Feb7-12 - condition u ( x, 0) = f ( x ). This is a...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Feb. 7, 2012 1. In honor of the Exam on Thursday, the first part of today’s class will be a review of the course so far. 2. More on Inner Products. We will continue to discuss orthogonal projections with applications to the Method of Least Squares. For some details see: http://www.math.upenn.edu/ kazdan/260S12/notes/vectors/vectors6.pdf (Notes on inner products) 3. To see historically how Fourier series arose, we will discuss the temperature u ( x, t ) at a point x of a bar, say on 0 < x < L at time t with specified initial temperature u ( x, 0) = f ( x ). For this we will (very) quickly discuss partial derivatives and try to solve the heat equation ∂u ∂t = k 2 u ∂x 2 with boundary conditions u (0 , t ) = u ( L, t ) = 0 (ice cubes at both ends) and initial
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: condition u ( x, 0) = f ( x ). This is a homogeneous linear partial dierential equation , second order in x and rst order in t . Just as for ODEs like u 00 ( t ) + 4 u = 0, we will seek special solutions and take a linear combination in order to get the general solution. We will then attempt to match the initial condition. The added complication is that there will be innitely many special solutions and force us to nd a Fourier series. The same teachiques can be used to discuss the motion of a vibrating string . See, for example, http://www.math.upenn.edu/ kazdan/260S12/notes/math21/math21-2012-2up.pdf (Math 21 Lecture Notes , Chapter 8.3, p. 369384) [Last revised: February 13, 2012] 1...
View Full Document

This note was uploaded on 03/06/2012 for the course MATH 260 taught by Professor Staff during the Spring '12 term at UPenn.

Ask a homework question - tutors are online