presentation of 2.8.

Sept 14: We further discuss the approximation properties of Fourier series: the mean square error leading to Bessel's inequality and Parseval's identity plus the proof of the uniform convergence theorem. Then we recall basic properties of linear partial differential equations. The material roughly correponds to Asmar, sections 2.5, the second part of 2.6, and the core of 2.9, followed by 3.1.

Sept 20: We start with the classification of second-order linear PDE into elliptic, parabolic and hyperbolic problems. Then the solution of the one-dimensional wave equation for given initial conditions is discussed, combining separation of variables with Fourier series. The topics correspond to Asmar, sections 3.1 and 3.3. However, the solutions described in the examples in this book are "generalized solutions". We provide sufficient conditions for having "classical" (i.e. twice differentiable) solutions. Supplementing notes by Gerd Grubb can be found here: http://www.math.ku.dk/~grubb/pdl06.pdf .

Sept 21: We recall the basic ideas of the Fourier-theoretic solution and its theoretical analysis. D´Alembert's simpler and more transparent solution of the above problem is presented afterwards. It allows the discussion of domains of dependence and the influence of inital singularities. The new material corresponds to section 3.4 from Asmar's book.

Sept 27: First we finished the discussion of d'Alembert's solution for the wave equation and use characteristic parallelograms to find simple expressions for the solution in subsets of the (t,x)-space. We then continued with the one-dimensional heat equation with various boundary conditions, again solved using separation of variables and Fourier series.

Sept 28: We discussed the two-dimensional heat and wave equations. Like for the wave equation, we also briefly addressed the convergence of the series solution in the one-dimensional case, which turned out to be simpler than for the wave equation. The material corresponds to the last bits of section 3.4, section 3.5, parts of 3.6 as well as 3.7 in Asmar's book. You can find an informative treatment of convergence issues in Gerd Grubb's notes http://www.math.ku.dk/~grubb/pdl06.pdf .

Oct 4: We solved the Laplace and Poisson (= inhomogeneous Laplace) equations in two dimensions with the help of Fourier series / eigenfunction expansions and briefly discussed convergence. At the end we briefly introduced Neumann and Robin boundary conditions. In Asmar's book, this roughly corresponds to most of 3.8 and 3.9 plus the relevant sections from Gerd Grubb's notes http://www.math.ku.dk/~grubb/pdl06.pdf.

Oct 5: We finished Chapter 3 by discussing a problem with mixed Dirichlet, Neumann and Robin boundary conditions. Afterwards we discussed the Laplacian in polar, cylindrical and spherical coordinates. We concluded with a whirlwind introduction to orthogonal expansions in Hilbert spaces (not relevant for any assignments or exams). In Asmar's book, the content corresponds to Sections 3.10 and 4.1 (plus my personal notes on Hilbert spaces - the material should be covered in any reasonable book dealing with Hilbert spaces).

Oct 11: We discussed the wave and Laplace equations on a two-dimensional disk in polar coordinates. For the wave equation, separation of variables leads to an ordinary differential equation in radial direction, which is solved by Bessel functions instead of our cherished sines and cosines. The basic properties of Bessel functions were presented. Most importantly, suitably normalized they form an orthonormal basis of a space of square-integrable functions. The associated generalized Fourier expansion allowed us to derive the solution of the wave equation. Solving the boundary value problem for the (homogeneous) Laplace equation in polar coordinates did not require special functions. We obtained a Fourier-like series for the solution. In Asmar's book, the covered material corresponds to (parts of) Sections 4.2-4.4 and selected background material from Sections 4.7-4.9.

Oct 12: We discuss the Laplace equation on a circular wedge and then started to discuss the Laplace equation in a three-dimensional ball. Legendre polynomials were needed to solve the equation in the theta variable, so we had to talk about their basic properties. In particular, we defined Legendre expansions and stated results concerning their convergence. In Asmar's book, the material corresponds to (parts of) Sections 4.4, 5.1-5.2 and selected background material from 5.5-5.7.

Oct 18: We finished our discussion of the Laplace equation in a ball, including the properties of Legendre functions and spherical harmonics. Then basic properties of the Fourier transform were discussed. In particular, we extensively discussed the Fourier representation theorem. In Asmar's book, the material about the Laplace equation corresponds to Sections 5.2-5.3 and some background material in 5.5-5.7. The material concerning Fourier transforms is contained in Sections 7.1, 7.2 and Gerd Grubb's notes http://www.math.ku.dk/~grubb/pdl06.pdf. Our proof of the representation theorem reappear on the sheet of exercises.

Oct 19: We continue to study the Fourier tranform. Topics include convolutions, Schwartz functions and applications to the explicit solution of partial differential equations such as the heat and Laplace equation. The material is contained in Chapter 7 and Gerd Grubb's notes http://www.math.ku.dk/~grubb/pdl06.pdf.

Remaining lectures: We discuss the differentiability of solutions to the initial value problem for the heat equation on the real line and the Dirichlet problem for the Laplace equation on the upper half plane. We then introduce the sine and cosine transforms on the half-line, extend the Fourier transform to n dimensions and consider applications to our model equations. The course concludes with an outlook towards general evolution equations (with time-independent coefficients), in particular the semigroup property of the solution operator and Duhamel's principle for inhomogeneous equations, and the solution and basic properties of the n-dimensional wave equation.