3. Consider the heat equation in a two-dimensional rectangular region 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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Please solve the following by hand and without the use of AI. Please be thorough and use detailed mathematical formulas to solve. Thank you.

Transcribed Image Text:3. Consider the heat equation in a two-dimensional rectangular region
0<x<L, 0 < y < H.
ди
J²u
J²u
= =k
It
მე-2
+
მყ2
subject to the initial condition
u(x, y, 0) = f(x, y).
Solve the initial value problem and analyse the temperature as t→ ∞o, if the
boundary conditions are
a. u(0, y, t) = 0, u(L,y,t) = 0, u(x, 0,t) = 0, u(x, H,t) = 0.
ди
b.
(0, y, t) = 0,
მე
ди
მე:
ди
ди
(L, y, t) = 0,
(x, 0,t) = 0,
(x, H,t) = 0.
მყ
მყ
Note:
You may assume without derivation that product solutions
u(x,y,t) = (x,y)h(t) = f(x)g(y)h(t) satisfy
dh
-Xkh,
dt
and the two-dimensional eigenvalue problem V2 + λ = 0 with further
separation
d²f
dx2
d²g
:-μf,
dy2
+(A -µ)g=0,
or you may use results of the two-dimensional eigenvalue problem.
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