lec3v2

# lec3v2 - 18.152 Introduction to PDEs Fall 2004 Prof...

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18.152 - Introduction to PDEs , Fall 2004 Prof. Gigliola Staﬃlani Lecture 3 - Initial and Boundary Value Problems Well Posed Problems Physical problems have in general three characteristics which should be reﬂected in the math- ematical equations we use to model them: 1. Existence - The phenomenon exists there exists a solution to the diﬀerential equation 2. Uniqueness - Physical processes are causal: given the state at some time we should be able to produce only one state at all later times 3. Stability - Small changes in the initial conditions should lead to small changes in the output. Examples of well posed problems: 1. Δ u = 0 u = f ( x ) Ω 2. ∂u = Δ u ∂t u ( x, 0) = g ( x ) The heat equation smooths things out. By contrast, the backwards heat equation, ∂u = Δ u ∂t u ( x, 0) = f ( x ) is ill posed. u = ± a n ( t ) sin( nx ) n 2 Δ u = ± n a n t sin( nx ) n ∂u = ± a ˙ n ( t ) sin( nx ) so ∂t n ∂u a ˙ n ( t ) = n 2 a n ( t ) ∂t = Δ u a n ( t ) = e n 2 t a n (0) 1

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so for the heat equation high frequencies decay much faster than low frequencies. For the backward heat equation
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## This note was uploaded on 10/02/2010 for the course MAT 18.152 taught by Professor Gigliolastaffilani during the Fall '04 term at MIT.

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lec3v2 - 18.152 Introduction to PDEs Fall 2004 Prof...

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