Let u be the solution to the initial boundary value problem for the Heat Equation, d,u(t, x) = 5 du(t, x), te (0, 00), x € (0,5); with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial condition 0, € [0, € [$., 15). u(0, x) = f(x) = {2, 0, XE XE x € [15,5]. XE
Let u be the solution to the initial boundary value problem for the Heat Equation, d,u(t, x) = 5 du(t, x), te (0, 00), x € (0,5); with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial condition 0, € [0, € [$., 15). u(0, x) = f(x) = {2, 0, XE XE x € [15,5]. XE
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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![Let u be the solution to the initial boundary value problem for the Heat Equation,
du(t, x) = 5 du(t, x),
t € (0, ∞0),
x € (0,5);
with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial
condition
0,
15
u(0, x) = f(x) = 2,
= [0, 2),
= [5, ¹5).
€ [15.5].
0,
XE
The solution u of the problem above, with the conventions given in class, has the form
u(t, x) = Ï Cn Un(t) wn(x),
n=1
with the normalization conditions U,(0) = 1 and wn (5) = 1. Find the functions
Un, Wn, and the constants Cn.
Un(t) =
wn(x) =
Cn =
x E
XE
M
M](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d01a12a-1ab5-4b5b-8ac9-ffcff6df37dc%2F28cb3a6e-9303-4805-9476-5dcb139b4be5%2Ftbos4o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let u be the solution to the initial boundary value problem for the Heat Equation,
du(t, x) = 5 du(t, x),
t € (0, ∞0),
x € (0,5);
with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial
condition
0,
15
u(0, x) = f(x) = 2,
= [0, 2),
= [5, ¹5).
€ [15.5].
0,
XE
The solution u of the problem above, with the conventions given in class, has the form
u(t, x) = Ï Cn Un(t) wn(x),
n=1
with the normalization conditions U,(0) = 1 and wn (5) = 1. Find the functions
Un, Wn, and the constants Cn.
Un(t) =
wn(x) =
Cn =
x E
XE
M
M
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