Let u be the solution to the initial boundary value problem for the Heat Equation, d,u(t, x) = 4 dzu(t, x), tE (0, 00), х € (0, 1); with Mixed boundary conditions dzu(t,0) = 0 and u(t, 1) = 0 and with initial condition u(0, x) = f(x), where f'(0) = 0 separation of variables, the solution, u, of this problem is 00 u(t, x) = C, Vn(t) w,(x),
Let u be the solution to the initial boundary value problem for the Heat Equation, d,u(t, x) = 4 dzu(t, x), tE (0, 00), х € (0, 1); with Mixed boundary conditions dzu(t,0) = 0 and u(t, 1) = 0 and with initial condition u(0, x) = f(x), where f'(0) = 0 separation of variables, the solution, u, of this problem is 00 u(t, x) = C, Vn(t) w,(x),
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:Let u be the solution to the initial boundary value problem for the Heat Equation,
d,u(t, x) = 4 dku(t, x), tE (0, 0),
х (0, 1);
with Mixed boundary conditions dzu(t, 0)
= 0 and u(t, 1) = 0 and with initial condition u(0, x) = f(x), where f' (0) = 0 and f(1) = 0. Using
separation of variables, the solution, u, of this problem is
00
u(t, x) = en Un(t) W,(x),
n=1
with the normalization conditions v,(0) = 1 and w,(0) = 1. Find the functions v, and Wn.
%3D
U,(t) =
Σ
w,(x) =
Σ
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