u = 2 uzz) 0 < < 6, t >0 with boundary/initial conditions: u(0, t) = 0, u(6, t) = 0, S 2, 0
u = 2 uzz) 0 < < 6, t >0 with boundary/initial conditions: u(0, t) = 0, u(6, t) = 0, S 2, 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We will solve the heat equation
=2ur
2 Usc
0<r<6, t> 0
with boundary/initial conditions:
(0, t) = 0,
2(6, t) = 0,
2, 0<x<3
and u(x, 0) = {
0, 3<x < 6
This models temperature in a thin rod of length L = 6 with thermal diffusivity a- 2 where the temperature at the ends is fixed at 0 and the initial temperature
distribution is u(x, 0).
For extra practice we will solve this problem from scratch.
Transcribed Image Text:Find Eigenfunctions for ).
The problem splits into cases based on the sign of A
(Notation: For the cases below, use constants a and b)
• Case 1: A = 0
X(x)
Plugging the boundary values into this formula gives
0 = X(0)
%3D
0 = X(6)
So X(x) =
which means u(x, t) =
We can ignore this case.
• Case 2: A =-y² <0
(In your answers below use gamma instead of lambda)
X(x) =
Plugging the boundary values into this formula gives
0 – X(0)
0 = X(6) =
So X(x) =
which means u(x, t) =
We can ingore this case.
• Case 3: A= >
X(x) =
(In your answers below use gamma instead of lambda)
Plugging in the boundary values into this formula gives
0 = X(0) =
%3D
0 = X(6)
Which leads us to the eigenvalues A, = Yn where Yn =
and eigenfunctions X,(z) =
(Notation: Eigenfunctions should not include any constants a or b.)
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