Heat Equation

The 1D heat equation describes how temperature $u(x, t)$ diffuses through a material:

$\frac{\partial u}{\partial t} = \alpha\, \frac{\partial^2 u}{\partial x^2}$

where $\alpha > 0$ is the thermal diffusivity.

Separation of Variables

For the rod $x \in [0, L]$ with Dirichlet boundary conditions $u(0,t) = u(L,t) = 0$ and initial condition $u(x, 0) = f(x)$, the solution is:

$u(x, t) = \sum_{n=1}^{\infty} B_n \sin\!\left(\frac{n\pi x}{L}\right) \exp\!\left(-\alpha\left(\frac{n\pi}{L}\right)^2 t\right)$

Fourier Coefficients

The Fourier sine coefficients $B_n$ are determined by projecting the initial condition onto the sine basis:

$B_n = \frac{2}{L} \int_0^L f(x) \sin\!\left(\frac{n\pi x}{L}\right) dx$

Each mode $n$ decays exponentially in time with rate $\alpha(n\pi/L)^2$. High-frequency modes decay fastest — this is why heat smooths out sharp features rapidly.

Numerical Integration

Given $f$ sampled on a uniform grid of $N_{\text{grid}}$ interior points $x_k = k \cdot L / N_{\text{grid}}$ for $k = 1, \ldots, N_{\text{grid}} - 1$, approximate the integral with the rectangle rule:

$B_n \approx \frac{2}{L} \sum_{k=1}^{N_{\text{grid}}-1} f(x_k) \sin\!\left(\frac{n\pi x_k}{L}\right) \Delta x$

where $\Delta x = L / N_{\text{grid}}$.

Example

For $f(x) = \sin(\pi x / L)$, only $B_1 = 1$ is nonzero (all other $B_n = 0$), giving:

$u(x, t) = \sin\!\left(\frac{\pi x}{L}\right) e^{-\alpha(\pi/L)^2 t}$

The temperature profile maintains its sinusoidal shape but decays uniformly.

Your Task

Implement heat_coefficient_Bn(f_values, L, n, N_grid=100) where f_values is a list of $N_{\text{grid}}-1$ interior sample values. Implement heat_solution(x, t, alpha, L, Bn_list) that evaluates the series given a precomputed list $[B_1, B_2, \ldots]$. Implement heat_series(x, t, alpha, L, f_values, N_terms=10) that computes coefficients and evaluates the solution in one call.