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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
so for the heat equation high frequencies decay much faster than low frequencies. For the backward heat equation
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online