Chapter 2: The Heat Equation¶
Modeling Heat Flow¶
How does temperature change over time when heat flows through a material?
The Setup¶
Temperature distribution: T(x, t)
x = position along a rod
t = time
Question: How does T change?
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from matplotlib.animation import FuncAnimation
from IPython.display import HTML
sns.set_style('whitegrid')
plt.rcParams['figure.figsize'] = (14, 10)
np.set_printoptions(precision=3, suppress=True)
Deriving the Heat Equation¶
Key Physical Insight¶
Heat flows from hot to cold. The rate depends on the temperature gradient.
Fourier’s Law: $\( q = -k\frac{\partial T}{\partial x} \)$
where q = heat flux, k = thermal conductivity
The Result¶
Interpretation: Where there’s curvature in space, there’s change in time!
def visualize_heat_intuition():
"""Show why curvature causes change in time."""
x = np.linspace(0, 10, 100)
# Three temperature profiles
fig, axes = plt.subplots(3, 1, figsize=(12, 10))
scenarios = [
("Concave (d²T/dx² > 0)", lambda x: -0.1*(x-5)**2 + 3, "Temperature increases"),
("Linear (d²T/dx² = 0)", lambda x: 0.2*x + 1, "Temperature constant"),
("Convex (d²T/dx² < 0)", lambda x: 0.1*(x-5)**2 + 1, "Temperature decreases")
]
for ax, (title, func, result) in zip(axes, scenarios):
T = func(x)
ax.plot(x, T, 'b-', linewidth=3)
ax.axhline(T[50], color='r', linestyle='--', alpha=0.5, label='Center point')
ax.set_title(f'{title} → {result}', fontweight='bold', fontsize=12)
ax.set_xlabel('Position x')
ax.set_ylabel('Temperature T')
ax.grid(True, alpha=0.3)
ax.legend()
plt.tight_layout()
plt.show()
print("Heat Equation Intuition:")
print(" Concave: Heat flows IN → Temperature rises")
print(" Linear: No curvature → No change")
print(" Convex: Heat flows OUT → Temperature drops")
visualize_heat_intuition()
Simulating Heat Diffusion: From Equation to Code¶
To see the heat equation in action, we discretize the spatial domain into a grid of \(n_x\) points and advance the temperature forward in time using a finite difference scheme. The second spatial derivative \(\frac{\partial^2 T}{\partial x^2}\) is approximated by the central difference formula:
At each time step, we update the temperature using \(T_i^{new} = T_i + \alpha \frac{\Delta t}{(\Delta x)^2}(T_{i+1} - 2T_i + T_{i-1})\), where \(\alpha\) is the thermal diffusivity. The simulation starts with a “hot spot” in the middle of the rod and shows how the temperature profile smooths out over time as heat diffuses outward. This numerical approach – solving a PDE by discretizing it on a grid – is the same idea behind physics-informed neural networks (PINNs), where a neural network learns solutions to PDEs by enforcing the equation as a loss constraint.
def heat_diffusion_demo():
"""Simulate 1D heat diffusion."""
# Parameters
L = 1.0 # Length
nx = 50 # Grid points
alpha = 0.01 # Thermal diffusivity
dt = 0.001
nt = 500
x = np.linspace(0, L, nx)
dx = L / (nx - 1)
# Initial condition: hot in the middle
T = np.zeros(nx)
T[nx//2-5:nx//2+5] = 100
# Store snapshots
snapshots = [T.copy()]
times = [0]
for n in range(nt):
T_new = T.copy()
for i in range(1, nx-1):
T_new[i] = T[i] + alpha * dt / dx**2 * (T[i+1] - 2*T[i] + T[i-1])
T = T_new
if n % 50 == 0:
snapshots.append(T.copy())
times.append(n * dt)
# Plot evolution
fig, ax = plt.subplots(figsize=(12, 6))
for i, (T_snap, t) in enumerate(zip(snapshots[::2], times[::2])):
alpha_val = 0.3 + 0.7 * i / len(snapshots[::2])
ax.plot(x, T_snap, alpha=alpha_val, linewidth=2,
label=f't = {t:.3f}' if i % 2 == 0 else '')
ax.set_xlabel('Position x', fontsize=12)
ax.set_ylabel('Temperature T', fontsize=12)
ax.set_title('Heat Diffusion Over Time', fontweight='bold', fontsize=14)
ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()
print("Heat diffuses from hot center to cooler edges!")
heat_diffusion_demo()
Summary¶
The Heat Equation¶
Key Insight: Curvature in space drives change in time
Next: How to solve this equation!