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Unformatted text preview: MthSc 208 (Spring 2011) Worksheet 7a MthSc 208: Differential Equations (Spring 2011) Inclass 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 steadystate 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 ...
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This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.
 Spring '09
 Staufeneger
 Differential Equations, Equations

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