# PS3 - y t solves the 2D heat equation w t = k w xx w yy 5...

This preview shows page 1. Sign up to view the full content.

MATH 412: Problem Set 3 due Thursday, February 21, 2008 1. Consider the following diﬀerential equation for u ( x, t ): ± u xx = βu, x Ω , u = 0 , x Ω . Although this is strictly speaking an ODE, use an energy argument to show without solving that for “interesting” solutions ( u 6 = 0) to exist, β must be negative. 2. Strauss 2.4 #16 - diﬀusion equation with decay 3. Strauss 2.4 #17 - diﬀusion equation with time-dependent decay 4. (rewrite of Strauss 2.4 #19) If u ( x, t ) and v ( x, t ) are any two solutions to the 1D heat equation, then show that w ( x, y, t ) = u ( x, t ) v
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( y, t ) solves the 2D heat equation w t = k ( w xx + w yy ) 5. Show explicitly that each of the following is a solution to the 1D heat equation: (a) the heat kernel S ( x, t ) = 1 √ 4 πkt exp(-x 2 / 4 kt ) (ignore the singularity at t = 0) (b) the separation of variables solution u n ( x, t ) = sin( nx ) exp(-n 2 kt ) Note that t appears in two diﬀerent places in the two exponentials! 6. Strauss 3.1 #1 -heat equation on the half line 7. Strauss 3.2 #5 -wave equation on the half line...
View Full Document

## This note was uploaded on 07/23/2008 for the course MATH 412 taught by Professor Belmonte during the Spring '08 term at Penn State.

Ask a homework question - tutors are online