This is a heat equation problem with specific boundary and initial conditions. Let’s solve it step by step.

Problem:

We are solving the heat equation: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad t > 0, \, 0 \leq x \leq L$

Boundary Conditions:

$\frac{\partial u}{\partial x}(t, 0) = 0, \quad \frac{\partial u}{\partial x}(t, L) = 0$

Initial Condition:

$u(0, x) = \cos\left(\frac{n \pi x}{L}\right)$

Solution:

Step 1: Assume a solution of the form:

$u(t, x) = X(x)T(t)$

Step 2: Substitute into the heat equation:

$X(x) \frac{dT}{dt} = T(t) \frac{d^2 X}{dx^2}$

Divide both sides by $X(x)T(t)$: $\frac{1}{T(t)} \frac{dT}{dt} = \frac{1}{X(x)} \frac{d^2 X}{dx^2} = -\lambda$

Here, $\lambda$ is the separation constant.

Step 3: Solve for $X(x)$:

$\frac{d^2 X}{dx^2} + \lambda X = 0$

Using the boundary conditions $\frac{dX}{dx}(0) = 0$ and $\frac{dX}{dx}(L) = 0$, the solution for $X(x)$ is: $X(x) = A \cos\left(\frac{n \pi x}{L}\right)$ where $\lambda = \left(\frac{n \pi}{L}\right)^2$.

Step 4: Solve for $T(t)$:

$\frac{dT}{dt} = -\lambda T \quad \Rightarrow \quad T(t) = B e^{-\lambda t}$

Step 5: Combine the solutions:

$u(t, x) = \sum_{n=0}^\infty C_n e^{-\left(\frac{n \pi}{L}\right)^2 t} \cos\left(\frac{n \pi x}{L}\right)$

Step 6: Use the initial condition $u(0, x) = \cos\left(\frac{n \pi x}{L}\right)$:

From the initial condition, $C_n = 1$ for $n = 1$, and all other terms vanish.

Thus, the final solution is: $u(t, x) = e^{-\left(\frac{n \pi}{L}\right)^2 t} \cos\left(\frac{n \pi x}{L}\right)$

Let me know if you'd like further clarification or more detailed steps.

Related Questions:

1. How would the solution change if the boundary conditions were Dirichlet ($u(t, 0) = u(t, L) = 0$)?
2. Can you derive the eigenfunctions and eigenvalues for this problem independently?
3. How does the initial condition $u(0, x)$ affect the coefficients $C_n$?
4. How would the solution look for a non-homogeneous heat equation?
5. What physical phenomena does this mathematical model represent in real-world scenarios?

Tip: Always check that the boundary conditions are satisfied by the final solution!