Calculus 13: Partial Differential Equations (PDEs) — Simulating the Wave Equation

“PDEs let us model how information, energy, or probability spreads in space and time.”

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1 · What Are PDEs?

A partial differential equation involves unknown functions of several variables and their partial derivatives.

Example: The 1D Wave Equation

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

where $u(x, t)$ describes the wave (e.g., string displacement), and $c$ is the speed.

• Initial condition: shape of the string at $t=0$.
• Boundary conditions: what happens at the ends.

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2 · Why ML Engineers Care

• Physics-Informed Neural Networks (PINNs): Learn solutions to PDEs from data (used for modeling physical systems).
• Diffusion, Vision, and Graphs: Many image and diffusion operations are PDEs.
• Generative models: E.g., denoising diffusion models are essentially solving PDEs in data space.

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3 · Finite-Difference Simulation: 1D Wave Equation

We discretize both space and time.

• $x$ grid: $i = 0, 1, ..., N-1$
• $t$ steps: $n = 0, 1, ...$

The finite-difference form is:

$$u_i^{n+1} = 2u_i^n - u_i^{n-1} + C^2 (u_{i+1}^n - 2u_i^n + u_{i-1}^n)$$

where $C = c \frac{\Delta t}{\Delta x}$ (Courant number).

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4 · Python Demo: Animated Wave Equation

# calc-13-pde/wave_equation_animate.py
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

# --- Parameters
N = 180
dx = 1.0 / N
c = 1.0        # wave speed
dt = 0.8 * dx / c
steps = 240

# --- Initial state: bump in the middle
u = np.zeros(N)
u[N // 3:N // 3 * 2] = 1.0
u_prev = u.copy()  # Assume zero velocity (u_prev = u at t=0)

C2 = (c * dt / dx) ** 2

# --- Storage for frames
frames = []

for n in range(steps):
    # Compute new u using finite-difference
    u_next = np.zeros_like(u)
    u_next[1:-1] = (
        2 * u[1:-1] - u_prev[1:-1]
        + C2 * (u[2:] - 2 * u[1:-1] + u[:-2])
    )
    # Boundary conditions: fixed ends (u=0)
    u_next[0] = u_next[-1] = 0.0
    frames.append(u_next.copy())
    # Advance
    u_prev, u = u, u_next

# --- Animation
fig, ax = plt.subplots(figsize=(7, 3))
line, = ax.plot(frames[0])
ax.set_ylim(-1.2, 1.2)
ax.set_title("1D Wave Equation")

def animate(i):
    line.set_ydata(frames[i])
    ax.set_xlabel(f"Step {i}")
    return line,

ani = animation.FuncAnimation(
    fig, animate, frames=len(frames), interval=35, blit=True, repeat=True
)
plt.show()

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5 · Exercises

1. Initial Shape: Change the initial $u$ to a Gaussian bump. How does the propagation differ?
2. Reflecting vs. Absorbing Boundaries: Implement open (absorbing) boundaries (no reflection) and compare to the fixed (reflecting) boundaries.
3. Higher Courant Number: Try increasing $C$ (i.e., dt). What happens when $C > 1$? Why does the solution become unstable?
4. 2D Extension: Extend the simulation to a $32\times32$ grid, with a hot spot in the center. Animate the 2D wave propagation.

Put solutions in calc-13-pde/ and tag v0.1.

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Next: Calculus 14 — From Diffusion to Deep Generative Models: Bridging PDEs and Data.