Consider the problem of heat flow in a uniform wire of length 2 described by 0 < x < 2, t> 0, 4 Əx²' du du (2, t) = 0, t> 0 Əz (0, t). u(x, 0) = f(x) 0 < x < 2 a) If u(x, t) = X(x)T(t), derive and solve the eigenvalue problem obeyed by X(x). b) Find T(t), and write down the general solution for u(x, t) obeying the boundary conditions. Page 1 of 2 c) Find the formal solution for the temperature distribution in the wire at all times t > 0 if the initial temperature distribution in the wire is f (x) = g(x) + cos 3Tx with 1, 0< x < 1, g(x) = 2, 2< x < 2, Write the solution in sigma notation.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Consider the problem of heat flow in a uniform wire of length 2 described by
du
- 4
0 < x < 2, t > 0,
ди
(0, t) =
(2, t) = 0, t> 0
u(x, 0) = f(x) 0 < x < 2
a) If u(x, t) = X(x)T(t), derive and solve the eigenvalue problem obeyed by X(x).
b) Find T(t), and write down the general solution for u(x,t) obeying the boundary
conditions.
Page 1 of 2
c) Find the formal solution for the temperature distribution in the wire at all times
t > 0 if the initial temperature distribution in the wire is f(x) = g(x) + cos 37x
with
1, 0 < x < 1,
g(x) :
2, 2< x < 2,
Write the solution in sigma notation.
Transcribed Image Text:Consider the problem of heat flow in a uniform wire of length 2 described by du - 4 0 < x < 2, t > 0, ди (0, t) = (2, t) = 0, t> 0 u(x, 0) = f(x) 0 < x < 2 a) If u(x, t) = X(x)T(t), derive and solve the eigenvalue problem obeyed by X(x). b) Find T(t), and write down the general solution for u(x,t) obeying the boundary conditions. Page 1 of 2 c) Find the formal solution for the temperature distribution in the wire at all times t > 0 if the initial temperature distribution in the wire is f(x) = g(x) + cos 37x with 1, 0 < x < 1, g(x) : 2, 2< x < 2, Write the solution in sigma notation.
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