hw3S - Partial Differential Equations Math 442 C13/C14 Fall...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Partial Differential Equations Math 442 C13/C14 Fall 2009 Homework 3 Solutions 1. Here we will prove that solutions to the heat equation satisfy (some of) the invariance principles mentioned in class, or in the book in 2.4. That is, if u ( x,t ) is a solution to u t = ku xx for x R ,t > 0, then so are (a) u ( x- y,t ) for any fixed y , (b) u x ,u t , (c) v ( x,t ) = integraltext u ( x- y,t ) g ( y ) dy where g has finite support, (d) v ( x,t ) = u ( ax,at ) for any a > 0. Solution: (a) Let v ( x,t ) = u ( x- y,t ). Then v t ( x,t ) = u t ( x- y,t ) 1 = u t ( x- y,t ) , v x ( x,t ) = u x ( x- y,t ) 1 = u x ( x- y,t ) , 2 v x 2 ( x,t ) = x v x ( x,t ) = x u x ( x- y,t ) = 2 u x 2 ( x- y,t ) . Then v t ( x,t )- k 2 v x 2 ( x,t ) = u t ( x- y,t )- k 2 u x 2 ( x- y,t ) = 0 , since u solves the heat equation. (b) We compute for u x , the other is similar. Denoting v = u x gives v t = ( u x ) t = u xt , v xx = ( u x ) xx = u xxx . Then v t- kv xx = u xt- u xxx = u tx- u xxx = ( u t- u xx ) x = 0 x = 0 . (c) Since g has compact support, we can exchange derivatives and integration (see e.g. Theorem A.3.2 from Strauss), and thus we have v t = t integraldisplay u ( x- y,t ) g ( y ) dy = integraldisplay u t ( x- y,t ) g ( y ) dy and 2 v x 2 = 2 x 2 integraldisplay u ( x- y,t ) g ( y ) dy = integraldisplay 2 u x 2 ( x- y,t ) g ( y ) dy. But then v t- kv xx = integraldisplay u t ( x- y,t ) g ( y ) dy- integraldisplay 2 u x 2 ( x- y,t ) g ( y ) dy = integraldisplay parenleftbigg u t ( x- y,t )- 2 u x 2 ( x- y,t ) parenrightbigg g ( y ) dy = integraldisplay dy = 0 . 1 (d) We have v t ( x,t ) = a u t ( ax,at ) , v x ( x,t ) = a u x ( ax,at ) , 2 v x 2 ( x,t ) = a 2 u x 2 ( ax,at ) ....
View Full Document

This note was uploaded on 01/15/2010 for the course MATH 346 taught by Professor S during the Spring '09 term at University of Victoria.

Page1 / 6

hw3S - Partial Differential Equations Math 442 C13/C14 Fall...

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