Problem 5: Let L > 0 (again). Use separation of variables to solve the Neumann problem ,2 0²u əx² du Ət - a = 0 du du (0, t)= 0, (L, t)=0 əx u(x,0) = f(x) where 0 ≤ x ≤ L and t ≥ 0. Discuss the physical interpretation of the boundary and initial values of the problem.
To solve the Neumann problem for the heat equation:
∂u/∂t - a^2 ∂²u/∂x² = 0
∂u/∂x(0, t) = 0
∂u/∂x(L, t) = 0
u(x, 0) = f(x)
We'll use separation of variables. We assume u(x, t) can be written as a product of two functions, one depending only on x (X(x)) and the other depending only on t (T(t)):
u(x, t) = X(x)T(t)
Now, we apply separation of variables to the heat equation:
X(x)T'(t) - a^2 X''(x)T(t) = 0
Divide by X(x)T(t):
T'(t)/T(t) = a^2 X''(x)/X(x)
Both sides must be equal to a constant, which we'll call -λ²:
T'(t)/T(t) = -λ²
a^2 X''(x)/X(x) = -λ²
Now, we have two separate ordinary differential equations (ODEs).
1. T'(t)/T(t) = -λ² with solution T(t) = C1 * e^(-λ²t)
2. a^2 X''(x)/X(x) = -λ²
Now, we apply the boundary conditions ∂u/∂x(0, t) = 0 and ∂u/∂x(L, t) = 0:
For X(x):
∂X/∂x(0) = 0
∂X/∂x(L) = 0
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