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

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Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Feb. 14, 2012 1. In honor of the Exam last Thursday, the first part of today’s 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 ODE’s like u ( 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. With hindsight, the
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