Feb14-12 - Math 260, Spring 2012 Jerry L. Kazdan Class...

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Unformatted text preview: Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Feb. 14, 2012 1. In honor of the Exam last Thursday, the first part of todays class will discuss this exam, whose solutions are now posted at http://www.math.upenn.edu/%7Ekazdan/260S12/260S12Ex1s-solns.pdf 2. 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 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 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 infinitelyto match the initial condition....
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This note was uploaded on 03/06/2012 for the course MATH 260 taught by Professor Staff during the Spring '12 term at UPenn.

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