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lec2v2 - 18.152 Introduction to PDEs Fall 2004 Prof...

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18.152 - Introduction to PDEs , Fall 2004 Prof. Gigliola Staffilani Lecture 2 - First order linear PDEs and PDEs from physics I mentioned in the first class some basic PDEs of first and second order. Today we illustrate how they come naturally as a model of some basic phenomena. 1. u t + cu x = 0 Transport equation (simple transport) 2. u tt c 2 u xx = 0 Wave equation (vibrating string) 3. u t = k Δ u Parabolic equation (heat equation and diffusion) 4. Δ u = 0 Elliptic equation (stationary wave and diffusion) 5. iu t = Δ u Schr¨ odinger equation (Hydrogen atom) Derivation of (1): Imagine putting a drop of ink in the pipe. Let u ( x, t ) be the concentration or density of ink at point x at time t . But how do we describe it? Fix an interval [0 , b ]. ink in [0, b] = M = b u ( x, t ) dx time= t 0 Suppose the water moves at speed c , so at time h + t the same quantity of ink will be ch + b M = u ( x, t + h ) dx ch b ch + b u ( x, t ) dx = u ( x, t + h ) dx 0 ch Taking b u ( b, t ) = u ( b + ch, t + h ) Taking h h =0 0 = cu x ( b, t ) + u t ( b, t ) | 1
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Derivation of (2): A typical example of wave motion in a plane is the motion of a string with fixed end points. Why does it have this shape? We will see later. Let’s take a piece of it: We will ignore all the forces on the string except for its tension. We will consider a perfect string in which for all x we have T ( x, t ) = T v ( x, t ) (1) for constant T and v ( x, t ) a unit tangent vector to the string at ( x, t ). (1 , u x ) v ( x, t ) = 2 1 + u x We assume two facts: 1. The string has constant mass density ρ .
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