This problem describes the temperature $u(x,t)$ of a rod as governed by the heat equation:

$u_t = u_{xx}, \quad 0 < x < 1, \; t > 0,$

with the boundary conditions $u(0,t) = 0$, $u(1,t) = 0$, and the initial condition $u(x,0) = 8\sin(4\pi x)$.

To determine $M(t)$, the maximum temperature in the rod at any time $t$, we proceed as follows:

Step 1: General solution of the heat equation

The heat equation with Dirichlet boundary conditions has a general solution of the form:

$u(x,t) = \sum_{n=1}^\infty B_n e^{-n^2\pi^2 t} \sin(n\pi x),$

where $B_n$ are coefficients determined by the initial condition $u(x,0)$.

Step 2: Applying the initial condition

The initial condition is $u(x,0) = 8\sin(4\pi x)$. Comparing this with the general solution at $t = 0$, we find that only the $n = 4$ term is nonzero. Thus, the solution becomes:

$u(x,t) = 8e^{-16\pi^2 t} \sin(4\pi x).$

Step 3: Maximum temperature $M(t)$

The maximum temperature occurs at the peak of $\sin(4\pi x)$, which is 1. Therefore:

$M(t) = 8e^{-16\pi^2 t}.$

Final Answer:

$M(t) = 8e^{-16\pi^2 t}.$

This expression represents the maximum temperature in the rod at any time $t$.

Would you like further clarification or more steps explained? Here are 5 related questions to expand on this topic:

1. How do boundary conditions affect the solution of the heat equation?
2. What is the significance of the exponential decay term $e^{-16\pi^2 t}$?
3. Can you generalize this solution for different initial conditions $u(x,0)$?
4. How does the frequency of the sine term $\sin(4\pi x)$ relate to the problem's solution?
5. What physical assumptions underlie the derivation of the heat equation?

Tip: The exponential decay $e^{-n^2\pi^2 t}$