s11_mthsc208_ws7a-HeatEqn

s11_mthsc208_ws7a-HeatEqn - MthSc 208 (Spring 2011)...

Info iconThis preview shows pages 1–4. 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

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: MthSc 208 (Spring 2011) Worksheet 7a MthSc 208: Differential Equations (Spring 2011) In-class Worksheet 7a: The Heat Equation NAME: We will solve for the function u(x, t) defined for 0 x and t 0 which satisfies the following initial value problem of the heat equation: ut = c2 uxx u(0, t) = u(, t) = 0, u(x, 0) = x( - x), (a) Carefully descsribe (and sketch) a physical situation that this models. (b) Assume that u(x, t) = f (x)g(t). Compute ut and uxx , and derive boundary conditions for f (x). (c) Plug u = f g back into the PDE and separate variables by dividing both sides of the equation by c2 f g. Now set this equal to a constant , and write down two ODEs: one for f (x) and one for g(t). Written by M. Macauley 1 MthSc 208 (Spring 2011) Worksheet 7a (d) Solve the ODE for g(t). (e) Solve the ODE for f (x) (including boundary conditions), and determine . Consider separately the cases when = 0, = 2 > 0, and = - 2 < 0. Written by M. Macauley 2 MthSc 208 (Spring 2011) Worksheet 7a (f) Find the general solution of the PDE. As before, it will be a superposition (infinite sum) of solutions un (x, t) = fn (x)gn (t). (g) Find the particular solution to the initial value problem by using the initial condition. The following information is useful: The Fourier sine series of x( - x) is n=1 - 4 (1 - (-1)n ) sin nx. n3 Written by M. Macauley 3 MthSc 208 (Spring 2011) Worksheet 7a (h) Consider the following initial/boundary problem for heat equation. ut = c2 uxx u(0, t) = u(, t) = 0, u(x, 0) = 3 sin 2x + 5 sin 6x. The only difference between this problem and the previous is in the initial condition, thus it will have the same general solution. Repeat part (g) to find the particular solution. (i) What is the steady-state solution to the function u(x, t), both in part (g), and in part (h)? Give a physical interpretation for this quantity. Written by M. Macauley 4 ...
View Full Document

This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.

Page1 / 4

s11_mthsc208_ws7a-HeatEqn - MthSc 208 (Spring 2011)...

This preview shows document pages 1 - 4. Sign up to view the full document.

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