a²u , a?u k x² ди at ax2 ' ay?, ||

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Consider the heat equation in a two-dimensional rectangular region
0 <x < L,
0 <h<H,
ди
= k
at
a?u
+
ду?
Subject to the initial condition u(x, y, 0) = a(x, y), and boundary conditions:
u(0, y, t) = u(L, y,t) = 0
%3D
ди
ди
(x, H,t) = 0
ду
%3D
dy (x, 0, t)
ду
a.
Derive, step by step, an expression for u(x, y, t).
b. Discuss the solution for temperature as t -→ 0o.
Transcribed Image Text:Consider the heat equation in a two-dimensional rectangular region 0 <x < L, 0 <h<H, ди = k at a?u + ду? Subject to the initial condition u(x, y, 0) = a(x, y), and boundary conditions: u(0, y, t) = u(L, y,t) = 0 %3D ди ди (x, H,t) = 0 ду %3D dy (x, 0, t) ду a. Derive, step by step, an expression for u(x, y, t). b. Discuss the solution for temperature as t -→ 0o.
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