Chapter 2: Fourier series In this chapter we deal with expansions of functions in terms of sines and cosines, as they occured in the last section of Chapter 1. In many respects it is simpler to work with the complex exponential function eix instead of the
Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 14 December 2007 4 January 2008 , Solution Exercise: Analyse the vibrations of an elastic solid cylinder occupying the
region 0 r 1, 0 z 1, in cylindrical coordinates if its top and b
Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 14 January 2008 18 January 2008 1. (5 points) Show F ea|x| = 2a( 2 + a2 )1 . Hint : Use F (x2 + a2 )1 = and the Fourier inversion formula. 2. Use the Plancherel Theorem, f , g = 2 f ,
Chapter 3: Orthogonal sets of functions
In this chapter we investigate the following problem: Given a xed set of functions cfw_fn , can we expand other functions f into a series
f (x) =
n=0
cn fn (x) ?
As we learned from the previous chapter, the converge
Chapter 4: Sturm-Liouville problems
4.1. Regular Sturm-Liouville problems
In the previous applications we arrived via separation of variables to some linear homogeneous ODE of second order on an interval (a, b), together with some homogeneous boundary con
Chapter 5: Some boundary value problems
In this chapter we apply our results and techniques to some boundary value problems.
5.1. SturmLiouville problem with periodic boundary conditions
Assume l > 0, and consider the Sturm-Liouville problem with periodic
Chapter 6: Bessel functions
6.1. Motivation
The 2-dimensional wave equation in polar coordinates is vr v vtt = c2 vrr + + 2 . r r If we apply separation of variables, v = R(r)()T (t), the wave equation becomes T R R = + +2. 2T c R rR r Both sides must be
Chapter 7: Fourier transform
7.1. Preliminaries
The goal of this chapter is to provide tools on how to solve some PDE using the Fourier transform. The idea is the following: the Fourier transform allows us to replace a given PDE by a simplier, merely alge
Math 212
Introductory Partial Dierential Equations
Course notes , Fall 2007 Instructor: Dr. Friedemann Brock Chapter 1: Introduction 1.1. Notation I N I R C I In R natural numbers real numbers complex numbers Euclidean space of dimension n, = cfw_x = (x1
Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 310 October 2007 1. Show that u(x, y, t) = (1/t)exp[(x2 + y 2 )/(4kt)] satises the heat equation ut = k (uxx + uyy ). 2. Show that u(x, y, z ) = (x2 + y 2 + z 2 )1/2 satises Laplaces
Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 1724 October 2007 1. (a) Let f : IR C be 2L-periodic, (L > 0), and Riemann integrable. The I corresponding Fourier series are
+ 1 a0 + (an cos(nx/L) + bn sin(nx/L) , 2 n=1
cn einx/L ,
Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 26 October2 November 2007 1. Consider the inhomogeneous heat problem () ut = kuxx + F (x), u(x, 0) = 0, (0 < x < l, t > 0),
(0 < x < l), (t > 0),
u(0, t) = u(l, t) = 0,
where l > 0, k
Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 6 13 November 2007 1. Parsevals equality says that if f is Riemann integrable on (, ) and
n1 (an
has Fourier series (a0 /2) + 1 2
cos nx + bn sin nx) =
+ inx , n= cn e
then
1 1 (|an
Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 14 December 2007 4 January 2008 Exercise: Analyse the vibrations of an elastic solid cylinder occupying the
region 0 r 1, 0 z 1, in cylindrical coordinates if its top and bottom are h