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Feb7-12

Feb7-12 - condition u x 0 = f x This is a homogeneous...

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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
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Unformatted text preview: condition u ( x, 0) = f ( x ). This is a homogeneous linear partial diﬀerential equation , second order in x and ﬁrst order in t . Just as for ODE’s 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 inﬁnitely many special solutions — and force us to ﬁnd 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. 369–384) [Last revised: February 13, 2012] 1...
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